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Thinking like a theorist: Are complex numbers real?

UPDATE: My answer to this problem. Which, at the end of the day, isn't really an answer at all.

As part of our explorations of "why this math" for aspects of physics, I pose an obvious and seemingly simple question: which type of numbers is required to describe the world around us?

We use different types of numbers in mathematics. For example, we have the integers, the rational numbers, the real numbers, and the complex numbers. Furthermore, there is a distinct hierarchy among number types - the hierarchy for the four types above is:

\(integers \subset rational ~numbers \subset real ~numbers \subset complex~ numbers\).

There are additional number types such as the hypercomplex numbers (which are fascinating if you've never run across them). Physics, however, should only need to use a certain type of number to describe the universe, and which type should be dictated by nature. Which number type is required for our current physical theories, and why?

A couple comments for this question:

  • Arguing that one number type is simply more convenient to describe phenomena is not a valid argument, as we are looking for what is required.

  • On its surface this seems like a very straightforward question, but I warn everyone that it is actually rather subtle.

  • There are different answers to this question, depending on what you believe about measurement and/or the fundamental structure of our world. Hence I hope everyone will provide a number of interesting viewpoints.

Finally, as always - only if enough interest and discussion is shown by the members of our community will I post my own answer. I and your peers on Brilliant want to hear everyone's thoughts!

Note by David Mattingly
4 years ago

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You asked us what number type is required for our current physical theories. Well, I can name certain theories where complex numbers are a necessity. Quantum Mechanics is such an example. It seems that you can't even approach Quantum Mechanics without introducing complex numbers.

In QM the probability of something happening is the square of the magnitude (absolute value) of the “probability amplitude”. This probability amplitude can be complex valued. For example if the probability amplitude is\(\frac{i}{\sqrt2}\), then the probability is \(|{\frac{i}{\sqrt2}}|^2=\frac{1}{2}\).In fact, the Schrödinger equation, which is arguably the backbone of Quantum Mechanics, has an '\(i\)' right at the beginning. You can't escape it! The equation of a single particle looks something like this:

\(i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t)+V(\mathbf{r})\Psi(\mathbf{r},\,t)\) [It took me a really long time to render this in LaTeX!].

I don't know much about Quantum Mechanics and most of what I wrote here is copied and pasted from things I found on the internet. So, I will stop talking about Quantum Mechanics now.

A classmate of mine once asked me, "Why are we studying about complex numbers anyway? They don't exist after all! So why make a fuss about things that don't even exist?"

In reply, I asked him another question, "What is the physical significance of multiplying something by \(-1\)? Multiplication by \(-1\) rotates something by \(180\) degrees. If something is heading towards north at a velocity of \(1 ms^{-1}\). This means that thing is heading towards south at a velocity of \(-1ms^{-1}\).

What if we could find a number that would rotate something by \(90\) degrees? Assume that we have such a number. What would happen if we multiplied something by this number twice? It would rotate that thing by \(90+90=180\) degrees. This is remarkable! Because that's very thing multiplying by \(-1\) does. In other words,

\(something\times new\) \(number \times new\) \(number = something \times (-1)\)

Or in other words, \(new\) \(number^2=-1\).

And this is how the imaginary unit \(i\) is defined. Just because you can't count \(i\) chickens doesn't mean complex numbers are any less real. By this definition, even negative numbers don't exist (you can't actually count \(-5\) chickens)."

My answer was able to convince my classmate.

Another point: we know that the distance between two points in Euclidean space is \(\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2}\) [this is just the Pythagorean theorem].

When relativity came along, we realized that space and time were very closely related. And the distance [this is also known as the space-time interval] between two points (events) in space-time is:

\(\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2 -c^2t^2}\).

What does that have to do with complex numbers? Watch again: the formula is

\(\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2 +(ict)^2}\).

So \(i\) also creeps up here! So physical theories need numbers and sometimes those numbers happen to be complex. But as you have said, there are different answer to this question and a lot of people will come up with different viewpoints. According to me, in order to understand and describe everything [I'm putting a lot of stress on this] in this universe, complex numbers are necessary.

Look at the size of this comment! I think I'll stop now.

EDIT: there have been a couple of comments recently that say that complex numbers are not actually necessary for describing the universe. They are absolutely right!

But this raises another question:

Do we even need numbers to describe the universe around us?

Any physical theory is a model of the universe. Numbers are a tool to describe a theory. They are not a property of the theory itself. Numbers have properties and we use those properties as a tool to try to describe a theory that in turn describes the universe to some extent. We can have theories that don't use numbers at all!

There are certain things that we observe in the universe and we try to capture those things with numbers and end up using those numbers in theories. Complex numbers exist in nature in the same way other numbers exist. They have certain properties that we can experience, perceive and observe. I tried to demonstrate that with the example of rotation. I tried to illustrate current physical theories that use the properties of complex numbers to describe natural phenomenon without getting too philosophical about it. The universe doesn't care if we use numbers to understand it. Numbers are merely a tool for us.

I understand that this has gotten a little bit more philosophical than I would have wanted. So, I'm stopping here. Mursalin Habib · 4 years ago

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@Mursalin Habib Although I agree with Mursalin and have nothing to add about QM, I do object to his observation that \(i\) also creeps up in special relativity when considering the interval ("distance") between two points in spacetime. Unfortunately, properly explaining why this interpretation is not suitable, requires quite a lengthy explanation.

First off, vectors with complex components are used in QM for instance. The norm squared of such a vector is (considered to be) equal to the inproduct with its conjugate rather than itself, so even with complex components you still get something strictly positive.

The four-vector as used in special (and general) relativity is not a proper vector in the mathematical sense, or at least not in the normal four-dimensional vector space. The scalar product of two four-vectors is defined as

\( \mathbf{A} \cdot \mathbf{B} = -A_0 B_0 + A_1 B_1 + A_2 B_2 + A_3 B_3 = \sum_{\mu,\nu=0}^{3} \eta_{\mu\nu} A^\mu B^\nu \),

where \( \eta \) is the Minkowski (or flat-space) metric tensor with \( \eta_{00} = -1, \eta_{11} = \eta_{22} = \eta_{33} = 1 \) and all other components zero. It should be noted that sometimes the scalar product (and hence \( \eta \)) is defined with the exact opposite sign; there is no ironclad convention for this. Also, typically the summation sign is omitted; summation is implicit whenever an index is repeated (that is ironclad).

At this point I cannot resist but demonstrate what makes four-vectors and their special scalar product so useful. The fundamental principle (or postulate, if you like) behind (special) relativity is that it does not matter what (inertial) frame of reference you use to describe something. This is nicely reflected in this scalar product: the scalar product of two four-vectors is invariant with respect to a change of reference frame. That goes for the norm of the spacetime four-vector \( (ct, x, y, z) \),

\( \eta_{\mu\nu} x^\mu x^\nu = -c^2 t^2 + x^2 + y^2 + z^2 \),

which gives you the spacetime interval. But it also works for other four-vectors such as the energy-momentum four-vector \( (E/c, p_x, p_y, p_z) \), for which we have

\( \eta_{\mu\nu} p^\mu p^\nu = -E^2/c^2 + p_x^2 + p_y^2 + p_z^2 = -m^2 c^2 \),

where \( m \) is the invariant restmass. You might recognize the above formula for the special case \( \vec{p} = 0 \), but this is the more general form.

Going back to the definition of a scalar product, it may seem that using \( \eta \) is just overly complicated; why introduce a 16-component tensor for a sum of four products? And why not use imaginary components to deal with the minus sign? I suppose the most convincing answer (which is what I have been working towards) is that, when you go to general relativity, the scalar product generalizes to

\( \mathbf{A} \cdot \mathbf{B} = g_{\mu\nu} A^\mu B^\nu \),

where \( g \) is still real-valued but may now have nonzero off-diagonal components.

Finally! Sorry for the long post. Thomas Beuman · 4 years ago

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@Mursalin Habib I agree with you. It's like algebra, that one use to describe a system, and same system can be described by different algebras, but it doesn't mean that algebraic model exists physically. As we have different tools like matrices system, similarly complex numbers and it's properties are used to describe systems, it doesn't have to do with physical existence. Greendragons X · 4 years ago

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Hey all. Sorry I haven't been able to monitor this discussion as closely as I'd like. We got hit by lightning over the weekend and it took out our internet connection. However, it's a great discussion. I'm not going to reply to everyone's comments as I'll post my overall take on this question on Wednesday (after I consult Mariam B's tarot cards). The comments on QM are right in a certain way of looking at it, and the comments about deriving complex numbers from simpler number types are also right. I'll try and distill these viewpoints into my own response.

There is a physics question besides QM involved too, that no one has incorporated yet. I'll toss this question into the ring as well:

Is there a fundamental limit as to how much information one can squeeze into a certain region of space and time? If yes, then our universe, which is finite in extent, could be built out of a large but finite number of these small regions. How would this change the required number system?

Note, this goes a bit beyond the limitation "current physical theories", which was a deliberate choice. David Mattingly Staff · 4 years ago

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I like this question!

Some top answers have focused on the fact that complex numbers are required because basic equations describing quantum mechanics rely on complex numbers. Others have countered that only natural numbers are required because from them we can construct a system that mimics the behavior of, e.g., complex numbers using only the naturals.

As others have pointed out the Natural numbers purists are kind of cheating - if you construct a system that behaves like complex numbers it doesn't matter that you have avoided using the symbol 'i', you are still using complex numbers. Why not go even further and claim that we don't need numbers at all, just the concept of and axioms of a set (from which we can reconstruct all math)... On the other hand, I don't think that those on the complex numbers side of the debate go far enough.

Let's examine the meaning of the phrase "a number system". A number system is a collection of items (which we will call numbers) and relations and operations (different ways to compare or combine these numbers).

Numbers are useful because we can talk about them in the abstract, replace an actual number with an unknown variable, and we know how the operations will be evaluated when we assign any specific value to the variable. The problem with this abstraction is that sometimes you can write down unsolvable questions. For example if your number system is the natural numbers, you can write down 4+x=3, but now there is no number that makes the equation true unless we introduce integers. Similarly if we want to solve all types of equations we must expand our number set to include rations, reals, and complex numbers to solve polynomial equations. The thing is, as soon as we have introduced 0, 1, the operations of addition and multiplication, and, crucially, the idea that we want to be able to 'undo' those operations. That is, we want to be able to 'do algebra'. Then we have magically already created all the complex numbers.

What about vectors? surely in a multi-dimensional world we need vectors to describe things. Vectors appear to just be a list of numbers, but they have different operations. For example, Maxwell's laws require the use of cross product. We need matrices to deal with vectors. And while the matrix is just an array of numbers, and the matrix operations can be described as a sequence of steps using basic number operations, the overall algebra of matrices has a different structure to that of 'numbers'. So I believe that we have to include matrices on our list of required number systems for the same reason that we have to include complex numbers.

It is a false cop-out to claim that because we can describe a more complex system using the symbols of a simpler system, we don't really need the complex system at all.

Are there other number systems needed? Surely. I don't pretend to know any details about cutting edge particle physics theory, but I know that the ways that these particles interact can be quite complicated, and cannot be modeled within the algebra of basic numbers.

On the very cutting edge of what might not even be real physics after all, string theorists make predictions about reality using even more exotic algebraic structures.

At the root, numbers are an abstraction that allow us to makes predictions about the universe. For any complicated way that objects in the universe can interact, we need a number system that incorporates and models that kind of interaction. As our knowledge of physics in incomplete, I'm sure that there is no final answer to the types of number systems that we need to describe the world around us. Colin Hinde · 4 years ago

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I find that many of the people here that argue for the coarser left side of David M.'s inclusion chain above argue something like this: "You don't need complex numbers because you can construct those from real numbers by algebraic closure" or like this: "You don't need real numbers because you can construct those from the rationals using Dedekind cuts or Cauchy sequences", If that is the case, then you're still using those numbers, you've just created them anew. Just call a spade a spade and admit it. Also, I think a few interesting options are left out: the algebraic numbers, the Gaussian integers and \(\mathbb Q(i)\)

For the current quantum theory, I believe we need a continuum of values (or, at least, something dense in the continiuum) that multiply without changing modulus. Thus we need values from all over the complex unit circle. Also, all rational numbers should be present, since ratios are a thing. So, at least all complex numbers with rational polar coordinates (with argument a rational multiple of \(\pi\)), extended to a field (Again, since ratios are a thing, I think we need to have a field). That would be \(\mathbb Q(i)\) extended with all possible values of \(\sin (q\,\pi),\; q\in \mathbb Q\). Some of these might be trancendental. I can't tell this late at night.

Do we really need \(\pi\) or \(e\)? How about all the other trancendental numbers? I don't know. I am not certain enough to give an answer, but I have told you what I believe to be a minimum of numbers needed. Arthur Mårtensson · 4 years ago

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I was quite impressed by Mursalin's answer to the question. I know some basics about quantum mechanics, so I would try to make the reason here:- The Schrodinger's equation gives the probability of finding an electron in a given amount of space. It is the square of the wavefunction, and the wavefunction itself consists of a complex number. In fact, even the Newtonian mechanics can be derived from the Schrodinger's equation after assigning some values and parameters to the equation. So this clearly shows that this equation is a much more basic way of describing the universe than the newton's laws itself. So the answer comes here, everything in this universe is described by its wavefunction, and this wavefuction is an imaginary quantity. Hence complex numbers are the more basic and essential number type required for describing the universe around us. I don't know anything about hypercomplex numbers, but as per my knowledge complex numbers may satisfy the requirements. Siddharth Kumar · 4 years ago

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I'd like to make another pitch for an answer that seems mostly neglected on this thread, which is the answer of INTEGERS!

One of the first issues with this answer is obviously the irrational numbers. However, I think that there is an easy way around this. Consider that no measurement is exact, that any measurement has a finite amount of significant digits in any measurement. Then, any measurement we could ever take has a finite number of decimal points, and can then be expressed as a fraction of two integers. In other words, it should never be necessary to use every infinite digit of pi to describe the circumference of a circle, or any other physical quantity. Only in pure math would you require these irrational numbers.

I am not so far in physics to be able to speak confidently about the issues of these claims in quantum mechanics, nor the role or necessity of imaginary numbers in quantum mechanics, so I will leave that alone. But perhaps there is a similar argument to be made there. Mark Brown · 4 years ago

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@Mark Brown I like it! Any applied science needs only enough digits to have the calculations turn out accurately enough. Bob Krueger · 4 years ago

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Mursalin gave an answer which I think everybody is liking. But honestly speaking I don't think that it is right. If you ask what type of numbers is required to describe universe around us then my answer would be Natural numbers! Yes.. It looks very stupid but let me explain. First of let me clear to you that it is not true that it is impossible to imagine quantum mechanics without complex numbers. This is a very famous misconception. Let us ask.. if nobody would have thought about complex numbers before the advent of quantum mechanics, wouldn't scientists have solved puzzles of blackbody radiation or photoelectric effect? Actually quantum mechanics can be constructed without using complex numbers, we just need to write equations for 2 variables instead of 1 with proper constraints for them. Complex number has two such quantities contained in it (magnitude and phase) and so if we use complex number then we need to write only single equation. Thus mathematically quantum mechanics or LCR circuit equations become easier if we use complex numbers.

Now what about 0,negative integers,rational numbers and irrational numbers? It is well known that all these quantities can be constructed mathematically. I think everybody knows about constructions of 0 and negative integers: they are constructed as solutions to certain equations. Last of all irrational numbers are constructed using well known procedures of Dedekind cuts or Nested intervals (you may read wikipedia articles about these).

Now the next question: suppose we don't construct any of these and only use natural numbers. Then is it possible to write say theory of general relativity or quantum mechanics? And answer is YES! Only problem would be that it would be too complicated since each variable would need lot of description! May be Kroneker had realized this long back and said "God made the natural numbers; all else is the work of man."

People who don't like my answer may also want to read about "Preintuitionism" on wikipedia. Snehal Shekatkar · 4 years ago

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@Snehal Shekatkar Though members of this thread have successfully acknowledged that the entities that represent rationals and irrationals are completely built from natural numbers (e.g. equivalence classes of ordered pairs for fractions and sequences for irrationals), one significant component of the construction of real numbers has been ignored: the operations on these new structures, addition and multiplication.

At each stage of construction, one must redefine addition and multiplication for ordered pairs, sequences, etc. For example, if the ordered pairs (a,b) and (c,d) are fractions, we have the rule, which we must define their sum by the rule (a,b) + (c,d) = (ad + bc, bd). So even if we only use ordered pairs and sequences of ordered pairs of natural numbers, we have to provide new axioms that define the structures of the real numbers. So even employing this construction, one is still using a system isomorphic to the real numbers, equipped with the entire structure of the continuum. Thus if physics uses any idea of the continuum, it uses real numbers, no matter how they are disguised Louis Esser · 4 years ago

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@Snehal Shekatkar I disagree on one point,can all irrational numbers be constructed from natural numbers?How would you construct pi or e? Aahitagni Mukherjee · 4 years ago

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@Aahitagni Mukherjee Well, one possible way is this:

\(\pi=4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}- \cdots\)

\(e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+ \cdots\) Mursalin Habib · 4 years ago

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@Mursalin Habib How do you know that L.H.S. is equal to R.H.S. here? You first need to construct an object which we call irrational number and only then you can think about representing them using infinite series. This is not the correct way. You can either use nested intervals or Dedekind cut. Snehal Shekatkar · 4 years ago

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@Snehal Shekatkar mursalin's way is obviously not a feasible construction-It will require an infinite number of terms to be calculated.Moreover,there are irrational numbers which can't be represented as such explicit series. If I've Understood the method of Dedekind cuts correctly,it requires you to construct a set of rational numbers where each element is less than the number to be constructed. My point is,in both of the above methods,we are talking about infinite sequences,and thus,not applicable in actual physics calculations,no matter how complicated a calculation or analysis we are ready to undertake.

But it has got really interesting.I read that there are many irrational numbers which can't be explicitly represented(by some root,ratio,or series).Do these numbers have any significance in physics? Aahitagni Mukherjee · 4 years ago

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@Snehal Shekatkar The first one comes from the obvious fact that \(tan\frac{\pi}{4}=1\) and the Taylor Expansion of \({tan}^{-1}x\).

The second one is a direct implementation of the Taylor Expansion of \(e^x\).

I'm trying to express them just from their definition and without using any idea of where they might be on the number line. But I think we're deviating away from the topic a little bit.... Mursalin Habib · 4 years ago

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@Mursalin Habib Dear Mursalin, I think you are not getting my point. May be this is because yet you didn't study analysis. Read first what I said and then you will understand. Snehal Shekatkar · 4 years ago

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@Aahitagni Mukherjee Read about Dedekind cut on wikipedia or in any book on analysis. Snehal Shekatkar · 4 years ago

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@Snehal Shekatkar What you've said is absolutely right! But that wasn't what I was going for. See the EDIT part of my comment. Mursalin Habib · 4 years ago

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I think that complex numbers are necessary, because they make a complete system. This means all equations can be described. This is important to mathematics (I don't know about QM and that stuff). And for all those people who say "everything can be constructed by integers", that doesn't matter. For example, some people said that fractions are simply integers over integers. Yes, but they aren't INTEGERS, they're RATIONAL NUMBERS. If you need to construct something to use it, face it, you need it. Now real numbers. I'm sure we all love calculating out infinite fraction sequences (not really), but for all purposes, it's better to use the irrational numbers. Complex numbers are similarly useful, even if they can't describe actual amounts. If you only count amounts, then sure, natural numbers are fine. But the universe is obviously more complex than that. And like shown dozens of times, physical equations use complex numbers. So I think that complex numbers are necessary, if not in a too obvious way, to our physical system. Bob Bob · 4 years ago

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Interesting, i find this kind of discussion fascinating, and would really like to see what Davids own answer is... One method of reasoning is that all rational numbers can be expressed using integers - simply use fractions.... real numbers also tend to be able to be expressed using integers - taylor expansions etc, although i do not know if it would be possible to express all real numbers in this way... maybe not.... but the ones necessary for current physical theories, quite possibly. Complex numbers such as 'i' can just be expressed as 'root(-1)', using integers and common notation.

The possible flaw in this method is that although you can express these using integers, overall what you have expressed isnt actually an integer - if you replace i with 'root(-1)' surely you're still using complex numbers - I guess it just depends on how you like to think about things.

If you ignore this flaw and continue using this reductionist method, you can actually dispose of "numbers" altogether, and instead use symbols and notation - sort of like 'typographical number theory', it is probably possible to express current physical theories using pure logic. It would be horrific - but I think its possible.

On the topic of beyond current physical theories, if we assume that the universe is finite, and that in a certain region there can only be a certain finite amount of information, I think that this information can be expressed/approximated in lots of different ways, to varying degrees of accuracy - different constructs are required for different approximations. I think if there was an ultimate fundamental "theory of the universe", it is likely to be simple in nature, in that it shouldn't require as a necessity certain man-made constructs (which I believe numbers essentially are), but would have complex implications - think fractals and chaos theory, however I also feel that such a theory would also be impossible... because surely any such system must be either incomplete or inconsistent - according to Godel. If this is applicable to 'the universe' or what implication it would have if it was is beyond me.

Whichever which-way you look at it, the "fundamental structure of our world" seems to be so far beyond human understanding that all we can do is approximate using crude models based on artificial constructs which seem to agree with our observations, which are also crude. Ben Blayney · 4 years ago

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@Ben Blayney Fascinating view. Bob Krueger · 4 years ago

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I have another question.... Are quaternions real? Taehyung Kim · 4 years ago

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All number systems are essentially a mathematical construct to explain the space, e.g. 1-D space can be described by a real number, 2-D space by a complex number or an ordered pair of two real numbers, 3-D space by vectors, and 4-D by quaternions, etc. Sunny Prajapati · 4 years ago

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Really, all we ever need is \(1\), \(-1\), addition, and multiplication. Because with these few tools and a little algebra, you can construct all other numbers. However, I did realize that transcendental numbers aren't the easiest to construct with this system, so we should also include helpful numbers like \(\pi\), \(e\), or anything else like that. (Here is a more detailed explanation of this construction system and transcendental numbers) Whereas real and complex numbers are convenient for most physical applications, they are not necessary. Furthermore, if you consider \(1\) and \(-1\) as "real," then undoubtably, all these other numbers are real. Bob Krueger · 4 years ago

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I think it should be complex numbers. Though I have not seen them(purely complex numbers) being used so often, but I know one field where it is used. My teacher had showed me using it in solving parallel LCR circuits. I don't know whether I am going as per the requirements of the discussion. Nishant Sharma · 4 years ago

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@Nishant Sharma Aah, so this is a good example of ease of description. Is it necessary to have complex numbers for LCR circuits, or is it merely a convenient description because it makes the math easier? David Mattingly Staff · 4 years ago

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@David Mattingly From what I can tell, in classical wave analysis, complex numbers are introduced as a way to simplify representations of wave equations, using Euler's formula. In the example of LCR circuits, one sets up the differential equation describing the system; an oversimplified method of seeing how complex numbers relate to this is by showing that the solution of the differential equation may take the form of \(e^{i(wt-c)}\), where w is angular frequency and c is phase shift. This is convenient, as it is represents the wave equations involving sine and cosine, so we know intuitively that we're on the right track like this. When you plug it all back into the original differential equation and solve with initial conditions and such, we find that, indeed, all the imaginary terms from the \(e^{ix}\) term disappear by the time we arrive at the final solution. Thus, it becomes apparent that the complex numbers were only introduced to help solve the equation, and are devoid of physical meaning. Quantum mechanics is a different story, however... John X · 4 years ago

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@David Mattingly Well I think most of the things can be simplified out using one or more approaches. It depends on the effectiveness( I mean accuracy) of the technique being employed. Nishant Sharma · 4 years ago

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@David Mattingly It makes the math easier I guess. I remember that while learning LCR circuits, I would end up with a second order differential equation and solving them isn't included in High School mathematics. Instead, we were introduced to phasors and use vectors(?) to solve the problems. Pranav Arora · 4 years ago

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I like the book of Paul J. Nahin, "An Imaginary Tale: The Story of i" (you can see on Amazon) :) Virgilius Teodorescu · 3 years, 11 months ago

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First of all focussiong on what numbers are? guyz they are nothing except something used as a magnitude to describe teh physical quantities if the universe.ANd according to Model delpendent realism it occurs in our brain.Numbers are only in our brain and are create dfrom logic and i dont know the numbers which will decribe the universe but i know is the last and frist number of this number system- the numbers are -0 and Infinity...and guyzz thay are same Nabasindhu Das · 4 years ago

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Seems like this answer has not come up: real numbers. Complex numbers, can actually be represented as an ordered pair, and just make the operations just like the ones in complex numbers. I do not have much time to build this up but I believe it is possible. In my opinion, complex numbers is just a tool for convenience. Other stuff can just replace it. It is obvious that rational numbers are needed, otherwise how are you going to evaluate \(P=\frac{F}{A}\) (just an example)? So we have left why we need irrational numbers. Again I think this is obvious, as many constants are irrational and we need them. \(\pi\) is significant enough. Well I think this is not the answer, just my opinion, so I hope someone out there can tell me my misconceptions, mistakes, gaps or whatsoever. Yong See Foo · 4 years ago

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I think complex numbers should be used , though we can only use the reals and the tasks where complex numbers were used should be assciated with pairs of real numbers. Ahmed Taha · 4 years ago

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I think COMPLEX NUMBERS is required to describe the whole world around us.

Reason being , Complex numbers exists in Conjugate pairs . In the same way , In physics Forces exists in conjugate pairs (Fundamental theory of classical mechanics : newtons 3rd law). We can't move a inch without forces .SO, forces are there in every fundamental structure of the world and exists in pairs (analogous to complex numbers) .

Secondly , In physics we use vectors everywhere which is analogous to complex numbers.

Thirdly , We can't compare complex numbers. We can't say whether 3 + i or 4 - i greater. Can we? In the same way in physics , We can't compare different measuring units in physics. For example , we can't compare ' Kilo grams and pascals ' etc. etc. ( somewhat analogous )

(I think my 3rd reason makes no sense , But I am just trying to prove it )

Fourthly , Maths in the mother of all sciences. It is required in every walk of life and physics. In physics we use mathematical equations instantly. There is always a tendency to get a purely complex number as a root of the equation. And there are infinitely many pure complex numbers and infinitely many real numbers (or integers etc) .SO the probability of getting a pure complex number is equal to the probability of getting a real number. So , we can say , complex are used in almost every equations of maths (as integers⊂rational numbers⊂real numbers⊂complex numbers. )

Combining above four conditions , I can say that COMPLEX NUMBERS is required to describe the whole world around us. Vinay Pandey · 4 years ago

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I think that the whole process of classifying number types has begun from simple systems such as integers to intricate ones like complex numbers or hypercomplex numbers (quaternions, tessarines, coquaternions, etc. ). Now one can question what would be the need for newer (or more complex) number systems and the answer to that lies in the fact that these new number systems provide the appropriate mathematical formalism to realize ideas and theories. Now why are they appropriate and the answer to that would be if any other number system be used to formalize a theory it's be insufficient to describe the nuances of that theory. This brings us to question as to why advanced mathematics like lie algebra, or group theory, etc. is needed to do advanced physics. One would not be able to get far along if one continues to abandon complex numbers to do Quantum Mechanics ( as someone rightly pointed out). Coming back to number systems, the process with which new number systems were developed tells us an important fact as we made progress with the advent of complex numbers and hypercomplex numbers, our understanding of the universe improved. Does this not hint towards the fact that there is a possibility of an all encompassing number system which will qualify as the singular type of system needed for all physical theories. Definitely this number system can't be integers, real numbers, or even complex numbers because these prove to be parts of a bigger better system for theories. Taking a crude example, we can always use complex numbers instead of real numbers or integers, but let's say does it serve any purpose to use complex numbers in simple arithmetic. The point here is these are all subsets of what is required. Striving for a better, bigger, "all encompassing" and universal number system is what would be required for our current physical theories. And as complex numbers is in a sense a subset of hypercomplex numbers. So "currently" my answer would be hypercomplex numbers. And yeah, i could be totally wrong but this is what I feel should be. Hope I have conveyed my idea properly. Rahul Mehra · 4 years ago

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I think complex numbers must be the numbers used for describing the universe around us. Siddharth Kumar · 4 years ago

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@Siddharth Kumar Can you give a reason? David Mattingly Staff · 4 years ago

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@David Mattingly yes sir ,in real life situations like free fall under all the effects of nature like drag, viscosity, etc..the energy dissipated by the falling object is in the form of complex number magnitudes.. Chitres Guria · 4 years ago

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I think all we need is real numbers.other things are all structures constructed from real numbers.A complex number is only an ordered pair of real numbers. An n-tuple of real numbers constitute an n-dimensional vector,a matrix of real numbers is an n cross n tensor,and so on.If necessary,we can also add another dimension and get a 3 dimensional analogue of a tensor,which will have n cross n cross n elements.So I think faith in 'reality' should be restored :) Aahitagni Mukherjee · 4 years ago

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hmmm Christian Baldo · 4 years ago

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[Deleted for irrelevance- Peter] Mariam Baurice · 4 years ago

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@Mariam Baurice Internet preachers are everywhere! Unbelievable! Mursalin Habib · 4 years ago

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@Mariam Baurice What this has to do with the original question?? LOL Snehal Shekatkar · 4 years ago

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@Mariam Baurice !!!!??? Heli Trivedi · 4 years ago

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@Mariam Baurice We are interested in your 'situation'....please tell about it Aahitagni Mukherjee · 4 years ago

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Harvard Egyptian Students Association

Purpose Statement

HESA is a social platform for Egyptians as well as anyone interested in Egypt and the Egyptian culture to get together in a variety of cultural and social events. HESA provides opportunities for Egyptian undergrads to network with Egyptian alumni with a common vision towards making the post-revolution Egypt a better place. HESA aspires to reach out to Egyptian high school students and provide them with information about the Harvard College application process as well as life at Harvard. HESA aims to provide a forum for students to actively contribute to the debates that are going on in Egypt and the Arab World through a wide range of activities (including guest speakers, workshops, etc.) HESA aims to mobilize the incomparable academic and intellectual resources we have at Harvard and the surrounding institutions for the purpose of creating a vibrant academic and intellectual forum for Egyptians and Arabs at Harvard and the greater Boston Area.

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| Published in Interesting Facts | Written by Thomas Stroup

Rolf Schulmeister. How Much Time Has to Be Spent on Education?

Wondering why your academic score is so low compared to other students? How much time do you spend on average preparing for your classes? Is it 9, 23, 50 hours per week? It’s not all about time, but this issue still matters. A new study conducted by famous German scientist, Professor Rolf Schulmeister, proves that the key to student’s performance is motivation.

The professor of education from Hamburg spend three years investigating the attitude of German students towards the process of studying. The results which have been published recently, equally surprised both other scientists and community. It turns out that there is no direct link between academic performance and time spent on the educational process.

If the diligence does not guide the academic success, what is the point? "The key to success has nothing to do with the intellect or talent. Motivation is the clue," Shulmeister claims. The motivation has a great impact on the dedicated time as well.

Rolf held his research, titled "Overloaded" (Zeitlast-Studie), since early 2009. According to his findings, German students spend on the average only 23 hours per week on tuition. It does not prevent them from complaining of stress. These 23 hours include almost every aspect of learning process: attendance, studying on your own, and even the time spent on the road to the educational institution. There are students who spend no more than nine hours per week. The record time was 50 hours.

All of the students are divided into several groups (according to Schulmeister). Those students who can self-define the rhythm and know how to focus perceive the process of studying in the easiest way. They manage to accomplish much more than many of their classmates within an hour. At the same time, those who avoid difficult tasks experience some hard times. "As soon as it becomes difficult or unclear, they resort to distracting methods like TV or social networks," he states.

There is also a number of external factors that affect the academic performance. "Those are numerous training programs", Rolf Shulmeister says. German universities tend to offer too many different training activities within just one week. As a result, students cannot decide on what theme to work." Within a short period, students have to assimilate a lot of information before the exams and perform numerous essay writing tasks. However, even in case of success all knowledge quickly disappears.

As an experiment, it was proposed to teach the students under modular system conditions. For some technical specialties, the instructors staged the four-week seminars so that the students could focus on one topic. It was a success. (You can read also about: Harvard Primal Scream: Instruction to Freshmen)

For instance, IT students from one of the German universities passed the exam with the better average score than they used to a year before. However, the cost of education has also increased. Instead of the average index, they spend 34 hours a week.

"There are many departments that offer a lot of optional subjects in addition to the primary, so it would be good to introduce a modular approach," Rolf Shulmeister believes. I.e. the classes might be held applying modular system three days a week. The rest of the time must be devoted to the elective classes.

The only remaining question is whether the Bologna system needs an educational reform


Freemasons of West Boylston
The search for truth is has its own reward. Since 1876 local Freemason's learn here how to earn the right to wear the ring!
What is a Freemason?
A Freemason is a member of the world's largest fraternal organization. He not only enjoys the friendship and brotherhood of other Freemasons in his community, but also is welcomed by Freemasons anywhere in the world.
Freemason is a man of faith, who uses tools of moral and ethical principles to serve mankind. He binds himself to like-minded men in a brotherhood that transcends all religious, ethnic, social, cultural and educational differences. He shows the aspirations and obligations of men of good will who seek to make themselves better than they are - not better than others.
Freemason is dedicated - he recognizes his responsibility for justice, truth, charity, honesty and integrity. Freemasons work building their lives and character, just as a carpenter works at building a house.
Freemason believes that there is such a thing as honor, and that a man has a responsibility to act with honor in everything he does. A Freemason teaches that principle. He believes that a life not founded on honor is hollow and empty - that a man who acts without honor is less than a man.
Freemason believes that it is not only more blessed to give than to receive, it's also more rewarding. He becomes involved with problems and needs of others because he knows it gives each of us a good feeling to help, a feeling unlike any other.
Freemason believes that every person should strive to be a good citizen and that he has a moral duty to be true to the country in which he lives, for loyalty to one's country is essential.
He is expected to obey every lawful authority, obey the laws of the country in which he lives and promote its general welfare. In no way do these requirements interfere with a Freemason's civil right to protest and seek legislative changes by lawful means.
Freemasons and the Masonic family contribute not only their money, but also an incredible amount of time to various charities. Internationally the Masonic Family donates over three million dollars per day to various charitable works. Much of our help is given anonymously. We're not after gratitude. We're more than rewarded by that feeling which comes from knowing we have helped another person overcome some adversity, so that their life can go on.
Without The Guidepost Of Principles,
What Do You Have?

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World Series Profiles
Take a look inside the stories of the people, places, and events that have made history in the Babe Ruth Softball World Series. Through diverse authors follow the struggles, triumphs, the grass roots efforts and the profound moments that have led to the 2017 Softball World Series.
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East Tennessee Civil War History

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Tennessee and the Civil War

Tennessee and the Three Grand Divisions

East Tennessee Civil War History Map
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East Tennessee Railroads and the Civil War Map

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Civil War History

East Tennessee is the name applied to approximately the eastern third of the state of Tennessee. Unlike the names bestowed to regions or portions of many U.S. states, the term East Tennessee can be precisely defined. It is the portion of the state located within the Eastern Time Zone and four counties in the Central Time Zone, namely Bledsoe, Cumberland, Marion, and Sequatchie counties. East Tennessee is noted for its mountains, particularly the Great Smoky Mountains portion of the Appalachian Mountains. The East Tennessee Region is also known for being the birthplace of country music and the State of Franklin.

Contrary to most geographic regions within the United States, the term East Tennessee has legal as well as socioeconomic status. East Tennessee, with Middle Tennessee and West Tennessee, comprises one of the state's three Grand Divisions. According to the Tennessee State Constitution, no more than two of the Tennessee Supreme Court's five justices can come from any one Grand Division. A similar rule applies to certain commissions and boards, to prevent them from reflecting a geographic bias.

The major cities of East Tennessee are Knoxville and Chattanooga. Other important cities include the "Tri-Cities" of Bristol, Johnson City, and Kingsport located in the extreme northeastern area of the state. It is an area previously and traditionally referred to by residents as Upper East Tennessee, although today the term Northeast Tennessee is preferred by non-residents.

Greeneville, Tennessee
Greeneville Tennessee Map.jpg
Location of Greeneville, TN, Map

During Secession, Andrew Johnson was the only Southern United States Senator who refused to resign.

During the secession crisis, Andrew Johnson remained in the United States Senate even when Tennessee seceded, which made him a hero in the North and a traitor in the eyes of most Southerners. In 1862, President Lincoln appointed him Military Governor of Tennessee, and Johnson used the state as a laboratory for Reconstruction. In 1864 the Republicans, contending that their National Union Party was for all loyal men, nominated Johnson, a Southerner and a Democrat, for Vice President. (Right) Greeneville, located in East Tennessee, was the hometown of President Andrew Johnson and it was also known for its pro-Unionism.
During the Civil War, East Tennessee was the poorest of the State's three Regions. Within Tennessee, East Tennessee had the least amount of slaves based on a percentage and numerical basis. Consequently, the Region witnessed divided loyalties, and bushwhackers, outlaws and lawlessness reigned. Many East Tennesseans served in Western North Carolina Civil War regiments. Bordered with Western North Carolina, the two Regions reflected many similarities: Western North Carolina was the poorest of North Carolina's three Regions; both were rugged mountainous Regions; within North Carolina, Western North Carolina possessed the least amount of slaves; both Regions experienced lawlessness and anarchy during the Civil War; and they experienced many battles and skirmishes against the same Union commands.

Andrew Johnson residence in Greeneville
President Andrew Johnson's home.jpg
Library of Congress

It "was a running fight for ten miles. Two Federals were killed in the yard of Senator Patterson, the son-in-law of President Johnson. Twelve or fifteen others were killed. General Williams, while slowly retreating before [Union General] Burnside, heard our artillery open upon the enemy. Dashing forward at a gallop, he materially aided us in the achievement of one of the most brilliant retreats of the war." Lt. Col. William W. Stringfield while fighting Burnside's army at the Battle at Henderson's Mill on October 11, 1863

East Tennessee proved invaluable in the defense of the vital and strategic Saltworks and railroads. For example, while guarding the Strawberry Plains Bridge, the Thomas Legion's Private James Keelan was posthumously awarded the rare Confederate Medal of Honor. The East Tennessee and Western North Carolina mountains were also in striking distance of several major Civil War battles. See also Tennessee Civil War History.

The East Tennessee area has four seasons with mild Springs and Falls. Summers can be hot and humid with highs in the 80s and 90s. Winters are usually mild with daily averages in the 30s, but temperatures can drop to the 20s on occasion.

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Recommended Reading: War at Every Door: Partisan Politics and Guerrilla Violence in East Tennessee, 1860-1869. Description: One of the most divided regions of the Confederacy, East Tennessee was the site of fierce Unionist resistance to secession, Confederate rule, and the Southern war effort. It was also the scene of unrelenting 'irregular,' or guerrilla, warfare between Union and Confederate supporters, a conflict that permanently altered the region's political, economic, and social landscape. In this study, Noel Fisher examines the military and political struggle for control of East Tennessee from the secession crisis through the early years of Reconstruction, focusing particularly on the military and political significance of the region's irregular activity. Continued below...

Fisher portrays in grim detail the brutality and ruthlessness employed not only by partisan bands but also by Confederate and Union troops under constant threat of guerrilla attack and government officials frustrated by unstinting dissent. He demonstrates that, generally, guerrillas were neither the romantic, daring figures of Civil War legend nor mere thieves and murderers, but rather were ordinary men and women who fought to live under a government of their choice and to drive out those who did not share their views.

Recommended Reading: The Last Confederate General: John C. Vaughn and His East Tennessee Cavalry (Hardcover). Description: John Crawford Vaughn was one of the most famous men in Tennessee in the mid-nineteenth century. He was the first man to raise an infantry regiment in the state--and one of the very last Confederate generals to surrender.  History has not been kind to Vaughn, who finally emerges from the shadows in this absorbing assessment of his life and military career.  Making use of recent research and new information, Larry Gordon’s biography follows Vaughn to Manassas, Vicksburg and other crucial battles; it shows him as a close friend of Jefferson Davis, and Davis’s escort during the final month of the war. Continued below…
And it considers his importance as one of the few Confederate generals to return to Tennessee after Reconstruction, where he became President of the State Senate.  Gordon examines Vaughn’s (hitherto unknown) location on the field of crucial battles; his multiple wounds; the fact that his wife and family, captured by Union soldiers, were the only family members of a Confederate general incarcerated as hostages during the Civil War; and the effect of this knowledge on his performance as a military commander.  Finally, the book is as valuable for its view of this little understood figure as it is for the light it casts on the culture of his day. Reviews: Modern reappraisals of maligned Civil War figures like Confederate General John Crawford Vaughn often attempt to even the score by either going too far in the other direction or attacking the subject's critics. Fortunately, Charles Larry Gordon's The Last Confederate General avoids such tactics, and instead attempts a balanced, 'warts and all' portrait of the man…The Last Confederate General is a largely sympathetic, yet even handed, appraisal of the checkered military career of John C. Vaughn. In addition to the narrative account of the general's military service, the lengthy biographical sketch is useful, both as a portrait of a neglected Civil War figure and a personalized example of the deep divisions within East Tennessee politics and society.--Civil War Books & Authors. The Last Confederate General is a fine read for anyone: Action, adventure, love, drama, war and perseverance. What more can you ask for in a book. Five stars for Larry Gordon for a job well done…Larry Gordon seemed to have nailed both recreational reading and historically accurate statistics in one read….a fascinating story of courage, determination and self worth.--Our History Project

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Omega Chi Epsilon
The National Honor Society for Chemical Engineering

What is Omega Chi Epsilon?

The key (see image top left) represents the objectives of Omega Chi Epsilon. The four arms of the Maltese cross represent the first four objectives:

1. RECOGNITION to recognize excellence in chemical engineering
2. INVESTIGATION to promote original investigation and innovation in chemical engineering
3. SERVICE to provide service to the chemical engineering department or school and its student body
4. COMRADESHIP to promote comradeship among chemical engineers

These four are bound intimately together by the fifth,

5. PROFESSIONALISM to promote honesty, integrity and social responsibility: the hallmarks of professional ethics

represented by the circular maroon crest in the shape of a globe, reminding us of the world-wide scope of chemical engineering. The gold Greek letters, Omega, Chi, Epsilon, are inscribed on a white band encircling the globe. The crossed retorts, the integral sign, and the bolt of lightning represent the three main tools of the chemical engineer: chemistry, mathematics, and physics.

WCE is a member of the Association of College Honor Societies. The Society also has many common goals with the American Institute of Chemical Engineers (AIChE); we support the AIChE with funds for student awards. We further share many goals with the American Society for Engineering Education. WCE has over 60 active university chapters and total membership of nearly 20,000 men and women. While the chapters are primarily composed of Junior and Senior students, it is also open to graduate students and faculty. In recent years, 2-4 new chapters have been chartered each year.


The file you have requested: Modifications / Vehicles / I to J Titles / Lamborghini Murcilago SHARK TUNING - 1.19mb is not available. Perhaps try browsing all files for the game Grand Theft Auto: San Andreas and upload any files you want to share or are missing. This is the only information we could locate:

******************************************************Lamborghini Murcilago SHARK TUNING************************************************************************ GTA San Andreas ******************************** **** Here is the Lamborghini Murcilago, a detailed and well made car that will either replace the hotring or any other car of your choice within GTA San Andreas.

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Opening times for The Co Op Pharmacy in Accrington

The Co-operative Pharmacy - Accrington - 406 Blackburn Road is the name of a chemist's belonging to the The Co-operative Pharmacy group. Its usual opening times are: 07:00-23:00, 07:00-23:00, 07:00-23:00, 07:00-23:00 and 07:00-23:00, from Monday to Friday respectively. On Saturday and Sunday: 08:00-22:00 and 10:00-16:00. Drugstore's address is 406 Blackburn Road, Accrington, Blackburn, post code BB51SA. Its telephone number is 01254 238 378. There are other towns located in vicinity like Accrington, Altham, Baxenden, Church.


6-Feb-2012 8:46 AM

US House of Representatives passses long-term FAA reauthorisation

US House of Representatives passed (03-Feb-2012) HR 658, the 'FAA Modernization and Reform Act of 2012' by a vote of 248 to 169. The bill is a four-year infrastructure measure to reform Federal Aviation Administration (FAA) programmes, improve US aviation infrastructure and modernise the air traffic control system. House and Senate conferees had reached agreement on the measure last week. The Senate is now expected to consider the conference report this week. [more - original PR]


Analysing Works of Hyperfiction

Analysing Works of Hyperfiction

In undertaking an examination of the special affordances to writing that hyperfiction allows, an overview of the development of hyperfiction through the study of central hypertexts would prove very valuable. These texts signalled in the new writing form, and in their interactive capabilities, as well as non-linearity and multi-perspectives, represent a challenge to the sequentiality of printed text. The hypertext titles under examination are Afternoon: a story by Michael Joyce,[1] Patchwork Girl by Shelley Jackson,[2] and Victory Garden by Stuart Moulthrop.[3] Surveying the development of hyperfiction narrative techniques through these hypertexts, helped to inform what shape my hyperfiction would eventually take.

These hyperfiction works are different from the proto-hyperfiction examined in the previous chapter in that they are born-digital. This means that the same narrative is not fully reproducible in another medium, such as printed text. An important rubric throughout this chapter has been questioning whether or not the hypertexts under examination would function the same in printed form. This has been useful in my own hyperfiction-writing process also. The fact is important because it shows a concerted effort on the part of the author to engage fully with the hypertext form. They made the decision that the best way to communicate their text was through hyperfiction, and designed the text accordingly. All of the hypertexts that will be looked at demonstrate a deep engagement with the mode. These hypertexts adopt non-linearity and multi-perspectives not just in form but in content as well, making them very successful examples of hyperfiction-writing.

Afternoon: a story

Afternoon: a story first appeared as a hypertext in 1987, but was re-engineered in 1990 for the hypertext-writing software Storyspace, which the author Michael Joyce helped to develop. It was one of the very first hypertexts, containing 539 separate sections or lexias, held together with over 900 hyperlinks. The user begins with the section “I Want to Say”, which is pertinent, as it prefigures the difficulty the narrator has in addressing certain painful subjects. The user clicks on different links with each lexia or fragment, which transports them to various different strands and narrative actualities.

Figure 1: afternoon: a story title page (Source: Eastgate Systems).

Afternoon is a very influential hypertext as it was one of the first to be made widely available. It also has a built-in feature which enables first-time users to come to grips with the links in hyperfiction. If they are unsure which hyperlink to click on, they can abdicate the decision by pressing the <enter> key. This returns the default narrative, which in a cursory way gives the user a sense of the plot and characters, only to arrive at an abrupt and unsatisfying conclusion. Jill Walker opted for the default narrative for her first of four eventual readings, each additional one revealing a new, often contradictory, narrative retelling: “This default sequence eases the new reader into reading Afternoon. Reading this sequence gave me enough background information to start enjoying the leaps between story lines, and to understand connections where I’d earlier only been confused” (Walker).[4]

The competing narratives become available through re-reading of the hypertext, with some lexia remaining closed-off until the second or third read-through. With this additional information and careful re-reading, details concerning the main character slowly start to collect. The new, supplementary information then manages to alter the meaning of the lexia read originally: “Repetition is one of the most important rhetorical figures in Afternoon. We re-read nodes, re-interpret nodes read in a new context or we try frantically to escape from relentless loops of re-reading” (Walker). This is something I would incorporate into my own hypertext, where competing, parallel narratives would offer additional detail to one telling of events, in the aim of altering the reader’s initial impression of a character.

The uncertainty surrounding the branching narratives serves to decentre the hyperfiction’s overall structure. The way in which two competing narratives are held at the same time denotes a degree of simultaneity also. As Janet H. Murray points out, “Multiform narrative attempts to give a simultaneous form to these possibilities, to allow us to hold in our minds at the same time multiple contradictory alternatives” (Murray 37-38).[5] Further to the subject of synchronicity in hypertexts, is the speed at which hyperlinks operate. Flitting between lexia through hyperlinks occurs almost simultaneously – not that it takes long for readers to turn a page or cast their eye to the next new paragraph in a printed text. Programmatically at least the individual lexia exist simultaneously, but there is a slight delay on the part of the reader, for reasons of comprehension.

The branching, overlapping structure of Afternoon also proves very ripe for different uses of multi-perspectives. This can be seen in the jumping between the 539 different sections that make up the text. Not only does this allow for wholly different and competing narrative events, but also multiple viewpoints of the same event. This is similar to the multiple perspectives in Ulysses and Mrs Dalloway, as well as the works of Faulkner. In addition the branching narrative allows for the possibility of competing, contradictory narratives. As Bolter points out, the hypertext form enables the author to achieve what up to now was almost prohibitively complex – multiple and conflicting narrative potentialities: “[Afternoon] does openly and with ease what experimental writers in print could do only with great difficulty. It offers a narrative that encompasses contradictory possibilities” (Bolter 143).[6] This proved very useful in my experience of writing a hypertext, with the competing parallel narratives and viewpoints something I wished to replicate.

An interesting question in relation to these branching narratives of varying length, where multiple re-readings are necessary, is how is closure actually achieved? With a text that offers so much opportunity for re-reading, how much is sufficient, how will the reader know when exactly, without a clear declaration of “The end”? Joyce banks on one of two things occurring with the reader, that they either grow bored and give up, or through the endlessly recursive reading strategy, will have collected what feels to them sufficient information to induce feelings of closure. Either way, the decision is in the hands of the reader: “Joyce makes the responsibility for closure, for stopping, entirely the reader’s” (Landow 229).[7] This degree of reader autonomy is tied to interactivity – the reader can at any point return to the start screen and pursue a different route, taking another stab at the hypertext, very much like being in a video game, hitting reload, and starting the same level again. Also connected to this is nonsequentiality, as this approach subverts the traditionally-held importance of beginning, middle and end – especially the end, in this case.

Joyce problematizes the sense of closure: it does not have to be reached by the normal end to a text, but at any point in a reading. Joyce is quoted saying as much: “‘Closure is, as in any fiction, a suspect quality, although here it is made manifest. When the story no longer progresses, or when it cycles, or when you tire of the paths, the experience of reading it ends’” (Landow). George Landow also observes the difficulty in concluding a hypertext such as Afternoon, and attributes this to its branching structure: “the problems that hypertext branching create for narrativity appear with particular clarity in the matter of beginning and ending stories” (226).

The structure of Afternoon is highly variable, with the different hyperlinks transporting the reader to different lexias, narrators, and chronologies. The highly confused narrative, taken as a whole, resembles in several ways the perception of time by the character Benjy in The Sound and the Fury. Like with Faulkner’s text, re-reading is necessary to make sense of Joyce’s varying plots and narratives, as well as its choppy chronology.

Some critics do question to what degree Joyce’s branching, fragmented narrative is multi-linear, however. The act of reading is inherently linear – the user is strictly capable of reading one text at a time, even if that text exhibits multi-linear characteristics. As George Landow argues in “Is Hypertext Fiction Possible?”, it could be that readers have been culturally conditioned to expect linear instead of non-linear narratives: “[An] explanation might center on the claim that human beings in all times and cultures, including our own, depend on linear narrative. ‘We tell stories’, Joan Didion points out, ‘in order to live.” Perhaps linear narrative has too much human importance to abandon’” (265).

Bolter, who worked closely with Joyce on developing the Storyspace programme that Afternoon is written with, argues that despite the clearly branching, non-linear structure of the text, the order that the plot adopts has not evolved in the same way: “Paths can have narrative force, as when Peter tries to find out whether his wife and son have been in an automobile accident. Following this trajectory makes the story quite linear. Peter fights his way through various telephone calls, in an effort to get information” (Bolter 126).

Connected to this, the structure of Afternoon cannot be said to be entirely multi-linear. These narrative cycles that readers find themselves trapped inside, combined with a prescriptive albeit multiple narrative structure, make the hypertext arguably more instead of less linear. Certain nodes can only be accessed on the third or fourth reading, which closes off potential avenues of reading on the first time round. As Espen Aarseth writes, in this instance at least the traditional codex may be more instead of less open: “Hyperfictions written in Storyspace, like [Afternoon], do not allow its readers free browsing, unlike any codex fiction in existence. The reader’s freedom from linear sequence, which is often held up as the political and cognitive strength of hypertext, is a promise easily retracted and wholly dependent on the hypertext’s system in question” (Aarseth 75).

Only once the reader has read through the hypertext fragments in the preferred sequence does the narrative then reveal itself. Here it becomes a matter not of understanding, but of participation – the user must stay the course, and at least click on the links in the correct order, though an informed decision based on careful readings would expedite things. Hayles finds the power that Joyce holds over the narrative to be almost tyrannical: “[Joyce’s] control over these sequences is palpable, for several of them do not allow any exit… until the reader has clicked through the entire sequence, creating an oppressive sense of being required to jump through the same series of hoops numerous times” (Hayles, Electronic Literature, 62-63).[9]

In this way the reader participates in what links or narrative threads to pursue, and is in effect critiquing the narrative, interpreting and passing judgement on it. As mentioned in the introduction, this degree of participation helps to author in the text, with the reader’s thoughts informing the text’s overall structure: “In Afternoon we may get lost in Peter’s compelling narrative of his search for his son, but the need to make choices keeps pulling us back to the fact that we as readers are participating in the making of a fiction. We are constantly critiquing the nature of these choices as we read” (Bolter 167).

While Afternoon may restrict its reader’s passage in ways that other hypertexts explored here do not, this is in fact an important part of the story, and reflects the main character’s unsuccessful attempt to avoid confronting his past. The initial confusion experienced by the reader simulates the emotional turmoil of the main character Peter: “The needs of the reader to struggle with the story mirrors the struggle that the character goes through” (126).

The events that affect the main character Peter are drip-fed to the reader, and depending on the chosen hyperlinks, can have vastly different outcomes. But the struggle on the reader’s part to understand the narrative is reflected by Peter within the plot. The equivocal nature of Peter’s thought processes as he narrates events is indicative of a form of post-traumatic self-denial. Hidden within the narrative strands is a painful truth Peter tries to evade rather than face. Janet Murray observes, “[Joyce] conceals a key section in a way that mirrors the protagonist’s self-deceit. Only after repeated evasions can readers reach the lexia in which Peter will call his therapist and face his memory of his own culpability in the accident” (Murray 58).

The hypertext’s asymptotic shape, which circles these facts but generally avoids them, can be seen as an act of sublimation on the narrator’s part, in an effort of avoiding these disturbing realisations: “the reader discovers that Peter [may be responsible for his ex-wife and son’s deaths]. This discovery explains the approach-avoidance pattern Peter displays in attempting to find out where his son is; he does not want to face what in some sense he already knows” (Hayles, Electronic Literature, 61). The labyrinthine shape of the winding structure can also be compared to the “Wandering Rocks” chapter in the other Joyce’s Ulysses.

Pursuing the question can Afternoon be refashioned into a printed text and still function the same, like the proto-hypertexts analysed in the previous chapter, the answer is no. The high levels of interactivity, the branching, parallel narratives, and the need for the user to revisit certain sections several times before being allowed to progress, are not transferrable. Afternoon could only exist as a hypertext. Aspects of Afternoon I would in turn borrow for my hypertext were the branching narrative and different chronologies, in addition to the rejection of an ending or closure. However one aspect or element I was unable to reproduce, was the variable narrative that alters depending on what links are pursued. This seemed like a logistical step too far for my hypertext.

Patchwork Girl

While Afternoon was a successful iteration of interactive fiction, it was not sufficiently open or dynamic – while parallel narratives were possible, the route that the reader might take through the text was ultimately quite restricted. However this was not the case with Patchwork Girl. Produced by Shelley Jackson in 1995, using the same programme as Joyce with Afternoon, the Storyspace system, another key difference between them is that Patchwork Girl is heavily visual.

The hypertext presents the user with visual hyperlinks in the form of parts of a female body. Jackson compounds Mary Shelley’s Frankenstein with The Patchwork Girl of Oz by L. Frank Baum to produce the highly recursive and referential text. By clicking on a body part or organ, the user is whisked away to connected narrative fragments: “Patchwork Girl engaged the [Storyspace] tool in significantly different ways. In an important innovation, it drew connections between the electronic text and the female monster’s fragmented body” (Hayles, Writing Machines, 37-38).[10]

The reader is presented with an anatomical diagram of a human female, and encouraged to click on any of the portioned, differently-labelled sections of her body (Figure 2). Patchwork Girl is visual where Afternoon is not, and presenting the reader with the individual parts all at once confronts the reader with new narrative capabilities. It can be a lot to stomach for the reader at first; like a dinner plate, or an open-world video game, all of the individual ingredients are available from the off, which encourages the reader to pursue whatever looks the most appetising. Similar to Afternoon though is the uncertainty surrounding closure: some users will pick and choose, leaving the text alone once they have reached their fill, while completists will feel the need to consume every shred and morsel on offer.

Patchwork Girl

Figure 1: Patchwork Girl menu (Source: Eastgate Systems).

The theme of form mirroring content, seen in the delayed closure for both main character and narrative in Afternoon, is present in Patchwork Girl also. The main character, based on the female companion Frankenstein assembles for his monster, is a collection of parts from other woman, the same as the text is a collection of parts from different texts. Landow observes the same, that the collection of different narrative fragments parallels the construction of the main character: “This digital collage-narrative assembles Shelley Jackson’s (and Mary Shelley’s and Victor Frankenstein’s) female monster, forming a hypertext Everywoman who embodies assemblage, concatenation, juxtapositions, and blurred, recreated identities” (Landow 237).

The disfigurement of the narrator, and the scars decorating her body, operate on multiple levels of signification: joins of different human parts, cloth fragments collaged and sewn together, and separate lexias tied using hyperlinks: “Just as the monster finds pleasure and identity in her scars, good hypertext works are defined and distinguished by their unique linking structures” (239).

Under no circumstances could Patchwork Girl be downgraded to a printed text, the precise textual structure is so bound to the visual, interactive elements of the computer display that it could not operate in any other medium: “Patchwork Girl could only be an electronic text because the trace of the computer interface, penetrating deeply into its signifying structures, does more than mark the visible surface of the text; it becomes incorporated into the textual body” (Hayles, “Flicking Connectivities”).[11]

Owing to the open and immediate availability of the different pieces of text to Patchwork Girl, this not only disrupts the overriding narrative structure, but the chronology of the text as well. Afternoon emphasised re-reading and decision-making in order to progress, and in this way the hard-to-reach lexias resembled passing levels in a computer game. But as seen in Patchwork Girl, with no clear-cut route available, it falls to the reader to decide what links to click on and what order to progress.

This reader autonomy is something I wished to grant in my hyperfiction also, where although there is a finite number of sections in Patchwork Girl, the route that readers take passing through them is changeable and dynamic. But this also succeeds in fragmenting the narrative structure and the chronology. The timeframe in the hypertext, like Afternoon, leaps between the past and the future, which serves as Hayles writes to underline the present: “chronology is inherently tenuous because linking structures leap across time as well as space… Patchwork Girl locates its performance of subjectivity in the individual lexia. Since the past and the future can be played out in any number of ways, the present moment, the lexia we are reading right now, carries an unusually intense sense of presence” (Hayles, “Flicking Connectivities”).

This privileging of the present, and the way different lexias exist side-by-side, carrying the same potentialities, also shows a degree of synchronicity. Owing to the speed of computer programmes mentioned earlier, all of the lexias are evenly located in relation to each other, but as Hayles points out, it falls to the reader to apply a sequence to the hypertext in order to understand it: “Sequence is constructed by accumulating a string of present moments when the reader clicks on links… [In contrast] is the simultaneity of the computer program. Within the non-Cartesian space of computer memory, all addresses are equidistant (within near and far memory, respectively), so all lexias are equally quick to respond to the click of the mouse” (Hayles, “Flicking Connectivities”).

Useful features in Patchwork Girl that I wanted to reproduce for my own hypertext were its open, non-linear structure, and also the reader autonomy in determining what shape the narrative could take. However one important aspect of Patchwork Girl I could not see to including was the visual component. Also important hypertext techniques that were underrepresented in Patchwork Girl were multiple perspectives and synchronicity. These were present in the next hyperfiction by Stuart Moulthrop, named Victory Garden.

Victory Garden

As mentioned in the previous chapter, while the short fiction of Jorge Luis Borges contains ideas that are incredibly influential to authors of hyperfiction, Borges has never technically written examples of proto-hyperfiction himself, choosing instead to describe instances of such in a more conventional way. This can be seen in the short stories “An Examination of the Work of Herbert Quain”, “The Aleph”, and “The Garden of Forking Paths”. One reason for this is that if the texts concocted by Borges did exist, they would be absolutely unreadable: “Borges can imagine such a fiction, but he cannot produce it… [He] never had available to him an electronic space, in which the text can comprise a network of diverging, converging, and parallel times” (Bolter 139).

However Stuart Moulthrop identified how Borges’ “Garden of Forking Paths” could conceivably be reconfigured as a hyperfiction: “Borges’ text already contains pathmarks for some of these divergent routes… a hypertextual fiction would take it upon itself to explore these and other excluded possibilities, along with various networks of narrative that might pertain to them” (Moulthrop 123-124).[12] As a classroom experiment, Moulthrop and his students set about attempting to produce a hypertext version of “Garden of Forking Paths”, by drawing links and nodes from the original text, as well as narrative branches that run parallel to each other: “In a rather limited and tentative way, the electronic “forking paths” represents a hypertextual realization of Borges’ textual theorizing” (124). This tutorial with students contained the germ for Victory Garden, produced by Moulthrop in 1992.

As with the previous examples, asking if the hypertext can be successfully translated into a printed text, for Victory Garden like all the others the answer is no. This is due to the fact that Victory Garden was clearly intended to adapt and open-up printed fiction to the newer form. Many competing and contradictory accounts are held simultaneously and describe the same events in an instance of multiple perspectives. Similar to Patchwork Girl, the hypertext relies heavily on visuals. Beginning Victory Garden, the reader is presented with several different paths to progress through the hypertext, either the visualisation of the garden, the list of different interlinking paths (Figure 3), or by hyperlinks scattered throughout the text.

Like in “Garden of Forking Paths”, Victory Garden contains both events and actions, coupled with their alternatives. For example, the central character Emily is serving in the Gulf War. The competing events that transpire are that she dies in one eventuality, and survives in another. Each outcome is determined by the decision which hyperlink is clicked on by the user, and shows a high level of interactive variability.

A glimpse at the plan of the narrative structure to Victory Garden gives an impression of just how complex the hypertext is. Further, the level of intricacy plotting the hyperlinks dwarfs Afternoon in its labyrinthine structure: “With its 2,804 links connecting 993 lexia through a variety of paths planned over long stretches rather than from node to node… Victory Garden takes Daedalian architecture to a level of complexity that probably no reader has fully appreciated” (Ryan).[13]

Victory Garden

Figure 2: Victory Garden plan (Source: Ryan).

Some criticism has been levelled at Victory Garden for not fully achieving what Borges set out in his short story, a book containing all of the potentialities of an event, that it does not go quite far enough: “Moulthrop’s Victory Garden does not outline multiple destinies for its heroes, but rather traces many pathways into a reasonably solid and chronologically organized narrative core” (Ryan). In Borges’s story however, the forked garden or maze, and the labyrinthine novel, are one and the same, making Moulthrop’s hypertext more successful than Ryan credits.

The truth of the matter is that such a text would be unreadable, first requiring superhuman intelligence on the author’s part, and ceaseless patience from the reader. Slatin highlights the difficulties in both writing and reading a hypertext; for an accurate realisation of Borges’s story, the complications would be much worse: “the author of a hyperdocument has a hard time trying to predict where the reader will go from any given point. The reader who activates a link often has a hard time too because it can be so difficult to predict what the result will be” (Slatin 164). This is something I also had to address while writing my hypertext, how an overabundance of parallel narratives, and interactivity, threatens the cohesiveness of the hypertext. It was a careful balancing of the different hypertext features in order to make the hypertext as easily understood as possible.

The same as in previous examples, the labyrinthine structure and contradictory narratives actually reflects the mental life of several characters, in an example of form mirroring content. In one possible eventuality, the main character Emily dies in armed combat. How this is represented on the screen is, Moulthrop uses normal text in the run-up to the event, followed by images of shattered text. The effect is to simulate “as if the enemy shell itself had landed on the previous block of writing… The effect of moving from the intact lexia to the shattered one is like an animation of the landing of the shell” (Murray 83).

The narrative techniques first located in terms of modernist fiction and proto-hypertexts in the previous chapter, have been shown in use in these examples of hyperfiction, where they are not bound or limited by the constraints of printed texts. Such important features as interactivity, non-linearity, multi-perspectives and synchronicity, operate to make the reading experience far more participatory and dynamic for the user.

Afternoon: a story by Michael Joyce stands as one of the first and best examples of hyperfiction. Utilising hundreds of hyperlinks allows the text to hold competing narratives simultaneously. The multiple narratives reward re-reading, carrying great variability between each strand. The capabilities for storytelling that hypertext allows are of central importance to Afternoon, not just in structure but in theme as well. The breakdown of Afternoon’s central character, and his shirking of emotional baggage, is represented by the fragmented, non-linear narrative. While the structure of Afternoon does allow a more open style of reading, it is still quite constrained, particularly when compared to Patchwork Girl.

Patchwork Girl by Shelley Jackson boasts a completely open structure, providing the user with every available section to read from the beginning. Unlike Afternoon the reader encounters no obstacles to progressing through the text. What also separates Patchwork Girl from Afternoon is its visual nature, which it makes good use of, presenting the reader with an image of a body and inviting them to click on whichever part stands out. Related to the form of hypertext mirroring content, in Patchwork Girl the scars embroidering the narrator’s flesh represent both hyperlinks connecting different lexias, and body parts from a wide assortment of different women. Stuart Moulthrop’s Victory Garden, while still visual like Patchwork Girl, was very structurally complex. Begun initially as a class experiment to translate Borges’s “Garden of Forking Paths” into a hypertext, the links between its lexia are heavily plotted, resulting in several narratives running parallel to each other, which describe contrary actions.

[1] Joyce, Michael, Afternoon: a story.

[2] Jackson, Shelley, Patchwork Girl.

[3] Moulthrop, Stuart, Victory Garden.

[4] Jill Walker, “Piecing Together and Tearing Apart: Finding the Story in Afternoon

[5] Janet H. Murray, Hamlet on the Holodeck.

[6] Jay David Bolter, Writing Space.

[7] George P. Landow, Hypertext 3.0: Critical Theory and New Media in an Era of Globalization.

[8] Espen J. Aarseth, Cybertext: Perspectives on Ergodic Literature.

[9] N. Katherine Hayles, Electronic Literature: New Horizons for the Literary.

[10] Hayles, N. Katherine, Writing Machines

[11] N. Katherine Hayles, “Flickering Connectivities in Shelley Jackson’s Patchwork Girl: The Importance of Media-Specific Analysis”.

[12] Stuart Moulthrop, “Reading from the Map: Metonymy and Metaphor in the Fiction of Forking Paths”.

[13] Marie-Laura Ryan, “Multi-Variant Narratives”.

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The Human Memory Essay

Custom Student Mr. Teacher ENG 1001-04 3 April 2016

The Human Memory

Memory refers to processes which are used to obtain, store and later retrieve information. Encoding, storage, and retrieval are the three main processes involved in memory. Encoding is how we form new memories by changing information into a usable form. After the information has been encoded, it then will be stored for later use. Most of the stored memories will lie outside of our own awareness until we later need to use it. The retrieval process is what allows us to bring the memories back to a conscious awareness. (Cherry, 2013) The stage model of memory was initially proposed in 1968 by Atkinson and Schifrin. Their theory outlined three separate stages of memory. These stages were: sensory memory, short-term memory, and long-term memory. This stage model is used to describe the function and basic structure of memory. (Cherry, 2013) The earliest stage of memory is the sensory memory. This is the stage where sensory information from the environment is stored briefly. (For visual information, no longer than half a second, and for auditory information, no more than three or four seconds.)

Some of this information is passed to the next stage of memory. This next stage of memory is called the short-term memory. This is the information in which we are thinking about or are currently aware of. Many short-term memories are forgotten quickly. (Most info only kept 20 to 30 seconds.) However, attending to this information allows it to move into the next stage of memory. This stage of memory is known as the long-term memory. It is referred to the continuing storage of information. The information is mostly outside of our own awareness, but is able to be called into working memory as it is needed. (Cherry, 2013) There are several factors that impede or enhance the information flow during each stage of the memory. Attention is one of the key factors in the sensory memory stage. Paying attention to the stimulus or event will enhance the information flow during this stage, whereas not paying attention will impede the flow. Focusing on information will be a factor in the short-term stage because activation is high only when focusing on the information.

Last, a long direction is a factor that will impede the information flow in the long-term memory stage. Moving information from short-term to long-term memory involves connecting new information to prior knowledge. Stress also profoundly influences memory, depending on the timing of the stress determines which stage of memory that is affected. (Payne, 2007) The theory on why we forget things is known as interference. Two types of interference are proactive and retroactive interference. Proactive interference is when we lose current information due to it mixing with previously learned or similar information. Retroactive interference is where the information works backwards to interfere with previous information. Failure to store is another type of forgetting where encoding failures will sometimes stop information from entering the long-term memory stage. Motivated forgetting also works to forget memories. Motivated forgetting has two forms: suppression and repression.

Suppression is a conscious form of forgetting, while repression is an unconscious form of forgetting, (Cherry, 2013) There are many strategies to improve memory consolidation and retrieval. One way to improve this is through cues such as acronyms. Another way is by reviewing the material before going to sleep. Becoming a more active reader is another way to improve memory. This includes highlighting, underlining, or jotting key words in margins when reading chapters. Over-learning new material has also proven to enhance memory. One last way to improve memory is through providing retrieval practice. One could do this by creating a practice test of the material or even having other students make up questions for other students to answer. (Thorne, 2006)

Cherry, K. (2013). What is memory. Retrieved from Payne, J. D. (2007). Learning memory. Retrieved from Cherry, K. (2013). Explanations for forgetting. Retrieved from Thorne, G. (2013, May 06). 10 strategies to enhance students memories. Retrieved from

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Similar to constructing dykes and levees that separate land from water, data protection is increased by placing barriers between IT resources. Cloud resource segmentation is a process by which separate physical and virtual IT environments are created for different users and groups. For example, an organization's WAN can be partitioned according to individual network security requirements. One network can be established with a resilient firewall for external Internet access, while a second is established without a firewall since its users are internal and unable to access the Internet.

Resource segmentation is used to enable virtualization by allocating a variety of physical IT resources to virtual machines. It needs to be optimized for public cloud environments, since organizational trust boundaries from different cloud consumers overlap when sharing the same underling physical IT resources.

The cloud-based resource segmentation process creates cloud-based security group mechanisms that are determined through security policies. Networks are segmented into logical cloud-based security groups that form logical network perimeters. Each cloud-based IT resource is assigned to at least one logical cloud-based security group. Each logical cloud-based security group is assigned specific rules that govern the communication between the security groups.

Multiple virtual servers running on the same physical server can become members of different logical cloud-based security groups (Figure 1). Virtual servers can further be separated into public-private groups, development-production groups, or any other designation configured by the cloud resource administrator.

Cloud-based security groups delineate areas where different security measures can be applied. Properly implemented cloud-based security groups help limit unauthorized access to IT resources in the event of a security breach. This mechanism can be used to help counter the denial of service, insufficient authorization, and overlapping trust boundaries threats, and is closely related to the logical network perimeter mechanism.

Figure 1 - Cloud-Based Security Group A encompasses Virtual Servers A and D and is assigned to Cloud Consumer A. Cloud-Based Security Group B is comprised of Virtual Servers B, C, and E and is assigned to Cloud Consumer B. If Cloud Service Consumer A’s credentials are compromised, the attacker would only be able to access and damage the virtual servers in Cloud-Based Security Group A, thereby protecting Virtual Servers B, C, and E.


Anhia Santana Gives the Iconic Rugrats a Perplexing New Look

Artist Anhia Santana is once again showcasing her twisted talent with her collection of distorted Rugrats prints.

Going by the alias 'Distorted,' Santana is no stranger to perplexing artwork. The Dominican artist has made a name for herself within the visual art scene for her use of various colors and dark imagery. This collection of prints features 90s toddlers and tots, the Rugrats, reimagined in an eerie light. The lovable Tommy and gang are illustrated to have one eye replaced with a symbolic spiral commonly used by the artist. These prints also include odd accents such as immensely long tongues and gloomy backdrops.

Santana explained that her cryptic work stems from past experiences. Refusing to let negativity consume her, Anhia Santana expresses herself through her art "no matter how ugly" it may seem to some.


The Certificate IV in Training and Assessment

The current version of the Certificate IV in Training and Assessment is the normal requirement for staff who teach and assess in the Australian Vocational Education and Training (VET) sector. You should do this course if you do not already have a higher qualification in adult education and:

If you did a previous version, it is no longer recognized for teaching and assessing purposes in VET colleges.

The units | Prerequisite skills | The ACAS approach

If you already teach in VET, try this self-assessment

Frequently asked questions

What if you have a school teaching qualification?

What different ways can you do the course?

Why is it called "The TAE"?


Imani Colclough| Executive Chef/Owner

Chef Imani Colclough resides in Charlotte, North Carolina.  “My roots run deep in the south because of my heritage,” says the Chef.  His interest in food comes from a family, known for their soulful dishes and decadent desserts.  As a child, he lived for Sunday dinners and holiday feasts.  Years later, he started improvising family recipes for his own soulful culinary creations.  Presently, Chef Imani’s “fusion” style combines ethnic spices with soulful dishes.

          After attending The Culinary Institute at USC, Chef Imani continued to create dishes with new combinations of herbs and spices in ways that excite the palate and soothe the soul.  The Chef‘s specialty dishes include bread pudding.  His bread pudding, an old family favorite, includes a layer of pudding, sauce, and other spices not included in the original family recipe.  An appreciation of foods from other cultures, Chef Imani expresses a style that is sure to win the affection of many food lovers.  He showcases a soulful “fusion” style without leaving behind those decadent desserts that are surely worth a tooth ache.


The rel-san-vek is an ancient Vulcan ceremony that is performed to disolve a bonding between two Vulcans.

Following Spock's bonding to Saavik following the unusual circumstances of his resurrection in the year 2285, both parties eventually returned to planet Vulcan to have the ceremony performed. (TOS - Crucible novel: The Fire and the Rose)

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