I’ve been giving the problems in my post “division by zero” a lot of thought, and I’ve come to some interesting conclusions.
When Isaac Newton developed The Calculus, the primary purpose was to work around the problems of dealing with infinitesimal values by creating a set of rules which, when applied to functions, could transform them in meaningful ways normally blocked by the problems that arise when you work with zero (or numbers of infinitesimal size. I think they’re equivalent).
This, obviously, worked extremely well. Setting this aside, the ideas in my “division by zero” post allow a different path to some of the solutions that can be reached with calculus, namely derivatives. Instead of using a set of defined rules to emulate the effects of working with intractably small values, you can simply change the rules of arithmetic a bit, make zero a fundamental, very useful number, and reach the solution by actually using the infinitesimals
Let’s take the simplest nonlinear equation there is: x^2. To derive this using calculus, you would follow the rule that, in order to derive x^n, the result is nx^(n-1). Using zero, however, you can arrive at the same answer using nothing but the basic rules of algebra (with a few modifications, of course). The trick is to model it as the slope of a line (not a curve - just rise over run) between the point one-half of an infinitesimal less than x, and one-half of an infinitesimal more than x (I found that using whole infinitesimals gave you an approximate result only half as accurate, due to the fact that it spans more than one infinitesimal in the run). Here’s the basic equation, unsimplified:
((x + 0.5 * 0)^2 - (x - 0.5 * 0)^2)÷0
From there, you can apply the new rule that x÷0 = x*∞:
∞((x + 0.5 * 0)^2 - (x - 0.5 * 0)^2)
Next, you can square the terms in parentheses (simple algebra - nothing new here except the special treatment of zero):
∞(x(x + 0.5*0) +0.5*0(x + 0.5*0) - (x - 0.5*0)^2)
∞(x^2 + 0.5*0x + 0.5*0x + 0.25*0^2 - (x - 0.5*0)^2)
∞(x^2 + 0x + 0.25*0^2 - (x - 0.5*0)^2)
∞(x^2 + 0x + 0.25*0^2 - (x(x - 0.5*0) - 0.5*0(x - 0.5*0)))
∞(x^2 + 0x + 0.25*0^2 - (x^2 - 0.5*0x - 0.5*0x + 0.25*0^2))
∞(x^2 + 0x + 0.25*0^2 - (x^2 - 0x + 0.25*0^2))
∞(x^2 - x^2 + 0x + 0x + 0.25*0^2 - 0.25*0^2)
Okay, now I’m going to apply an extra rule I came up with since the previous post, which is that a - a = 0^∞ (this is because zero actually holds infinitesimal value, so, to get the “true” zero that you should, when subtracting a number from itself, you must multiply it by zero an infinite number of times):
∞(0^∞ + 2*0x + 0^∞)
∞(2*0^∞ + 2*0x)
Now distribute the infinity over the terms:
2*0^(∞-1) + 2x
For all intents and purposes (I.E. ignoring the infinitesimally small infinitesimal), you get the exact same answer that you would arrive at with The Calculus, 2x, without anything but standard algebra and a little bit of special treatment for zero. So far, I’ve also tested this on x^3 and 1/x, and the derivatives are all perfect (with a few of those 0^∞ terms left over, but that has no bearing on the actual value unless you’re dealing with ∞^∞). Conclusion: derivatives (and maybe some other parts of calculus) can be replicated in a much simpler way using this technique of dealing with zero.
I know how to answer this question for people now.
Me: Would anyone like to volunteer to read?
*moment of silence*
Student jumps up *in a dramatic voice*: I volunteer! I volunteer as tribute!
I don’t care how unbelievably nerdy or weird this is, I love this plot of (-1)^x. Math is so pure and beautiful.
I’ve been keeping track in my mind recently of how much time in class and at home is actually spent learning as a direct result of my school’s actions. While I go to a great private school, and think very highly of both my teachers and peers, I see that the amount of inefficiency in both my school and the American school system in general is unreasonably high, and that the entire modern teaching paradigm for elementary through high school is pretty fundamentally flawed. I realize that education is a very difficult thing to change, and that many of the ideas in this post might be impractical, but I think I’ve come up with a few simple rules that, if applied, could drastically improve the way that this generation acquires knowledge and skills.
First of all, I’d like to point out that school has deviated pretty far from its original purpose. When the concept of an educational institute was first proposed I-don’t-know-how-many-millennia ago, it had one goal and one goal only: teach the students everything they need, and maybe even more if they’re interested.
Over the centuries, and especially in recent decades in America, school has become a catch-all kid-raising center. A parent can simply send their child to the school, and the school will do the work of providing the young man or woman with a social life, an athletic activity, a little bit of musical experience, some (hopefully) good work habits, an ability to follow rules and instructions, and maybe a little academics where they can squeeze it in. This is even more true of private schools than public, and although the private schools are considered “better” due to this fact, whether this quality is a good thing or a bad thing is debatable.
While all of these functions of school I’ve enumerated are undeniably important parts of an individual’s younger years, I don’t think that school is the right place for them. I’d like to see a school focussed entirely on learning - not friends, sports, art, or homework.
Given this premise, I’ve devised a list, a constitution if you will, of this school’s philosophies, aimed at a more college-style learning experience. This might be crazy and it’s definitely controversial, but I think it might just be crazy enough to work:
1. Everything digital
The most common causes of strife in the classroom I attend (and my life as a whole, actually), stem from humanity’s infatuation with the archaic art of applying pigments to slices of dead trees’ pulp. In this day and age, paper is an unnecessary complication that can almost always be solved by switching to a more electronic medium. I would like all homework to be composed and turned in online, tests to be administered (and sometimes graded) by a computer, and outlines of lessons to be posted online by teachers (for the more visual learners out there). These solves the constant problem of “Randal, where is your textbook?” - “I left it at home,” or “Timmy, where is your essay?” - “I forgot to print it out.” Electrons all the way.
2. All homework optional
Far too often do I find myself irritated by a long homework assignment proving my knowledge of a topic or proficiency in a skill that I mastered in the first hour of learning it. I believe that homework should be optional in the sense that, for every homework assignment not turned in, rather than the points of that assignment becoming a zero in the gradebook, the value of the assignment being added to the value of your test grade for the unit. This way, people (like me) who receive no benefit from the majority of the homework assignments given to them can simply not complete the assignments, and prove their understanding through the test (which, with little homework turned in, would count for something like 80% of your grade). On the other hand, more learn-by-doing oriented students who require further practice to master the topic, or students who break down under the pressure of a test, can choose to complete nearly all the homework, lowering the value of their test to a measly five or ten percent and proving their knowledge through work.
3. Attending class optional - notes from lesson posted day before
This one is probably the most controversial, but I think it’s also the rule that would make the biggest difference (it’s my personal belief that this difference would be positive). Quite often during class I find myself realizing that, while a few tidbits from the lesson are new and worthwhile, the majority of the explanation serves the purpose of further cementing knowledge of material that has already firmly taken hold in my mind, and the minds of many of my classmates. For almost every class I attend, I could glean a 1-hour lesson’s worth of understanding by spending 10 minutes reading and internalizing a bullet-point summary of the day’s teachings. If teachers could simply post well-made notes from lessons online the day before they give the lessons, students could decide whether or not they even need to attend that given class. For the instances that a student needs further information or explanation about a topic, they can choose to attend that day’s lesson only. Again, this is a framework that provides students with the freedom to make their own decisions about their own learning styles, and empowers them to take control of their education.
That’s it. I believe that with those three straightforward rules, plus the philosophy of schools being single-purpose establishments, education could be drastically improved. The biggest causes of inefficiency in my classroom tend to result from a newton’s third law of motion, in a metaphorical sense. Easily 25 percent of my school day is taken up by students pushing against teachers, and the teachers pushing back. Neither the students nor the teacher are to blame here; it’s the system that caused the students to push and the teachers to have to push back that causes the constant confrontations. If students could simply be given the opportunity to do what they like with their own education, the world would be a wonderful place.
Dividing by zero can’t be that bad, can it? Let’s just try… *KABOOM!*
Naw, just kidding. It’s supposed to be impossible, but’s lets see what happens if you suppose one simple rule:
1÷0 = ∞
Okay, steady as she goes. How about numbers other than 1?
x÷0 = x * (1÷0) = x * ∞
Still works, although a little strangely (more than one kind of infinity!?).
Now, here comes the weird part:
1÷0 = ∞ ; 1÷0*0 = ∞*0 ; ∞*0 = 1
0∞ is one? Okay, now that’s flat out weird. Zero times anything should be zero, right? Actually, not necessarily. Think about this:
Imagine you have a perfectly flat plane (physically impossible, but whatever). This plane, by definition, is of zero depth. This means that if you stacked another one on top of it, you’d still have zero depth. No matter how many you stack (how many multiples of zero you have), the depth is still zero. In order to achieve any depth whatsoever, you’d need an infinite number! If you agree with this idea (which you might not), then you agree that ∞ * 0 does have a value greater than zero. Actually, it has a value of exactly one.
If you’ve somehow managed to follow my insane mathematical ramblings so far, then it won’t come as a surprise to you that:
1÷0 = ∞ ; 1 = 0*∞ ; 1÷∞ = 0
In fact, that is the most sensible part of this whole post. You’re splitting one up into an infinite number of parts, so each part would be of no size at all. Makes sense.
Now to address the problems that normally go along with division by zero. Here’s the classic:
0 * 1 = 0 * 2
(0 * 1) ÷ 0 = (0 * 2) ÷ 0
1 = 2
Obviously one doesn’t equal two, so something’s wrong here.
My solution to this problem, and any more complex ones that follow the same principle, is this: In order for zero and infinity to be valid where all other numbers would be in an equation (even in division), they must be treated similarly to the imaginary unit i or -1 - they cannot be simplified with other numbers while multiplied.
Although most numbers can be multiplied together and simplified, like so:
2 * 2 = 4 (you don’t say?)
There are a choice few who must remain how they are:
97 * i
In this expression, i and 97 cannot be reconciled. i * 97 is just i * 97, and nothing more. I must stand on its own like a variable, even though it is a constant.
I am proposing that to be able to give zero the inalienable rights that all numbers deserve (are you seeing a possible movement here: the “Numerical Rights Movement?”), you must treat it similarly. x * 0 is no longer 0, it is just x * 0. Therefore:
0 * 1 ≠ 0 * 2
problem solved! This prevents the potential for information loss that always bothered me in algebra. This rule extends to division by infinity:
x ÷ ∞ = x * (1 ÷ ∞) = x * 0
so you can’t say
1÷∞ = 2÷∞ ; (x÷∞) * ∞ = (2÷∞) * ∞ ; 1 = 2
Because the first statement is equivalent to:
1 * 0 = 2 * 0
Which is not okay.
So far, division by zero and infinity both don’t seem that hard. You can you use them just like any other numbers, and everything seems to work fine. There is, however, one issue. Consider simplifying these fractions:
(1 ÷ 0) * (1 ÷ 0)
It seems easy enough, but there are in fact two ways to do it, both of which seem equally valid, but which give you different results. The first way is to treat both fractions as infinity:
(1 ÷ 0) * (1 ÷ 0) = ∞ * ∞ = ∞^2
Fine, but you can also multiply the fractions, then divide!
(1 ÷ 0) * (1 ÷ 0) = 1 ÷ 0 = ∞
∞ = ∞^2? That doesn’t seem right… Either exponentation of infinity does nothing, or there are restrictions placed on infinity.
Don’t worry - this isn’t necessarily a death-blow to the idea. Imaginary numbers have similar problems with expressions like √-1 * √-1:
√-1 * √-1 = √(-1 * -1) = √1 = 1
√-1 * √-1 = i * i = i^2 = -1
-1 most certainly does not equal 1, but that doesn’t mean imaginary numbers aren’t incredibly useful. Using them just imposes certain restrictions - it shows that, under extreme conditions, proceses that seem equivalent actually have subtle differences.
There’s one last issue, then you can begin reading sane posts again. What is (-1)^∞? Is it positive? negative? imaginary? complex? Is ∞ even, or odd, or even an integer? (-1)^∞ is like to trying to predict what number a gameshow wheel will land on if you spin it for eternity then stop it.
For now, let’s just say that it’s 3, because that seems harmless enough. :)
Well, I guess it’s about time to become involved in social media.
We’re going to have to use… *puts on sunglasses* …Tumblr.