Negative numbers are something new and interesting to think about. What do they mean?
They arose from changing the order in which we subtracted two numbers. While we usually think of a “difference” by starting our thought process with the larger number, when we fail to do that and try to “take away” more than we have, the answer is a negative quantity. So a negative answer implies we do not have enough; it represents the absence of a quantity, a quantity that is a debt.
states that we owe 2 items, while
states that we own two items, and lastly
states that we own no items and we owe no items. We have nothing.
Once we begin playing around with the order in which we subtract two numbers, we must eventually bump into an answer of zero (as well as positive and negative answers).
When you stop and think about it, zero is a rather interesting concept. Books have even been written about it. Most of our lives are spent reacting to things in our environment (positive numbers), or their absence (negative numbers). If we have never conceived of a particular “thing”, we would have no reason to conceive of its absence either… in other words, we would not be thinking about it at all. We have no thing, neither its presence nor its absence, we have nothing: zero.
The simple universe of counting numbers that we started out with when contemplating addition could not represent all possible answers that arise when using subtraction. Subtraction forced us to expand our universe of numbers from the “counting numbers”, to “the integers”.
Our original universe of the counting numbers (1, 2, 3, etc.) could represent the answer to all addition problems based on the counting numbers. But they could not represent the answer to all subtraction problems based on the counting numbers… subtraction problems expanded our universe of numbers to what we call the integers, which can represent the answer to all addition and subtraction problems based on the integers. The integers include: all the counting numbers, their corresponding negative numbers, and zero.
There are many situations in our lives where we pose a question that seeks a positive quantity as an answer. A question such as “what is the distance between Rome and Naples?” expects a positive quantity as an answer. We do do not care if the distance is measured from Rome to Naples, or from Naples to Rome. We only care about the total distance. We expect a non-negative number as an answer.
If we use subtraction to answer the distance question, there is a risk we could set the problem up “improperly” and end up with a negative answer. Let’s compare the results we get by setting up such a problem “properly” and “improperly”:
What do you notice about the two answers? If you visualize them on a number line, they are both the same distance from zero, one in the positive direction, the other in the negative direction.
A number of math problems can turn out this way: if they are set up the wrong way around, the answer will have the wrong sign, but still have the correct distance from zero. The idea of absolute value fixes this. By asking for the absolute value of the answer, we are asking for the magnitude of its distance from zero, regardless of direction. Negative two is just as far from zero as positive two… they both have a “distance from zero” of two.
The result of taking an “absolute value” will always be either a positive number or zero. The notation used to indicate “absolute value” is two vertical bars on either side of the quantity that should be zero or positive. These vertical bars are considered “grouping” symbols, somewhat similar to parentheses, and instruct us to evaluate the expression inside them completely, and then drop any negative sign from the answer:
A frequent mis-conception about absolute value notation arises when an expression is included inside the absolute value signs:
This problem asks you to first evaluate what is inside the absolute values signs completely, then drop any negative sign… not the other way around. Note that if there is a variable inside the absolute value signs, you cannot evaluate the inside completely… so you will have to leave the absolute value signs in place until you know the value of the variable. Absolute value signs do not instruct you to “make everything inside positive”. So the answer to the above is:
Note that as you drop any negative sign, you should also drop the absolute value signs at the same time – the result is positive or zero, so they are no longer needed.
Two numbers that have the same absolute value but opposite signs are considered opposites. They are on “opposite” sides of zero, and equal distances from zero.
Opposites have an interesting property: adding opposites always results in an answer of zero, which turns out to have a number of uses in algebra.
Posts that continue this series:
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions