Sunday, September 20, 2020

Division by 0

[Once again, I have written something for my class that I think will be valuable for others.]

Big question: What are the values of LaTeX: \frac{3}{0}, LaTeX: \frac{0}{3}, and LaTeX: \frac{0}{0}?


We want to be able to look at each of these fractions, know what it equals, and understand why. This becomes vital in calculus. [Note: Many students have trouble with this. It may be because elementary teachers are often uncomfortable with division, and teach it by memorization, instead of as something deep to understand. Or it may be that this is deep, and our brains need more time to really make sense of it.]


To help ourselves understand this, we tie it to something simpler that we understand better. Division is the inverse of multiplication (ie they undo each other). So it will help to explore how the two operations are connected.

We start with a very concrete and simple problem: LaTeX: \frac{6}{3}=2

[Note: One notational problem with division is that it's written in different ways that place the numbers in opposite orders. LaTeX: \frac{6}{3}=6\div3, but these are also equal to. When I was young, I had trouble keeping track of which was which, so I would write down an easy problem, like this one, to help me remember.]

Now we consider the multiplication problem that goes with this division problem: LaTeX: \frac{6}{3}=2\Longleftrightarrow3\cdot2=6, and we can say that 6 divided by 3 is 2 because LaTeX: 3\cdot2=6.


Let's use T for top, B for bottom, and A for answer, and rewrite this equivalence of a division problem and its associated multiplication problem, in a way that will always be true: LaTeX: \frac{T}{B}=A\Longleftrightarrow B\cdot A=T

In the fraction (or division), we have top over bottom gives answer, and that gives us a multiplication problem where the original bottom times the answer from the division gives us the original top.  [Note: I am purposely avoiding the proper terms: numerator or dividend, denominator or divisor, and quotient (for the answer). For anyone who gets those terms mixed up, it's easier just to focus on position for the moment.]

Now we are ready to consider each of the three original questions, using this correspondence.

1. Let's think about the multiplication associated with LaTeX: \frac{3}{0}:

LaTeX: \frac{3}{0}=A\Longleftrightarrow0\cdot A=3

So what do we multiply 0 by to get 3? Hmm. It seems that nothing works. There is no number that can multiply with 0 and give us 3. So the division problem (or fraction) has no solution, and we say that LaTeX: \frac{3}{0} is undefined.  This is why we say "division by 0 is undefined".


2. LaTeX: \frac{0}{3}=A\Longleftrightarrow3\cdot A=0. Ahh, this one is easier. LaTeX: 3\cdot0=0 so the answer is 0.


3. LaTeX: \frac{0}{0}=A\Longleftrightarrow0\cdot A=0. Hmm, this time A could be any number, and the multiplication would be correct. This is still division by 0, so it is still undefined, but it is very different from the first case. We call it indeterminate. We can see why by looking at a rational function example.

Example: LaTeX: y=\frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}

When x= -2 or 2, this function will be undefined (because we have division by 0). But the function's behavior for x values very close to -2 is very different from its behavior for x values very close to 2.

LaTeX: x=-2 is a vertical asymptote for the graph. This means that as x approaches -2, the y values approach LaTeX: \pm\infty. (This can be written "as LaTeX: x\longrightarrow-2,\:y\longrightarrow\pm\infty".) You can verify this by trying these x values: -2.1, -1.9, -2.01, -1.99,... (You can also use desmos to view the function.)

What happens near LaTeX: x=2? We see that the y value does not depend on the factor LaTeX: \left(x-2\right), because it cancels. So, as long as LaTeX: x\ne2, LaTeX: y=\frac{\left(x-1\right)}{\left(x+2\right)}. At LaTeX: x=2, this would equal 1/4. The function is not defined here, but now we can see that as LaTeX: x\longrightarrow2,\:y\longrightarrow\frac{1}{4}.

So why was LaTeX: \frac{0}{0} called indeterminate? Because the value associated with it in a particular function is determined by other parts of the function. Although LaTeX: \frac{0}{0} is undefined, we saw that, in this particular function the value of the function got close to 1/4 as the x value got close to 2, which is the number that would give us LaTeX: \frac{0}{0}. This concept goes with the concept of limits, one of the 3 major topics in calculus.



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