Why You Can't Divide By Zero

If you type 1/0 into a calculator and hit =, you will get an error. Here's why. 

Let's start with the equation: 

y = 1/x

If we pick 10 for x, y = 1/10. We can make a chart of values for x and y like so:

x                    y

0.001            1000

0.01              100

0.1                10

1                   1

10                 0.1

100               0.01

1000             0.001   

We can see that as x gets bigger, y gets smaller and vice-versa. Part of learning math is drawing your own pictures, so now I ask that you get a pen and paper and plot these points on a graph with y as the vertical axis and x as the horizontal axis. Connect the points to make a curve. You should end up with a curve that looks like an L. You can also use the free online graphing calculator below to make a graph of y = 1/x. Just type "1/x" in the box on the left and hit "enter". 

Desmos | Graphing Calculator

There is a gap in the graph at x = 0 and another at y = 0. As x approaches 0 from the negative side (the left), y becomes an increasingly large negative number. As x approaches 0 from the positive side (the right), y becomes an increasingly large positive number. When this sort of thing happens, we say that the graph has an asymptote. In this case, the graph has two asymptotes at x = 0 and two more asymptotes at y = 0.

Another way of describing what is happening here is that the limit of 1/x as x approaches 0 is undefined. It's positive infinity when approached from the right and negative infinity when approached from the left. Limits are used to prove the basic rules of calculus, but that is a lesson for another time. 

This means that the curve of the equation y = 1/x never crosses either the x or y axis of its graph. To put it another way, no matter how close x gets to zero, y just keeps getting bigger. On older mechanical calculators, dividing by zero would cause it to grind away for a long time, like this:

The user typed in 1/0. The mechanical calculator here keeps trying larger and larger values to multiply with 0 in order to get 1, but since anything times zero is zero, it gets stuck in a loop. If the user had instead entered 1/2, the calculator would have stopped at 0.5, because 0.5 times 2 is 1 and 1 minus 1 is 0. 

Now you know why we can't divide by zero. On a side note, different calculators sometimes give different answers to the same input because of the way they are built, like so:

It's important to know how to do math by hand because calculators don't know when you've hit a wrong button, and they can't tell you the right buttons to push. 

Here's a nice song called Divide By Zero. See y'all next time.