Probably most of us who use a calculator or a computer to make mathematical calculations have encountered a problem when trying to divide some quantity by zero. On my Texas Instruments calculator, I get an ‘Error’ message when I try to divide a number by zero. On the calculator on my Galaxy tablet, I get the message ‘Cannot divide by zero’. The calculator on my smart phone doesn’t respond at all if I try to divide by zero. And in MS Excel, if I enter a formula in a spreadsheet cell that divides by zero I get ‘#DIV/0!’ appearing in the cell.

So exactly what is the problem with dividing by zero? After all, doesn’t zero represent nothing? Can’t I just ignore zero when it shows up in some calculation?

Well, not quite. Zero has some rather unique mathematical properties, especially when it is involved in a division operation.

In this article, we’ll look at two situations involving division by zero that should clarify the issue of why dividing by zero causes a problem. The first situation occurs when we divide (or try to divide) a non-zero number or quantity by zero and the second situation occurs when we divide zero by zero. In both situations, the correct result of the division by zero operation is that it is undefined, meaning that there is no definite, finite value that results from the division by zero. We will look at these two situations in the following discussion and explain why there is no defined result obtain by dividing by zero.

Situation 1: dividing a non-zero quantity by zero

Suppose that represents some non-zero quantity. The result of dividing by zero can be stated mathematically as: , an undefined, infinitely large number. To see why this is true, consider the following explanation.

Multiplication can be considered as repeated addition. For example, if we multiply 4 x 2, we can obtain the result by adding 2 to itself 4 times; that is, . Similarly, division can be considered as repeated subtraction. For example, if we divide 9 by 3, we can obtain the result by counting how many times 3 can be subtracted from 9; for example,

First subtraction: 9 − 3 = 6
Second subtraction: 6 − 3 = 3
Third subtraction: 3 − 3 = 0

So that dividing 9 by 3 is equivalent to subtracting 3 from 9 three (3) times, so that

Given that concept, then, we ask how many times can we subtract zero (0) from a non-zero number? The answer of course is that we can subtract zero from a non-zero number (7, for example) an infinite number of times, so that

So, the problem we have when dividing a non-zero number by zero on a calculator, for instance, is that the computation results in a numerical value so large that the physical memory size of the calculator cannot contain the value and an error message is generated.

The bottom line: dividing a non-zero quantity by zero gives an infinitely large or undefined (no definite value) result.

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Situation 2: dividing a zero by zero

This result can be state mathematically as: , an undefined, infinitely large number.

But, you might ask, isn’t dividing a number by itself equal to one, so that zero divided by zero should be one, shouldn’t it?

To address that question, let’s assume that dividing zero by zero gives some finite and definite value so that mathematically we can write the equation . If we now use the typical method for clearing fractions in the equation by multiplying both sides of the equation by the denominator of the fraction, we have that . But we know than multiplying zero times any value equals zero, so that could be anything and the equation would be satisfied. So can be any value and is therefore indefinite or undefined. It’s important to note that zero divided by zero is NOT one (1).

The bottom line: dividing zero by zero gives an undefined (no definite value) result.

So, whether we divide a finite known quantity by zero or we divide zero by zero, the result is undefined in both cases.

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