Why Can’t We Divide By Zero?

by Carson
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1 divided by zero

Everyone knows that dividing any number by zero produces an error, but why can’t we divide by zero? Let’s find out about that in this article.

Proof Using Multiplicative Inverse

To prove why dividing by zero is prohibited, we will learn about multiplicative inverses. Let an integer be n and its multiplicative inverse by m. In that case, m * n always equals 1. For instance, the multiplicative inverse of 3 is 1/3 since 3 * 1/3 = 1.

However, 0 has no multiplicative inverse. Although the multiplicative inverse of an integer n always gets greater as n gets smaller, 0 is at the very center of the number line. Therefore, no matter what number you multiply by zero, you still get precisely zero. This implies that no number n can satisfy the condition where 0n = 1, thus 0 has no multiplicative inverse.

Since any number n divided by another integer m is just (the multiplicative inverse of m) * n, we cannot divide by zero.

Another Proof

There is another way to prove there is no multiplicative inverse for 0 by inducing a contradiction. In case you don’t know how to prove something by contradiction, it means that assuming that your hypothesis is wrong and showing that the consequences of this are impossible, thus proving that your conjecture can only be correct.

Firstly, assume there is a multiplicative inverse of 0, known as the number n. Then, 0 * n = 1, which means that 0 * n + 0 * n = 2. Under the distributive law of multiplication, it means that 0 * n = 2. However, we’ve already shown that 0 * n = 1. That means 1 = 2, and all numbers are equal. This is not true on the number line, creating a contradiction. Thus, the multiplicative inverse of 0 is non-existent.

How Do Computers Handle Division By Zero?

Division by zero is something unique for computers. Take this small Python code snippet as an example:

print(1 / 0)

After running this, you will immediately see a ZeroDivisionError, which can only be induced by instructing the computer to divide something by zero. This is different from misspelling some keywords or missing some special characters, where you might get a NameError, ValueError, or SyntaxError.

However, JavaScript reacts differently to division by zero. Here is an HTML code snippet that displays the result of 1 / 0:

<!doctype html>
<html>
    <head>
        <title>Test</title>
    </head>
    <body>
        <p id="output"></p>
    </body>
</html>
<script>document.getElementById("output").innerHTML = 1 / 0</script>

If you run this HTML code on your browser, you should see “Infinity” instead of an error or an undefined value. However, note that the value of any number divided by zero is still undefined because you can prove that we can’t divide by zero in the previous sections of this article. If you want to learn more about that, please visit the webpages in the references below.

References and Credits

  1. (n.d.). Why can’t we divide by zero? Retrieved December 18, 2021, from https://www.math.utah.edu/~pa/math/0by0.html
  2. (n.d.). Dividing by Zero. Retrieved December 18, 2021, from https://www.mathsisfun.com/numbers/dividing-by-zero.html
  3. TED-Ed. (2018, April 23). Why can’t you divide by zero? – TED-Ed. Retrieved December 18, 2021, from https://www.youtube.com/watch?v=NKmGVE85GUU

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