Numbers for Words People

You may have heard that division by zero is not allowed. In fact, if you try to divide any number by zero on a calculator, you will get a version of “undefined”. This post will show why this is the case and relate one maritime mishap that happened as a result of an attempt to break this rule!

What is division anyway?

In primary school, or elementary school if you’re reading this in North America, you may have been taught about division as “sharing equally” or “sharing fairly”. Figure 1 below shows the operation $12 \div 4$ or 12 pencils shared equally among 4 students.

As can be seen in the figure, the result of sharing 12 pencils equally among 4 students is that each student receives 3 pencils. We write this as: $12 \div 4 = 3$.

How division and multiplication are related

Bear with me, we are building towards an understanding of why we cannot divide by zero.

Figure 2 below shows that 4 groups of 3 pencils amount to 12 pencils. It is literally figure 1 upside down!

We can see that:

Figure 3 below shows the equivalence between combining 3 groups of 4 pencils on the one hand, and dividing 12 pencils equally among 3 students, on the other. It shows that:

What we have seen so far is that division and multiplication are inverse operations. Each of them is a mirror image of the other!

Finally, why can’t we divide by zero?

When we ask, “What is $56 \div 8$?” We are asking what is the number, n, such that $n \times 8 = 56$? This follows from the inverse rule we saw in the previous section.

Likewise, when we ask, “What is $5 \div 0$?” We are asking what is the number, n, such that $n \times 0 = 5$? There is no such number, since any number times 0 yields a zero, not 5. In other words, if we have 0 groups of any number of pencils, we must have 0 pencils.

In fact, while we won’t cover it in this post, if division by 0 were allowed, we would be able to prove such statements as 2 = 1. This video demonstrates it to those with some facility with algebra.

We cannot divide by zero because it would break the rules of arithmetic and lead to absurd conclusions.

Division by zero and the disaster that ensued!

On September 21, 1997, while cruising off the coast of Virginia, the billion-dollar missile cruiser shuddered to a halt. Yorktown was dead in the water.
Warships are designed to withstand the strike of a torpedo or the blast of a mine. Though it was armored against weapons, nobody had thought to defend the Yorktown from zero. It was a grave mistake (Seife, 2000, p. 9)¹

This was the story of USS Yorktown, after a software upgrade. The program controlling the propulsion system had a bug that made it divide by zero. The result was three hours of outage, during which engineers had to restore the engines. Later, it took them a few days to locate the problem. Thankfully, they were not doing all this while fighting a war.

Summary

What we have learnt in this post is the following: We cannot divide by zero because no number is the answer to the question: What, times zero, equals 5? The question could be reworded, with 5 replaced by any other number other than 0. It would still not have a sensible answer.

Programs that divide by zero, a common mistake among computer science students, can crash and lead to serious and expensive consequences, as happened in the case of the USS Yorktown in 1997.

Question for the reader

Zero and infinity go hand in hand in many mathematical settings. Would you like my next post to be about infinity?

Also, if you are a “words person” that is reading this, then I really want your feedback on the clarity of my writing. Are the explanations truly “from first principles” or am I assuming, unconsciously, an unreasonable amount of background knowledge?

Reference

1. Seife, Charles (2000). Zero: The Biography of a Dangerous Idea. Viking. ↩︎