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Why don’t numbers end? – Reyhane, age 7, Tehran, Iran
Here’s a game: Ask a friend to give you any number and you’ll give back one that’s bigger. Just add “1” to whatever number they come up with and you’re guaranteed to win.
It’s because the numbers go on forever. There is no highest number. But why? As a professor of mathematics, I can help you find an answer.
First, you need to understand what numbers are and where they come from. You learned about numbers because they enabled you to count. Early humans had similar needs – whether to count animals killed in a hunt or to keep track of the number of days that had passed. That’s why they invented numbers.
But back then, the numbers were quite limited and had a very simple form. Often, the “numbers” were just notches on a bone, amounting to a few hundred at most.
When the numbers increased
Over time, people’s needs increased. Livestock herds had to be counted, goods and services traded, and buildings and shipping measured. This led to the invention of larger numbers and better ways of representing them.
About 5,000 years ago, the Egyptians started using symbols for different numbers, with a final symbol for million. Since they didn’t usually come in larger quantities, they also used the same final symbol to represent “a lot”.
The Greeks, starting with Pythagoras, were the first to study numbers for their own sake, rather than seeing them as direct counting tools. As someone who has written a book on the importance of numbers, I cannot express enough how crucial this step was for humanity.
By 500 BCE, Pythagoras and his disciples not only realized that the counting numbers – 1, 2, 3 and so on – were infinite, but also that they could be used to explain cool things like the sounds made when you pull a tightrope. .
Zero is a critical number
But there was a problem. Although the Greeks could mentally think of very large numbers, they had difficulty writing them down. This was because they did not know about the number 0.
Consider how important zero is in expressing large numbers. You can start with 1, then add more and more zeros at the end to quickly get numbers like a million – 1,000,000, or a 1 followed by six zeros – or a billion, with nine zeros, or a trillion, 12 zeros.
It was only around 1200 CE that zero, invented hundreds of years earlier in India, came to Europe. As a result of this we write numbers today.
This brief history shows that numbers were developed over thousands of years. And while the Egyptians didn’t have much use for a million, we certainly do. Economists will tell you that government spending is commonly measured in millions of dollars.
Also, science has brought us to a point where we need even bigger numbers. For example, there are about 100 billion stars in our galaxy – or 100,000,000,000 – and the number of atoms in our Universe could be as high as 1 followed by 82 zeros.
Don’t worry if you find it hard to picture such big numbers. It’s okay to think of them as “many,” like the Egyptians numbering over a million. These examples show one reason why the numbers must go on and on. If we had a maximum, it would surely be exceeded by a new use or discovery.
Exceptions to the rule
But in certain circumstances, numbers sometimes have a maximum because people design them that way for a practical purpose.
A good example is a clock – or clock numerology, where we only use the numbers 1 to 12. There is no 13 o’clock, because after 12 o’clock we don’t go back to 1 o’clock again. If you played the “larger number” game with a friend in clock numerology, you would lose if they chose the number 12.
Since numbers are a human invention, how do we construct them so that they continue without end? Mathematicians began looking at this question beginning in the early 1900s. What they came up with was based on two assumptions: that the starting number is 0, and when you add 1 to any number you always get a new number.
These assumptions immediately give us the list of counting numbers: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, and so on, a progression that continues without end.
One may wonder why these two rules are assumptions. The reason for the first one is that we don’t really know how to define the number 0. For example: Does “0” equal “nothing,” and if so, what exactly does “nothing” mean?
The latter may seem more strange. After all, we can easily show that adding 1 to 2 gives us the new number 3, just as 1 to 2002 gives us the new number 2003.
But note that we are saying that this must hold on any number. We cannot verify this very well for every single case, as there will be an infinite number of cases. As humans who can only take a limited number of steps, we must be careful whenever we make claims about an infinite process. And mathematicians, in particular, refuse to take anything for granted.
Here, then, is the answer to why numbers never end: It’s because of the way we define them.
Now, the negative numbers
How do the negative numbers -1, -2, -3 and more fit into all of this? Historically, people have been very suspicious of such numbers, as it is difficult to picture an apple or orange “minus one”. As late as 1796, mathematics textbooks warned against using negatives.
The negatives were created to address a calculation issue. The positive numbers are fine when you add them together. But when you come to subtraction, they can’t handle differences like 1 minus 2, or 2 minus 4. If you want to be able to subtract numbers at will, you also need negative numbers.
A simple way to create negatives is to imagine all the numbers – 0, 1, 2, 3 and the rest – drawn evenly on a straight line. Now imagine a mirror placed at 0. Next, define -1 as reflecting +1 on the line, -2 as reflecting +2, and so on. You will end up with negative numbers this way.
As a bonus, you will also know that since there are as many negatives as there are positives, the negative numbers must also go on endlessly!
Hello, strange children! Have a question you’d like an expert to answer? Ask an adult to send your question to CuriousKidsUS@theconversation.com. Please tell us your name, age and the city where you live.
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This article is republished from The Conversation, a non-profit, independent news organization that brings you facts and analysis to help you make sense of our complex world.
Written by: Manil Suri, University of Maryland, Baltimore County.
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Manil Suri does not work for, consult with, share in, or be funded by any company or organization that would benefit from this article, and has not disclosed any relevant affiliations beyond their academic appointment.
