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 return one that’s bigger. Just add “1” to whatever number they come up with and you’re sure to win.
The reason is that 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 keep track of how many days had passed. That’s why they invented numbers.
But back then, they were quite limited and had a very simple form. Often, the “numbers” were just notches on a bone, going up to a couple hundred at most.
When they got bigger
As time went on, people’s needs grew. Herds of livestock had to be counted, goods and services traded, and measurements made for buildings and navigation. This led to the invention of larger numbers and better ways of representing them.
About 5,000 years ago, the Egyptians began using symbols for various numbers, with a final symbol for one million. Since they didn’t usually encounter bigger quantities, they also used this same final symbol to depict “many.”
The Greeks, starting with Pythagoras, were the first to study numbers for their own sake, rather than viewing them as just counting tools. As someone who’s written a book on the importance of numbers, I can’t emphasize enough how crucial this step was for humanity.
By 500 BCE, Pythagoras and his disciples had not only realized that the counting numbers – 1, 2, 3 and so on – were endless, but also that they could be used to explain cool stuff like the sounds made when you pluck a taut string.
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.
Think of how important zero is in expressing big numbers. You can start with 1, then add more and more zeroes at the end to quickly get numbers like a million – 1,000,000, or 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 centuries earlier in India, came to Europe. This led to the way we write numbers today.
This brief history makes clear that numbers were developed over thousands of years. And though the Egyptians didn’t have much use for a million, we certainly do. Economists will tell you that government expenditures are commonly measured in millions of dollars.
Also, science has taken us to a point where we need even larger numbers. For instance, there are about 100 billion stars in our galaxy – or 100,000,000,000 – and the number of atoms in our universe may be as high as 1 followed by 82 zeros.
Don’t worry if you find it hard to picture such big numbers. It’s fine to just think of them as “many,” much like the Egyptians treated numbers over a million. These examples point to one reason why numbers must continue endlessly. If we had a maximum, some new use or discovery would surely make us exceed it.
Exceptions to the rule
But under certain circumstances, sometimes numbers do have a maximum because people design them that way for a practical purpose.
A good example is a clock – or clock arithmetic, where we use only the numbers 1 through 12. There is no 13 o’clock, because after 12 o’clock we just go back to 1 o’clock again. If you played the “bigger number” game with a friend in clock arithmetic, you’d lose if they chose the number 12.
Since numbers are a human invention, how do we construct them so they continue without end? Mathematicians started looking at this question starting in the early 1900s. What they came up with was based on two assumptions: that 0 is the starting number, 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.
You might 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: Is “0” the same as “nothing,” and if so, what exactly is meant by “nothing”?
The second might seem even more strange. After all, we can easily show that adding 1 to 2 gives us the new number 3, just like adding 1 to 2002 gives us the new number 2003.
But notice that we’re saying this has to hold for any number. We can’t very well verify this for every single case, since there are going to be an endless number of cases. As humans who can perform only a limited number of steps, we have to be careful anytime we make claims about an endless process. And mathematicians, in particular, refuse to take anything for granted.
Here, then, is the answer to why numbers don’t end: It’s because of the way in which we define them.
Now, the negatives
How do the negative numbers -1, -2, -3 and more fit into all this? Historically, people were very suspicious about such numbers, since it’s hard to picture a “minus one” apple or orange. As late as 1796, math textbooks warned against using negatives.
The negatives were created to address a calculation issue. The positive numbers are fine when you’re adding them together. But when you get 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 need negative numbers too.
A simple way to create negatives is to imagine all the numbers – 0, 1, 2, 3 and the rest – drawn equally spaced on a straight line. Now imagine a mirror placed at 0. Then define -1 to be the reflection of +1 on the line, -2 to be the reflection of +2, and so on. You’ll end up with all the negative numbers this way.
As a bonus, you’ll also know that since there are just as many negatives as there are positives, the negatives must also go on without end!
The author is a professor of mathematics and statistics, University of Maryland, Baltimore County.
This article first appeared on The Conversation. It is republished here under a Creative Commons License.
