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Why don’t the numbers end? – Reyhane, 7 years old, Tehran, Iran
Here’s a game: Ask a friend to give you any number, and you’ll give back the larger one. Just add “1” to the number they find and you can be sure to win.
This is because numbers go on forever. There is no highest number. But why? As a math professor, I can help you find an answer.
First you need to understand what numbers are and where they come from. You learned numbers because they made you count. Early humans had similar needs; counting animals killed in a hunt or keeping track of how many days have passed. That’s why they invented numbers.
But at that time the numbers were quite limited and it had a very simple structure. Most of the time, the “numbers” were just notches on a bone, numbering up to a few hundred at most.
When the numbers get bigger
As time passed, people’s needs increased. Herds had to be counted, goods and services had to be traded, and measurements had to be made regarding buildings and navigation. This led to the discovery of larger numbers and better ways to represent them.
About 5,000 years ago, Egyptians began using symbols for various numbers, with the final symbol being one million. Since they did not usually encounter larger quantities, they also used the same final symbol to depict “many.”
The Greeks, starting with Pythagoras, were the first to study numbers for themselves, rather than viewing them merely as counting tools. As someone who has written a book on the importance of numbers, I cannot emphasize enough how important this step is for humanity.
By 500 BC, Pythagoras and his students had realized not only that counting numbers (1, 2, 3, etc.) were infinite, but that they could also be used to explain wonderful things, such as the sounds made when you pull a taut wire. .
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 the number 0.
Think about how important zero is in expressing large numbers. You can start with 1, then add more and more zeros to the end to quickly get numbers like a million – 1,000,000, or 1 followed by six zeros – or a billion, nine zeros, or a trillion, 12 zeros.
Invented centuries ago in India, the zero only came to Europe around 1200 AD. This led to the way we write numbers today.
This brief history makes clear that numbers have evolved over thousands of years. Even though the Egyptians don’t have much use for a million, we certainly need it. Economists will tell you that government spending is often measured in millions of dollars.
Also, science has brought us to a point where we need even larger numbers. For example, there are approximately 100 billion stars (or 100,000,000,000) in our galaxy, and the number of atoms in our universe can be as high as 1 followed by 82 zeros.
If you have trouble imagining such large numbers, don’t worry. It’s okay to think of them as “many,” just as the Egyptians viewed numbers over a million. These examples point to one reason why numbers must continue forever. If we had a maximum value, a new use or discovery would certainly cause us to exceed it.
Exceptions to the rule
But under certain conditions sometimes numbers have a maximum because people design them that way for a practical purpose.
A good example of this is clock or clock arithmetic, where we only use the numbers 1 through 12. There is no 13 o’clock, because after 12 o’clock we go back to 1 o’clock. If you played the “big number” game of clock arithmetic with a friend, if your friend chose the number 12, you lost.
Since numbers are a human invention, how can we construct them to last forever? Mathematicians began investigating this question starting in the early 1900s. What they found was based on two assumptions: 0 was the starting number, and when you added 1 to any number you always got a new number.
These assumptions immediately give us a list of counting numbers: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, etc., an endless progression.
You may wonder why these two rules are assumptions. The first reason is that we don’t know exactly 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 may seem even stranger. After all, we can easily show that adding 1 to 2 gives us the new number 3, just as adding 1 to 2002 gives us the new number 2003.
But notice that we said this should be true for every number. We cannot verify this very well for every case because there will be an infinite number of cases. As humans who can perform a limited number of steps, we need to be careful when claiming an infinite process. And mathematicians in particular refuse to take anything for granted.
Here’s the answer to why the numbers never end: It’s because of the way we define them.
Now negative numbers
How do negative numbers -1, -2, -3 and more fit into all this? Historically, people were very skeptical of such figures because it was difficult to imagine a “minus one” apple or orange. As late as 1796, mathematics textbooks warned against the use of negatives.
Negatives were created to solve a computational problem. Positive numbers cause no problems when 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 as you wish, you also need negative numbers.
A simple way to create negatives is to imagine all the numbers (0, 1, 2, 3, and the rest) being drawn evenly spaced on a straight line. Now imagine a mirror placed at 0. Then define -1 as the reflection of +1 on the line, -2 as the reflection of +2, and so on. This way you will get all negative numbers.
As a bonus, you’ll know that there are negative numbers as well as positive numbers, and that negative numbers must continue indefinitely!
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This article is republished from The Conversation, an independent, nonprofit news organization providing facts and analysis to help you understand our complex world.
Written by: Manil Suri, University of Maryland, Baltimore County.
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Manil Suri does not work for, consult, own shares in, or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond his academic duties.