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Why don't numbers end? – Reyhane, 7 years old, Tehran, Iran
Here's a game: ask a friend to provide you any number and also you'll get a much bigger one in return. Just add “1” to the number you discover and you’re sure to win.
The reason is that the numbers go on eternally. There isn’t any maximum number. But why? As a Professor of mathematicsI can aid you find a solution.
First, it is advisable understand what numbers are and where they arrive from. You learned about numbers because they allowed you to count. Early humans had similar needs, whether it was to count animals killed while hunting or to maintain track of what number of days had passed. That's why they invented numbers.
But back then the numbers were quite limited and had a quite simple form. Often the “numbers” were just notches on a bone, numbering a couple of hundred at most.
As the numbers got greater
Over time, people's needs grew. Herds of livestock needed to be counted, goods and services needed to be traded, and measurements needed to be taken for buildings and shipping. This led to the invention of larger numbers and higher ways to represent them.
About 5,000 years ago The Egyptians began using symbols for various numbers, with a final symbol for 1,000,000. Since they didn’t normally encounter larger quantities, in addition they used the identical final symbol to represent “many.”
The Greeks, starting with Pythagoras, were the primary to check numbers for their very own sake, somewhat than viewing them simply as counting tools. As someone who’s wrote a book in regards to the meaning of numbersI cannot emphasize enough how crucial this step was for humanity.
Around 500 B.C. BC Pythagoras and his students had not only realized that counting numbers – 1, 2, 3 and so forth – were infinitebut in addition that they could possibly be used to elucidate cool things like that Noises produced when a stretched string is plucked.
Zero is a critical number
But there was an issue. Although the Greeks could mentally imagine very large numbers, they’d difficulty writing them down. That was because they didn't know anything about it the number 0.
Remember how essential zero is in representing large numbers. You can start with 1 after which add an increasing number of zeros at the top to quickly get numbers like 1,000,000 – 1,000,000 or 1 followed by six zeros – or a billion with nine zeros, or a trillion with 12 zeros.
Only around 1200 AD did it reach zero, Invented centuries earlier in India, got here to Europe. This led to the best way we write numbers today.
This short history makes it clear that numbers have been developed over hundreds of years. And while the Egyptians didn't have much use for 1,000,000, we actually do. Economists will let you know that government spending is often measured in hundreds of thousands of dollars.
Additionally, science has brought us to some extent where we want even larger numbers. For example, there are about 100 billion stars in our galaxy – or 100,000,000,000 – and the variety of atoms in our universe could possibly be as high as 100,000,000,000 1 followed by 82 zeros.
Don't worry if you might have a tough time imagining such large numbers. It's okay to easily consider them as “many,” just because the Egyptians treated numbers over 1,000,000. These examples point to a reason why numbers must go on indefinitely. If we had a maximum, we would definitely exceed it through some latest use or discovery.
Exceptions to the rule
But under certain circumstances, numbers sometimes have a maximum because people design them that way for practical reasons.
A great example is a clock – or clock arithmetic, where we only use numbers 1 to 12. There isn’t any 1 p.m. because after 12 p.m. we simply return to 1 o'clock. If you were to play the Bigger Numbers clock game with a friend, you’d lose in the event that they selected the number 12.
Since numbers are a human invention, how can we engineer them in order that they persist indefinitely? Mathematicians began to handle this query originally of the twentieth century. They got here up with two assumptions: 0 is the starting number, and adding 1 to any number all the time ends in a brand new number.
These assumptions immediately give us the list of counting numbers: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, etc., a development that continues endlessly.
You could also be wondering why these two rules are assumptions. The reason for the primary is that we don't really know the way to define the number 0. For example: Is “0” the identical as “nothing” and if that’s the case, what exactly is supposed by “nothing”?
The second might sound even stranger. Finally, we will easily show that adding 1 to 2 gives the brand new number 3, just as adding 1 to 2002 gives the brand new number 2003.
But note that we are saying this have to be true for each number. We cannot check this precisely for every individual case as there will probably be an infinite variety of cases. As humans who can only complete a limited variety of steps, we must all the time watch out when making claims about an infinite process. And mathematicians particularly refuse to take anything as a right.
So here's the reply to the query of why numbers don't end: it's due to the best way we define them.
Now to the negative numbers
How do the negative numbers -1, -2, -3 and more healthy into all of this? Historically, people have been very suspicious of such numbers since it is difficult to assume an apple or orange with “minus one.” Mathematics textbooks were still available in 1796 warned against using negatives.
The negatives were created to unravel a calculation problem. The positive numbers are nice while you add them together. But relating to subtraction, they’ll't handle differences like 1 minus 2 or 2 minus 4. If you must subtract numbers at will, you furthermore mght need negative numbers.
An easy method to create negatives is to assume that each one the numbers – 0, 1, 2, 3 and the remaining – are drawn at equal distances on a straight line. Now imagine a mirror placed at 0. Then define -1 because the reflection of +1 on the road, -2 because the reflection of +2, and so forth. This way you get all negative numbers.
As a bonus, you furthermore mght know that the negative numbers must go on eternally since there are only as many negative numbers as there are positive numbers!
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