You learn very early in school that there are different sets of numbers. You start with the natural numbers, the ones you use to count: 1, 2, 3, and so on. But you just have to come up with the idea, for example, subtract 3 from 2 or divide 1 by 2 to realize that there must be more numbers.
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This is how you discover integers, rational numbers, and later real and complex numbers. But the math doesn’t stop there. For example, this is also a number:
To understand what is meant by this formula, we need to look at the work of British mathematician John Conway. While examining the math for the strategy game Go, Conway discovered – as Conway said – “surreal numbers.” This number is defined by defining two quantities that define the number from the left and the right, so to speak. You can write it like this, for example: x = {L | R}. The set L contains only numbers smaller than x, and in R one finds only numbers greater than x.
Using this knowledge, you can build surreal numbers step by step. The simplest number is represented by {| } where L and R are each empty set. If we call this number 0, one can create new surreal numbers from it. For example x = {0 | }. x cannot be less than 0 and the first number that satisfies this condition is 1. If we now write x = {0 | 1}, we can call the result ½ – and so forth: the narrow theoretical derivation of surreal numbers is a bit more complicated, but the principle should be clear.
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