Update

number.md

@@ -12,6 +12,8 @@ Humans first started to use positive natural numbers (it seems as early as 30000
 
 Basically **anything can be encoded as a number** which makes numbers a universal abstract "medium" -- we can exploit this in both mathematics and programming (which are actually the same thing). Ways of encoding [information](information.md) in numbers may vary, for a mathematician it is natural to see any number as a multiset of its [prime](prime.md) factors (e.g. 12 = 2 * 2 * 3, the three numbers are inherently embedded within number 12) that may carry a message, a programmer will probably rather encode the message in [binary](binary.md) and then interpret the 1s and 0s as a number in direct representation, i.e. he will embed the information in the digits. You can probably come up with many more ways.
 
+But what really is a number? What makes number a number? Where is the border between numbers and other abstract objects? Essentially number is an abstract mathematical object made to model something about [reality](irl.md) (most fundamentally the concept of counting, expressing amount) which only becomes meaninful and useful by its relationship with other similar objects -- other numbers -- that are parts of the same, usually (but not necessarily) infinitely large set. We create systems to give these numbers names because, due to there being infinitely many of them, we can't name every single one individually, and so we have e.g. the [decimal](decimal.md) system in which the name 12345 exactly identifies a specific number, but we must realize these names are ultimately not of mathematical importance -- we may call a number 1, I, 2/2, "one", "uno" or "jedna", it doesn't matter -- what's important are the relationships between numbers that create a STRUCTURE. I.e. a set of infinitely many objects is just that and nothing more; it is the relationships that allow us to operate with numbers and that create the difference between integers, real numbers or the set of colors. These relatinships are expressed by operations (functions, maps, ...) defined with the numbers: for example the comparison operation *is less than* (<) which takes two numbers, *x* and *y*, and always says either *yes* (*x* is smaller than *y*) or *no*, gives numbers order, it creates the number line and allows us to count and measure. Number sets usually have similar operations, typically for example addition and multiplication, and this is how we intuitively judge what numbers are: they are sets of objects that have defined operations similar to those of natural numbers (the original "cavemen numbers"). However some more "advanced" kind of numbers may have lost some of the simple operations -- for example [complex numbers](complex_number.md) are not so straightforward to compare -- and so they may get more and more distant from the original natural numbers. And this is why sometimes the border between what is and what isn't a number may be blurry -- for example it can't objectively be said if infinity is a number or not, simply because number sets that include infinity lose many of the nicely defined operations, the structure of the set changes a lot. So arguing about what is a number ultimately becomes subjective, it's similar to arguing about what is and isn't a planet.
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 **[Order](order.md)** is an important concept related to numbers, we usually want to be able to compare numbers so apart from other operations such as addition and multiplication we also define the comparison operation. However note that not every order is total, i.e. some numbers may be incomparable (consider e.g. complex numbers).
 
 Here are some [fun](fun.md) facts about numbers: