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number.md

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 Numbers (from Latin *numerus* coming from a Greek word meaning "to distribute") are one of the most elementary [mathematical](math.md) objects, building [stones](rock.md) serving most often as quantitative values (that is: telling count, size, length, order etc.) and labels, in higher math also used in much more [abstract](abstraction.md) ways which have only distant relationship to traditional counting. Examples of numbers are minus [one](one.md) half, [zero](zero.md), [pi](pi.md) or [i](i.md). Numbers constitute the basis and core of mathematics and as such they sit almost at the [lowest level](low_level.md) of it, i.e. most other things such as algebra, [functions](function.md) and [equations](equation.md) are built on top of numbers or require numbers to even be examined. In modern mathematics numbers themselves do not reside on the absolute bottom of the foundations though, they are themselves built on top of [sets](set.md), as set theory is most commonly used as a basis of whole mathematics, however for many purposes this is just a formalism that's of practical interest only to some mathematicians (as the topic gets closer to the fringes of mathematics and at times rather pertains to [philosophy](philosophy.md)) -- on the other hand numbers just cannot be avoided anywhere, by a mathematician or just a common folk. The word *number* may be the first that comes to our mind when we say *mathematics*. The area of [number theory](number_theory.md) is particularly focused on examining numbers (though it's examining almost exclusively integer numbers because these seem to have the deepest pattern related e.g. to divisibility). Interest in numbers isn't exclusive to mathematics -- numbers also play an important role in [culture](culture.md) and religion for example; some even believe in "[magical](magic.md)" power of numbers (see [numerology](numerology.md)).
 
-Numbers must not be [confused](often_confused.md) with digits or figures (numerals) -- a number is a purely abstract entity while digits serve as symbols for numbers so that they can be written down. The same number may be represented with many different symbols, using one of many [numeral systems](numeral_system.md) (Roman numerals, tally marks, Arabic numerals of different [bases](base.md) etc.), for example 4 stands for a number than can also be written as IV, four, IIII, 8/2, 16:4, 2^2, 4.00 or 0b100. There are also numbers which cannot be exactly expressed with our traditional numeral systems, for some of them we have special symbols -- most prominent example is of course [pi](pi.md) whose digits in decimal expansion form an [infinite](infinity.md) series -- and there are even numbers lacking any symbolic representation, ones not well researched yet or not important enough, only described by equations to which they are the solution. Sure enough, a number by itself isn't too interesting and probably doesn't even make sense, it's only in context, when it's placed in relationship with other numbers (by ordering them, defining operations and properties based on those operations) that patterns and useful attributes emerge.
+Numbers must not be [mistaken](often_confused.md) for digits or figures (numerals) -- a number is a purely abstract entity while digits serve as symbols for numbers so that they can be written down. The same number may be represented with many different symbols, using one of many [numeral systems](numeral_system.md) (Roman numerals, tally marks, Arabic numerals of different [bases](base.md) etc.), for example 4 stands for a number than can also be written as IV, four, IIII, 8/2, 16:4, 2^2, 4.00 or 0b100. There are also numbers which cannot be exactly expressed with our traditional numeral systems, for some of them we have special symbols -- most prominent example is of course [pi](pi.md) whose digits in decimal expansion form an [infinite](infinity.md) series -- and there are even numbers lacking any symbolic representation, ones not well researched yet or not important enough, only described by equations to which they are the solution. Sure enough, a number by itself isn't too interesting and probably doesn't even make sense, it's only in context, when it's placed in relationship with other numbers (by ordering them, defining operations and properties based on those operations) that patterns and useful attributes emerge.
 
 To embark on the [history](history.md) a bit, humans first started to use positive natural numbers (it seems as early as 30000 BC), i.e. 1, 2, 3 ..., so as to be able to trade, count enemies, days and so on -- since then they kept expanding the concept of a number with more [abstraction](abstraction.md) as they encountered more complex problems. First extension was to fractions, initially reciprocals of integers (like one half, one third, ...) and then general ones. Around 6th century BC Pythagoras showed that there even exist numbers that cannot be expressed as fractions ([irrational numbers](irrational_number.md), which in the beginning was a controversial discovery), expanding the set of known numbers further. A bit later (around 100 BC) negative numbers started to be used. Adoption of the number [zero](zero.md) also took some time (1st use of true zero seem to be in 4th century BC), with it first just having a limited use as a mere placeholder digit. Since 16th century a highly abstract concept of [complex numbers](complex_number.md) started to appear, which was later (19th century) expanded further to [quaternions](quaternion.md). With more advancement in mathematics -- e.g. with the development of set theory -- more and more concepts of new kinds of numbers appeared and still appear to this day. Nowadays we have greatly abstract numbers, ones existing in many dimensions, capable of counting and measuring infinitely large and infinitely small entities, and it seems we still haven't nearly discovered everything there is to know about numbers.