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The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality. [citation needed] In 2013, its numerical value in decimal notation was computed to ten billion digits. [1] Its decimal expansion, written here to 65 decimal places, is given by OEIS: A002194:
The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.
In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number.
In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today. [8] The symbol used to indicate a vinculum need not be a line segment (overline or underline); sometimes braces can be used (pointing either up or down). [9]
√ (square-root symbol) Denotes square root and is read as the square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, √2. √ (radical symbol) 1. Denotes square root and is read as the square root of.
For example √ p+q is the square root of the sum. The bar is also a symbol of grouping in repeated decimal digits. A decimal point followed by one or more digits with a bar over them, for example 0. 123, represents the repeating decimal 0.123123123... . [2] A superscript is understood to be grouped as long as it continues in the form of a ...
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
For example, in duodecimal, 1 / 2 = 0.6, 1 / 3 = 0.4, 1 / 4 = 0.3 and 1 / 6 = 0.2 all terminate; 1 / 5 = 0. 2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1 / 7 = 0. 186A35 has period 6 in duodecimal, just as it does in decimal. If b is an integer base ...