Square root

The mathematical expression "The (principal) square root of x"
For example, 25 = 5, since 25 = 5 ⋅ 5, or 52 (5 squared).

In mathematics, a square root of a number a is a number y such that y2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or yy) is a.[1] For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16.Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by 9 = 3, because 32 = 3 · 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.

Every positive number a has two square roots: a, which is positive, and −a, which is negative. Together, these two roots are denoted as ±a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.[2]

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

History

The Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 BC, showing 2 and 2/2 = 1/2 as 1;24,51,10 and 0;42,25,35 base 60 numbers on a square crossed by two diagonals.[3] (1;24,51,10) base 60 corresponds to 1.41421296 which is a correct value to 5 decimal points (1.41421356...).

The Rhind Mathematical Papyrus is a copy from 1650 BC of an earlier Berlin Papyrus and other texts – possibly the Kahun Papyrus – that shows how the Egyptians extracted square roots by an inverse proportion method.[4]

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).[citation needed] A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[5] Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.

It was known to the ancient Greeks that square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC.[6] The particular case 2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus.[citation needed] It is exactly the length of the diagonal of a square with side length 1.

In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[7]

A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436–1476). An R was also used for Radix to indicate square roots in Gerolamo Cardano's Ars Magna.[8]

According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo in 1546.

According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm (ج), the first letter of the word “جذر” (variously transliterated as jaḏr, jiḏr, ǧaḏr or ǧiḏr, “root”), placed in its initial form () over a number to indicate its square root. The letter jīm resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician Ibn al-Yasamin.[9]

The symbol '√' for the square root was first used in print in 1525 in Christoph Rudolff's Coss.[10]

Other Languages
Afrikaans: Vierkantswortel
العربية: جذر تربيعي
asturianu: Raíz cuadrada
azərbaycanca: Kvadrat kökləri
বাংলা: বর্গমূল
башҡортса: Квадрат тамыр
беларуская: Квадратны корань
беларуская (тарашкевіца)‎: Квадратны корань
български: Квадратен корен
brezhoneg: Gwrizienn garrez
čeština: Druhá odmocnina
Cymraeg: Ail isradd
dansk: Kvadratrod
Deutsch: Quadratwurzel
eesti: Ruutjuur
español: Raíz cuadrada
Esperanto: Kvadrata radiko
euskara: Erro karratu
فارسی: ریشه دوم
føroyskt: Kvadratrót
français: Racine carrée
贛語: 平方根
ગુજરાતી: વર્ગમૂળ
한국어: 제곱근
हिन्दी: वर्गमूल
Bahasa Indonesia: Akar kuadrat
íslenska: Ferningsrót
italiano: Radice quadrata
latviešu: Kvadrātsakne
Lingua Franca Nova: Radis cuadral
lumbaart: Radis quadrada
മലയാളം: വർഗ്ഗമൂലം
मराठी: वर्गमूळ
Bahasa Melayu: Punca kuasa dua
Nederlands: Vierkantswortel
नेपाली: वर्गमूल
नेपाल भाषा: वर्गमूल
日本語: 平方根
norsk: Kvadratrot
norsk nynorsk: Kvadratrot
occitan: Raiç carrada
ਪੰਜਾਬੀ: ਵਰਗ ਮੂਲ
Patois: Skwier ruut
português: Raiz quadrada
sicilianu: Radici quatrata
Simple English: Square root
slovenščina: Kvadratni koren
српски / srpski: Квадратни корен
srpskohrvatski / српскохрватски: Kvadratni koren
Basa Sunda: Akar kuadrat
svenska: Kvadratrot
తెలుగు: వర్గమూలం
Türkçe: Karekök
українська: Квадратний корінь
Tiếng Việt: Căn bậc hai
吴语: 平方根
粵語: 開方根
中文: 平方根