Square root

In mathematics, a square root of a number x is a number y such that ${\displaystyle y^{2}=x}$; in other words, a number y whose square (the result of multiplying the number by itself, or ${\displaystyle y\cdot y}$) is x. For example, 4 and −4 are square roots of 16 because ${\displaystyle 4^{2}=(-4)^{2}=16}$.

Every nonnegative real number x has a unique nonnegative square root, called the principal square root or simply the square root (with a definite article, see below), which is denoted by ${\displaystyle {\sqrt {x}},}$ where the symbol "${\displaystyle {\sqrt {~^{~}}}}$" is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write ${\displaystyle {\sqrt {9}}=3}$. 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 case, 9. For non-negative x, the principal square root can also be written in exponent notation, as ${\displaystyle x^{1/2}}$.

Every positive number x has two square roots: ${\displaystyle {\sqrt {x}}}$ (which is positive) and ${\displaystyle -{\sqrt {x}}}$ (which is negative). The two roots can be written more concisely using the ± sign as ${\displaystyle \pm {\sqrt {x}}}$. 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.

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 the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.

The Yale Babylonian Collection clay tablet YBC 7289 was created between 1800 BC and 1600 BC, showing ${\displaystyle {\sqrt {2}}}$ and ${\textstyle {\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}$ respectively as 1;24,51,10 and 0;42,25,35 base 60 numbers on a square crossed by two diagonals. (1;24,51,10) base 60 corresponds to 1.41421296, which is correct to 5 decimal places (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.

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). A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Apastamba who was dated around 600 BCE has given a strikingly accurate value for ${\displaystyle {\sqrt {2}}}$ which is correct up to five decimal places as ${\textstyle 1+{\frac {1}{3}}+{\frac {1}{3\times 4}}-{\frac {1}{3\times 4\times 34}}}$. 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 integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they cannot be written exactly as ${\displaystyle {\frac {m}{n}}}$, where m and n are integers). This is the theorem Euclid X, 9, almost certainly due to Theaetetus dating back to c. 380 BC. The discovery of irrational numbers, including the particular case of the square root of 2, is widely associated with the Pythagorean school. Although some accounts attribute the discovery to Hippasus, the specific contributor remains uncertain due to the scarcity of primary sources and the secretive nature of the brotherhood. It is exactly the length of the diagonal of a square with side length 1.