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The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square
The square root symbol refers to the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers , and the square root symbol refers to the principal square root, the one with a positive imaginary part.
A square root of a number x is a number r which, when squared, becomes x: =. Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:
Square roots of negative numbers are called imaginary because in early-modern mathematics, only what are now called real numbers, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical.
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis. Although the Greek mathematician and engineer Heron of Alexandria is noted as the first to present a calculation involving the square root of a negative number, [6] [7] it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572.
Methods of computing square roots are algorithms for approximating the non-negative square root of a positive real number. Since all square roots of natural numbers , other than of perfect squares , are irrational , [ 1 ] square roots can usually only be computed to some finite precision: these methods typically construct a series of ...
Although there are no real square roots of −1, the complex number i satisfies i 2 = −1, and as such can be considered as a square root of −1. [2] The only other complex number whose square is −1 is − i because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra.
For every x,n (other than zero), the minus-n-th root of x, is one divided by the n-th root of x. HOTmag 17:22, 13 September 2016 (UTC) To explain why, the root of a number is the number raised to the inverse of the power. That is, the square root of x is x 1/2, or more generally, the "y" root of "x" is x 1/y. The zeroth root would thus be ...