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Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number. No square root can be taken of a negative number within the system of real numbers, because squares of all real numbers ...
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.
For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers. Operations can involve dissimilar objects: a vector can be multiplied by a scalar to form another vector (an operation known as scalar multiplication ), [ 13 ] and the ...
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix.
A third method drastically reduces the number of operations to perform modular exponentiation, while keeping the same memory footprint as in the previous method. It is a combination of the previous method and a more general principle called exponentiation by squaring (also known as binary exponentiation).
One value can be chosen by convention as the principal value; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2).
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.
In mathematics, a negative number is the opposite (mathematics) of a positive real number. [1] Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset.