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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 x , {\displaystyle {\sqrt {x}},} where the symbol " {\displaystyle {\sqrt {~^{~}}}} " is called the radical sign[2] or radix. For example, to express the fact that the principal square root of 9 is 3, we write 9 = 3 {\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 x 1 / 2 {\displaystyle x^{1/2}} .

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

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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."[14]

According to Jeffrey A. Oaks, Arabs used the letter jm/m (), the first letter of the word "" (variously transliterated as jar, jir, ar or ir, "root"), placed in its initial form () over a number to indicate its square root. The letter jm 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.[16]

The principal square root function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} (usually just referred to as the "square root function") is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.

The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers).

The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws.

A positive number has two square roots, one positive, and one negative, which are opposite to each other. When talking of the square root of a positive integer, it is usually the positive square root that is meant.

The square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and hence have non-repeating decimals in their decimal representations. Decimal approximations of the square roots of the first few natural numbers are given in the following table.

As with before, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore have non-repeating digits in any standard positional notation system.

Square roots of positive numbers are not in general rational numbers, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained.

The name of the square root function varies from programming language to programming language, with sqrt[23] (often pronounced "squirt" [24]) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python.

where sgn(y) is the sign of y (except that, here, sgn(0) = 1). In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative.

If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define A1/2 = B. In general matrices may have multiple square roots or even an infinitude of them. For example, the 2 2 identity matrix has an infinity of square roots,[28] though only one of them is positive definite.

Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: 1 and 3.

A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in Z / n 2 Z , {\displaystyle \mathbb {Z} /n^{2}\mathbb {Z} ,} any multiple of n is a square root of 0.

The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. But the square shape is not necessary for it: if one of two similar planar Euclidean objects has the area a times greater than another, then the ratio of their linear sizes is a {\displaystyle {\sqrt {a}}} .

A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is a b {\displaystyle {\sqrt {ab}}} , one can construct a {\displaystyle {\sqrt {a}}} simply by taking b = 1.

The perfect square of a number is defined as the squares of whole numbers. The square of the number is given as the number times itself. Some of the examples of the perfect squares are 1, 4, 9, 16, 25, 36, and so on.

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If 'a' is the square root of 'b', it means that a a = b. The square of any number is always a positive number, so every number has two square roots, one of a positive value, and one of a negative value. For example, both 2 and -2 are square roots of 4. However, in most places, only the positive value is written as the square root of a number.

To find the square root of a number, we just see by squaring which number would give the actual number. It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that can be expressed as the product of a number by itself. In other words, perfect squares are numbers which are expressed as the value of power 2 of any integer. We can use four methods to find the square root of numbers and those methods are as follows:

It should be noted that the first three methods can be conveniently used for perfect squares, while the fourth method, i.e., the long division method can be used for any number whether it is a perfect square or not.

This is a very simple method. We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of 16 using this method.

Prime factorization of any number means to represent that number as a product of prime numbers. To find the square root of a given number through the prime factorization method, we follow the steps given below:

Long division is a method for dividing large numbers into steps or parts, breaking the division problem into a sequence of easier steps. We can find the exact square root of any given number using this method. Let us understand the process of finding square root by the long division method with an example. Let us find the square root of 180.

Step 6: The quotient thus obtained will be the square root of the number. Here, the square root of 180 is approximately equal to 13.4 and more digits after the decimal point can be obtained by repeating the same process as follows.

The square root table consists of numbers and their square roots. It is useful to find the squares of numbers as well. Here is the list of square roots of perfect square numbers and some non-perfect square numbers from 1 to 10.

Any number raised to exponent two (y2) is called the square of the base. So, 52 or 25 is referred to as the square of 5, while 82 or 64 is referred to as the square of 8. We can easily find the square of a number by multiplying the number two times. For example, 52 = 5 5 = 25, and 82 = 8 8 = 64. When we find the square of a whole number, the resultant number is a perfect square. Some of the perfect squares we have are 4, 9, 16, 25, 36, 49, 64, and so on. The square of a number is always a positive number. ff782bc1db