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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.

I understand that this is a pretty math-y question, but how do programs get square roots? From what I've read, this is something that is usually native to the cpu of a device, but I need to be able to do it, probably in c++ (although that's irrelevant).

The reason I need to know about this specifically is that I have an intranet server and I am getting started with crowdsourcing. For this, I am going to start with finding a lot of digits of a certain square root, like sqrt(17) or something.

Use this calculator to find the principal square root and roots of real numbers. Inputs for the radicand x can be positive or negative real numbers. The answer will also tell you if you entered a perfect square.

The answer will show you the complex or imaginary solutions for square roots of negative real numbers. See also the Simplify Radical Expressions Calculator to simplify radicals instead of finding fractional (decimal) answers.

There are 2 possible roots for any positive real number. A positive root and a negative root. Given a number x, the square root of x is a number a such that a2 = x. Square roots is a specialized form of our common roots calculator.

"Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3, since (-3)2 = (+3)2 = 9. Any nonnegative real number x has a unique nonnegative square root r; this is called the principal square root .......... For example, the principal square root of 9 is sqrt(9) = +3, while the other square root of 9 is -sqrt(9) = -3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root."[1].

This calculator will also tell you if the number you entered is a perfect square or is not a perfect square. A perfect square is a number x where the square root of x is a number a such that a2 = x and a is an integer. For example, 4, 9 and 16 are perfect squares since their square roots, 2, 3 and 4, respectively, are integers.

When programmers talk about the cost of an operation they typically mean how many instructions are required to perform the operation. For example a multiplication would typically take three instructions, two reads and one write. For more complex operations (such as division) it often takes many more steps to calculate an accurate representation of the result, thus, the expense in the operation. While square root may have once been an extremely costly exercise I have a hunch that it is now a much less relevant piece of advice than it used to be. Additionally it leads newer programmers to focus on changing the way they write code in order to optimize said code as they go. I am a big believer in writing clean, verbose code and optimizing only when it is absolutely necessary. That can be hard as a new programmer where you often want to write cleaner and more efficient code than your last attempt at solving the same problem. It gives you a sign that you are progressing and allows you to tackle bigger problems.

If the prevailing opinion is that square root is slower than simply multiplying our target value by itself then it is obvious to pit those two calculations against each other. I chose to add the power function to my testing because it seems like a simple interchange to make. Instead of using square root I could instead square my target value by raising it to the power of two. ff782bc1db