Square Root Calculator .Co

• List of mathematical constants
• Square root of 3,
• Square root of 5,
• Gelfond–Schneider constant,
• Silver ratio, 1 +

According to historian of mathematics D.E. Smith, Aryabhata’s method for finding the square root was first introduced in Europe by Cataneo—in 1546.

Hurwitz’s theorem in Diophantine approximations states that every irrational number can be approximated by infinitely many rational numbers in lowest terms in such a way that

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa98a646684b6d37e6bf569742a7b010292b5931" data-alt=" \left|x — \frac{m}{n}\right|  

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dcff6cfaed4b6670fe70ffb5f185882b1544656" data-alt="{\displaystyle \left|\alpha -{p_{i} \over q_{i}}\right|<{1 \over {\sqrt {5}}q_{i}^{2}},\qquad \left|\alpha -{p_{i+1} \over q_{i+1}}\right|<{1 \over {\sqrt {5}}q_{i+1}^{2}},\qquad \left|\alpha -{p_{i+2} \over q_{i+2}}\right| 

The field   like any other quadratic field, is an abelian extension of the rational numbers. The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination of roots of unity:

The multiplicative inverse (reciprocal) of the square root of two (i.e., the square root of ) is a widely used constant.

  …   (sequence in the OEIS)

One-half of  , also the reciprocal of  , is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates

This number satisfies

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

(sequence in the OEIS).

• Gourdon, X.; Sebah, P. (2001), «Pythagoras’ Constant:  «, Numbers, Constants and Computation.
• The Square Root of Two to 5 million digits by Jerry Bonnell and Robert J. Nemiroff. May, 1994.
• Square root of 2 is irrational, a collection of proofs
• Grime, James; Bowley, Roger. «The Square Root   of Two». Numberphile. Brady Haran.
• Search Engine 2 billion searchable digits of , and

• 1 ()
• = 1.5 ()
• = 1.416… ()
• = 1.414215… ()
• = 1.4142135623746… ()

A simple rational approximation (≈ 1.4142857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than (approx. ).

The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a₀ = 1 () is too large by about ; its square is ≈ .

Records in computation

«Pythagoras’s constant» redirects here. Not to be confused with Pythagoras number.

The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as
or
, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.

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.

Series and product

The identity cos  = sin  = , along with the infinite product representations for the sine and cosine, leads to products such as

The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for gives

The Taylor series of √ with and using the double factorial gives

The convergence of this series can be accelerated with an Euler transform, producing

One interesting property of   is

This is related to the property of silver ratios.

  can also be expressed in terms of copies of the imaginary unit using only the square root and arithmetic operations, if the square root symbol is interpreted suitably for the complex numbers and :

  appears in Viète’s formula for :

This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of  . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.

Ancient Roman architecture

The square root of 5 can be expressed as the continued fraction

  (sequence in the OEIS)

The successive partial evaluations of the continued fraction, which are called its convergents, approach  :

Each of these is a best rational approximation of  ; in other words, it is closer to   than any rational with a smaller denominator.

When   is approximated with the Babylonian method, starting with x₀ = 2 and using x_n+1 = x_n + , the th approximant is equal to the th convergent of the continued fraction:

The Babylonian method is equivalent to Newton’s method for root finding applied to the polynomial  . The Newton’s method update,  , is equal to   when  . The method therefore converges quadratically.

Proof:
Let   shorter length and   longer length of the sides of a sheet of paper, with

  as required by ISO 216.

Let   be the analogous ratio of the halved sheet, then

 .

There are some interesting properties involving the square root of 2 in the physical sciences:

• The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music.
• The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of areas between two successive apertures is 2.
• The celestial latitude (declination) of the Sun during a planet’s astronomical cross-quarter day points equals the tilt of the planet’s axis divided by  .

• Dauben, Joseph W. (2007). «Chinese Mathematics I». In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
• Gel’fand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-8176-3677-3.
• Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
• Smith, David (1958). History of Mathematics. Vol. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
• Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Bibcode:2008ehst.book…..S, ISBN 978-1-4020-4559-2.

• Apostol, Tom M. (2000), «Irrationality of the square root of two – A geometric proof», American Mathematical Monthly, 107 (9): 841–842, doi:10.2307/2695741, JSTOR 2695741.
• Aristotle (2007), Analytica priora, eBooks@Adelaide
• Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.
• Flannery, David (2005), The Square Root of Two, Springer-Verlag, ISBN 0-387-20220-X.
• Fowler, David; Robson, Eleanor (1998), «Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context», Historia Mathematica, 25 (4): 366–378, doi:.
• Good, I. J.; Gover, T. N. (1967), «The generalized serial test and the binary expansion of  «, Journal of the Royal Statistical Society, Series A, 130 (1): 102–107, doi:10.2307/2344040, JSTOR 2344040.
• Henderson, David W. (2000), «Square roots in the Śulba Sūtras», in Gorini, Catherine A. (ed.), Geometry At Work: Papers in Applied Geometry, Cambridge University Press, pp. 39–45, ISBN 978-0-88385-164-7.

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.

where and ₁₀ are the natural and base-10 logarithms.

as it allows one to adjust the estimate by some amount and measure the square of the adjustment in terms of the original estimate and its square. Furthermore,   when is close to 0, because the tangent line to the graph of   at  , as a function of alone, is  . Thus, small adjustments to can be planned out by setting   to , or  .

1. Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.
2. Replace x by the average (x + a/x) / 2 between x and a/x.
3. Repeat from step 2, using this average as the new value of x.

That is, if an arbitrary guess for   is x₀, and x_{n + 1} = (x_n + a/x_n) / 2, then each x_n is an approximation of   which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If , the convergence is only linear.

Using the identity

The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

Another useful method for calculating the square root is the shifting nth root algorithm, applied for .

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 an integer are algebraic integers—more specifically quadratic integers.

The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since   only roots of those primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is

As decimal expansions

As expansions in other numeral systems

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.

The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers.

As periodic continued fractions

One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange c.. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.

Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u² − v² = (u − v)(u + v) is proved using the commutativity of multiplication. If and are square roots of the same element, then u² − v² = 0. Because there are no zero divisors this implies or u + v = 0, where the latter means that two roots are additive inverses of each other. In other words if an element a square root of an element exists, then the only square roots of are and . The only square root of 0 in an integral domain is 0 itself.

In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that −u = u. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.

Given an odd prime number , let q = p^e for some positive integer . A non-zero element of the field with elements is a quadratic residue if it has a square root in . Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.

The principal square root function   (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).

For all real numbers x,

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0640b9c1359d9e56c1047817a6abaeb9d12f0eb9" data-alt="
\sqrt{x^2} = \left|x\right| =
\begin{cases}
x, & \mbox{if }x \ge 0 \\
-x, & \mbox{if }x       (see absolute value)

For all nonnegative real numbers x and y,

The square root function is continuous for all nonnegative x, and differentiable for all positive x. If f denotes the square root function, whose derivative is given by:

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.

For example, this case of the Rogers–Ramanujan continued fraction:

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   of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3.

Another example is provided by the ring of quaternions   which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including , , and . In fact, the set of square roots of −1 is exactly

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   any multiple of is a square root of 0.

For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent.

Proof by infinite descent

1. Assume that   is a rational number, meaning that there exists a pair of integers whose ratio is exactly  .
2. If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
3. Then   can be written as an irreducible fraction   such that and are coprime integers (having no common factor) which additionally means that at least one of or must be odd.
4. It follows that   and  .   ( ()ⁿ =  )   ( a² and b² are integers)
5. Therefore, is even because it is equal to . ( is necessarily even because it is 2 times another whole number.)
6. It follows that must be even (as squares of odd integers are never even).
7. Because is even, there exists an integer that fulfills  .
8. Substituting from step 7 for in the second equation of step 4:  , which is equivalent to  .
9. Because is divisible by two and therefore even, and because  , it follows that is also even which means that is even.
10. By steps 5 and 8, and are both even, which contradicts step 3 (that   is irreducible).

Q.E.D.

Since we have derived a falsehood, the assumption (1) that   is a rational number must be false. This means that   is not a rational number; that is to say,   is irrational.

Proof by unique factorization

As with the proof by infinite descent, we obtain  . Being the same quantity, each side has the same prime factorization by the fundamental theorem of arithmetic, and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.

Let be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the Pythagorean theorem,  . Suppose and are integers. Let be a ratio given in its lowest terms.

Hence, there is an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs . These values are integers even smaller than and and in the same ratio, contradicting the hypothesis that is in lowest terms. Therefore, and cannot be both integers; hence,   is irrational.

Proof by Pythagorean triples

If , , and are coprime positive integers such that a² + b² = c², then is never even.^[20]

This lemma can be used to show that two identical perfect squares can never be added to produce another perfect square.

Suppose the contrary that   is rational. Therefore,

 

where   and  

Squaring both sides,

 

 

 

Here, (b, b, a) is a primitive Pythagorean triple, and from the lemma is never even. However, this contradicts the equation 2b² = a² which implies that must be even.

(See the section below for their geometrical interpretation as decompositions of a   rectangle.)

  then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio:

The series of convergents to these values feature the series of Fibonacci numbers and the series of Lucas numbers as numerators and denominators, and vice versa, respectively:

In fact, the limit of the quotient of the   Lucas number   and the   Fibonacci number   is directly equal to the square root of  :

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 root of a negative number. This is done by introducing a new number, denoted by i (sometimes by j, especially in the context of electricity where «i» traditionally represents electric current) and called the imaginary unit, which is defined such that i² = −1. Using this notation, we can think of i as the square root of −1, but we also have (−i)² = i² = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any nonnegative number, then the principal square root of −x is

The right side (as well as its negative) is indeed a square root of −x, since

For every non-zero complex number z there exist precisely two numbers w such that w² = z: the principal square root of z (defined below), and its negative.

Principal square root of a complex number

To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number   can be viewed as a point in the plane,   expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair   where   is the distance of the point from the origin, and   is the angle that the line from the origin to the point makes with the positive real ( ) axis. In complex analysis, the location of this point is conventionally written   If

then the principal square root of   is defined to be the following:

The principal square root function is thus defined using the nonpositive real axis as a branch cut.
If   is a non-negative real number (which happens if and only if  ) then the principal square root of   is   in other words, the principal square root of a non-negative real number is just the usual non-negative square root.
It is important that <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27eb4930922e649015e37b8d782b6b1ad7d55b9" data-alt="{\displaystyle -\pi   because if, for example,   (so  ) then the principal square root is but using   would instead produce the other square root  

The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above Taylor series for   remains valid for complex numbers   with <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e657241d23e0514c31745c2d302fffa61a77ed" data-alt="{\displaystyle |x| 

The above can also be expressed in terms of trigonometric functions:

where is the sign of (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.

For example, the principal square roots of are given by:

where <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aeabfa69185eedc1363619bb534927d46eb8aff" data-alt="{\displaystyle -\pi   and <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679df4a1684cbd40b9ff7ac9bbed56bffe6a2903" data-alt="{\displaystyle -\pi  .

•  
  Counterexample for the principal square root: z = −1 and w = −1
  This equality is valid only when <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616dd651f429490e09f3398a52a279be4c00a279" data-alt="{\displaystyle -\pi  
•  
  Counterexample for the principal square root: and z = −1
  This equality is valid only when <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd8739a5027cfcf839f6d2f8105a1e94a115622" data-alt="{\displaystyle -\pi  
•  
  Counterexample for the principal square root: z = −1)
  This equality is valid only when  

A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations logz + logw = log(zw) or log(z^*) = log(z)^* which are not true in general.

if the branch includes +i or

if the branch includes −i, while the right-hand side becomes

where the last equality,   is a consequence of the choice of branch in the redefinition of √.

A cube root of   is a number   such that  ; it is denoted  

If is an integer greater than two, a th root of   is a number   such that  ; it is denoted  

Given any polynomial , a root of is a number such that p(y) = 0. For example, the th roots of are the roots of the polynomial (in )  

Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of th roots.

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 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  , one can construct   simply by taking .

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Euclid’s second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales’ theorem and, as in the proof of Pythagoras’ theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don’t need that, but it is the essence of the proof of Pythagoras’ theorem) so that AH:CH is as HC:HB, i.e. a/h = h/b, from which we conclude by cross-multiplication that h² = ab, and finally that  . When marking the midpoint O of the line segment AB and drawing the radius OC of length (a + b)/2, then clearly OC > CH, i.e.   (with equality if and only if ), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of «Heron’s method».

Another method of geometric construction uses right triangles and induction:   can be constructed, and once   has been constructed, the right triangle with legs 1 and   has a hypotenuse of  . Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above.