Rationalize the Denominator

If N is an approximation to  , a better approximation can be found by using the Taylor series of the square root function:

The radical or root may be represented by the infinite series:

with  . This expression can be derived from the binomial series.

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.

An unknown Babylonian mathematician somehow correctly calculated the square root of 2 to three sexagesimal «digits» after the 1, but it is not known exactly how. The Babylonians knew how to approximate a hypotenuse using

The denominator in the fraction corresponds to the nth root. In the case above the denominator is 2, hence the equation specifies that the square root is to be found. The same identity is used when computing square roots with logarithm tables or slide rules.

the Lucas sequence of the first kind U_n(P,Q) is defined by the recurrence relations:

and the characteristic equation of it is:

it has the discriminant   and the roots:

so when we want  , we can choose   and  , and then calculate   using   and  for large value of  .
The most effective way to calculate   and  is:

then when  :

The initialization step of this method is

while the iterative steps read

Then,   (while  ).

The convergence of  , and therefore also of  , is quadratic.

The proof of the method is rather easy. First, rewrite the iterative definition of   as

 .

Then it is straightforward to prove by induction that

This can be used to construct a rational approximation to the square root by beginning with an integer. If   is an integer chosen so   is close to  , and   is the difference whose absolute value is minimized, then the first iteration can be written as:

Using the same example as given with the Babylonian method, let   Then, the first iteration gives

Likewise the second iteration gives

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.

It was once conjectured that all polynomial equations could be solved algebraically (that is, that all roots of a polynomial could be expressed in terms of a finite number of radicals and elementary operations). However, while this is true for third degree polynomials (cubics) and fourth degree polynomials (quartics), the Abel–Ruffini theorem (1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation

cannot be expressed in terms of radicals. (cf. quintic equation)

• 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.

• Applying Newton’s method to the equation   produces a method that converges quadratically using three multiplications per step:

   

• Another iteration is obtained by Halley’s method, which is the Householder’s method of order two. This converges cubically, but involves five multiplications per iteration:^[]

   , and

   .

• If doing fixed-point arithmetic, the multiplication by 3 and division by 8 can implemented using shifts and adds. If using floating-point, Halley’s method can be reduced to four multiplications per iteration by precomputing   and adjusting all the other constants to compensate:

   , and

   .

The first way of writing Goldschmidt’s algorithm begins

 

  (typically using a table lookup)

 

 

until   is sufficiently close to 1, or a fixed number of iterations. The iterations converge to

 , and

 .

Note that it is possible to omit either   and   from the computation, and if both are desired then   may be used at the end rather than computing it through in each iteration.

A second form, using fused multiply-add operations, begins

  (typically using a table lookup)

 

 

until   is sufficiently close to 0, or a fixed number of iterations. This converges to

 , and

 .

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 .

«Heron’s method» redirects here. For the formula used to find the area of a triangle, see Heron’s formula.

More precisely, if is our initial guess of   and is the error in our estimate such that S = (x+ ε)², then we can expand the binomial

and solve for the error term

  since  .

Therefore, we can compensate for the error and update our old estimate as

1. Begin with an arbitrary positive starting value (the closer to the actual square root of , the better).
2. Let x_{n + 1} be the average of and (using the arithmetic mean to approximate the geometric mean).
3. Repeat step 2 until the desired accuracy is achieved.

It can also be represented as:

This algorithm works equally well in the -adic numbers, but cannot be used to identify real square roots with -adic square roots; one can, for example, construct a sequence of rational numbers by this method that converges to +3 in the reals, but to −3 in the 2-adics.

To calculate , where = 125348, to six significant figures, use the rough estimation method above to get

Therefore, ≈ 354.045.

Suppose that x₀ > 0 and S > 0. Then for any natural number n, x_n > 0. Let the relative error in x_n be defined by

Then it can be shown that

And thus that

and consequently that convergence is assured, and quadratic.

Worst case for convergence

Thus in any case,

Rounding errors will slow the convergence. It is recommended to keep at least one extra digit beyond the desired accuracy of the being calculated to minimize round off error.

Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted
,
, or
) of a real number. Arithmetically, it means given
, a procedure for finding a number which when multiplied by itself, yields
; algebraically, it means a procedure for finding the non-negative root of the equation
; geometrically, it means given two line segments, a procedure for constructing their geometric mean.

The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.

The method employed depends on what the result is to be used for (i.e. how accurate it has to be), how much effort one is willing to put into the procedure, and what tools are at hand. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result).

Procedures for finding square roots (particularly the square root of 2) have been known since at least the period of ancient Babylon in the 17th century BCE. Heron’s method from first century Egypt was the first ascertainable algorithm for computing square root. Modern analytic methods began to be developed after introduction of the Arabic numeral system to western Europe in the early Renaissance. Today, nearly all computing devices have a fast and accurate square root function, either as a programming language construct, a compiler intrinsic or library function, or as a hardware operator, based on one of the described procedures.

Every complex number other than 0 has n different nth roots.

The two square roots of a complex number are always negatives of each other. For example, the square roots of are and , and the square roots of are

If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle:

A principal root of a complex number may be chosen in various ways, for example

which introduces a branch cut in the complex plane along the positive real axis with the condition , or along the negative real axis with .

Using the first(last) branch cut the principal square root   maps   to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like Matlab or Scilab.

Roots of unity

The number 1 has n different nth roots in the complex plane, namely

These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of  . For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1,  , −1, and  .

Every complex number has n different nth roots in the complex plane. These are

In polar form, a single nth root may be found by the formula

Here r is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; if the number can be written as a+bi then  . Also,   is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that     and  

Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth roots is the nth root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the nth roots is  , where   is the angle defined in the same way for the number whose root is being taken. Furthermore, all n of the nth roots are at equally spaced angles from each other.

If n is even, a complex number’s nth roots, of which there are an even number, come in additive inverse pairs, so that if a number r₁ is one of the nth roots then r₂ = –r₁ is another. This is because raising the latter’s coefficient –1 to the nth power for even n yields 1: that is, (–r₁)ⁿ = (–1)ⁿ × r₁ⁿ = r₁ⁿ.

As with square roots, the formula above does not define a continuous function over the entire complex plane, but instead has a branch cut at points where θ / n is discontinuous.

In mathematics, an nth root of a number x is a number r which, when raised to the power n, yields x:

where n is a positive integer, sometimes called the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an th root is a root extraction.

For example, 3 is a square root of 9, since 3² = 9, and −3 is also a square root of 9, since (−3)² = 9.

Any non-zero number considered as a complex number has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all positive integers , since 0ⁿ = 0. In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real complex numbers; if is even and is a negative real number, none of the th roots is real. If is odd and is real, one th root is real and has the same sign as , while the other () roots are not real. Finally, if is not real, then none of its th roots are real.

Roots of real numbers are usually written using the radical symbol or radix
, with
denoting the positive square root of if is positive; for higher roots,
denotes the real th root if is odd, and the positive nth root if is even and is positive. In the other cases, the symbol is not commonly used as being ambiguous. In the expression
, the integer n is called the index and is called the radicand.

When complex th roots are considered, it is often useful to choose one of the roots, called principal root, as a principal value. The common choice is to choose the principal th root of as the th root with the greatest real part, and when there are two (for real and negative), the one with a positive imaginary part. This makes the th root a function that is real and positive for real and positive, and is continuous in the whole complex plane, except for values of that are real and negative.

A difficulty with this choice is that, for a negative real number and an odd index, the principal th root is not the real one. For example,
has three cube roots,
,
and
The real cube root is
and the principal cube root is

Roots can also be defined as special cases of exponentiation, where the exponent is a fraction:

Roots are used for determining the radius of convergence of a power series with the root test. The th roots of 1 are called roots of unity and play a fundamental role in various areas of mathematics, such as number theory, theory of equations, and Fourier transform.

This is a method to find each digit of the square root in a sequence. This method is based on the binomial theorem and basically an inverse algorithm solving  . It is slower than the Babylonian method, but it has several advantages:

• It can be easier for manual calculations.
• Every digit of the root found is known to be correct, i.e., it does not have to be changed later.
• If the square root has an expansion that terminates, the algorithm terminates after the last digit is found. Thus, it can be used to check whether a given integer is a square number.
• The algorithm works for any base, and naturally, the way it proceeds depends on the base chosen.
• Inconveniences are that the algorithm becomes quite unhandleable for higher roots and that it is not allowing inaccurate guesses or inaccurate sub-calculations as they, unlike the self correcting approximations like with Newton’s method, lead to every following digit of the result being wrong. Furthermore this algorithm, even though being efficient enough on paper, is way too expensive for software implementations as the many calculations become larger and larger and load the memory while still only allowing digit by digit progressions leading the algorithm to become slower and slower with every following digit.

Napier’s bones include an aid for the execution of this algorithm. The shifting th root algorithm is a generalization of this method.

First, consider the case of finding the square root of a number , that is the square of a two-digit number , where is the tens digit and is the units digit. Specifically:

Now using the digit-by-digit algorithm, we first determine the value of . is the largest digit such that is less than or equal to from which we removed the two rightmost digits.

In the next iteration, we pair the digits, multiply by 2, and place it in the tenth’s place while we try to figure out what the value of is.

Since this is a simple case where the answer is a perfect square root , the algorithm stops here.

The same idea can be extended to any arbitrary square root computation next. Suppose we are able to find the square root of by expressing it as a sum of n positive numbers such that

 .

By repeatedly applying the basic identity

the right-hand-side term can be expanded as

This expression allows us to find the square root by sequentially guessing the values of  s. Suppose that the numbers   have already been guessed, then the m-th term of the right-hand-side of above summation is given by   where   is the approximate square root found so far. Now each new guess   should satisfy the recursion

such that   for all   with initialization   When   the exact square root has been found; if not, then the sum of  s gives a suitable approximation of the square root, with   being the approximation error.

For example, in the decimal number system we have

where   are place holders and the coefficients  . At any m-th stage of the square root calculation, the approximate root found so far,   and the summation term   are given by

Here since the place value of   is an even power of 10, we only need to work with the pair of most significant digits of the remaining term   at any m-th stage. The section below codifies this procedure.

Decimal (base 10)

Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into pairs, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the square. One digit of the root will appear above each pair of digits of the square.

1. Starting on the left, bring down the most significant (leftmost) pair of digits not yet used (if all the digits have been used, write «00») and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by 100 and add the two digits. This will be the current value c.
2. Find p, y and x, as follows:
   • Let p be the part of the root found so far, ignoring any decimal point. (For the first step, p = 0.)
   • Determine the greatest digit x such that  . We will use a new variable y = x(20p + x).
     • Note: 20p + x is simply twice p, with the digit x appended to the right.
     • Note: x can be found by guessing what c/(20·p) is and doing a trial calculation of y, then adjusting x upward or downward as necessary.
   • Place the digit   as the next digit of the root, i.e., above the two digits of the square you just brought down. Thus the next p will be the old p times 10 plus x.
3. Subtract y from c to form a new remainder.
4. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.

Find the square root of 152.2756.

  1 2. 3 4  / \/ 01 52.27 56 01 1*1 <= 1 < 2*2 x=1  01  y = x*x = 1*1 = 1 00 52 22*2 <= 52 < 23*3 x=2  00 44  y = (20+x)*x = 22*2 = 44 08 27 243*3 <= 827 < 244*4 x=3  07 29  y = (240+x)*x = 243*3 = 729 98 56 2464*4 <= 9856 < 2465*5 x=4  98 56  y = (2460+x)*x = 2464*4 = 9856 00 00 Algorithm terminates: Answer=12.34

Binary numeral system (base 2)

This section uses the formalism from the digit-by-digit calculation section above, with the slight variation that we let  , with each   or  .
We iterate all  , from   down to  , and build up an approximate solution  , the sum of all   for which we have determined the value.
To determine if   equals   or  , we let  . If   (i.e. the square of our approximate solution including   does not exceed the target square) then  , otherwise   and  .
To avoid squaring   in each step, we store the difference   and incrementally update it by setting   with  .
Initially, we set   for the largest   with  .

As an extra optimization, we store   and  , the two terms of   in case that   is nonzero, in separate variables  ,  :

  and   can be efficiently updated in each step:

 , which is the final result returned in the function below.

    "sqrt input should be non-negative"              // dₙ which starts at the highest power of four <= n       // The second-to-top bit is set. // Same as ((unsigned) INT32_MAX + 1) / 2.                     // if Xₘ₊₁ ≥ Yₘ then aₘ = 2ᵐ      // Xₘ = Xₘ₊₁ - Yₘ        // cₘ₋₁ = cₘ/2 + dₘ (aₘ is 2ᵐ)       // cₘ₋₁ = cₘ/2 (aₘ is 0)     // dₘ₋₁ = dₘ/4    

If S < 0, then its principal square root is

If S = a+bi where a and b are real and b ≠ 0, then its principal square root is

is the modulus of S. The principal square root of a complex number is defined to be the root with the non-negative real part.

1. There is no factor of the radicand that can be written as a power greater than or equal to the index.
2. There are no fractions under the radical sign.
3. There are no radicals in the denominator.

Simplifying radical expressions involving nested radicals can be quite difficult. It is not obvious for instance that:

The above can be derived through:

Let  , with and coprime and positive integers. Then   is rational if and only if both   and   are integers, which means that both and are nth powers of some integer.

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.

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.

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   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  

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   and  .

•  
  Counterexample for the principal square root: z = −1 and w = −1
  This equality is valid only when  
•  
  Counterexample for the principal square root: and z = −1
  This equality is valid only when  
•  
  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 √.

Expressing the degree of an nth root in its exponent form, as in  , makes it easier to manipulate powers and roots. If   is a non-negative real number,

Every non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands   and   are straightforward within the real numbers:

Subtleties can occur when taking the nth roots of negative or complex numbers. For instance:

  but, rather,  

Since the rule   strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.

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,

      (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.

Quadratic irrationals (numbers of the form  , where a, b and c are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion, and hence its rational approximation. Let S be the positive number for which we are required to find the square root. Then assuming a to be a number that serves as an initial guess and r to be the remainder term, we can write   Since we have  , we can express the square root of S as

By applying this expression for   to the denominator term of the fraction, we have

Proceeding this way, we get a generalized continued fraction for the square root as
 

Step 2 is to reduce the continued fraction from the bottom up, one denominator at a time, to yield a rational fraction whose numerator and denominator are integers. The reduction proceeds thus (taking the first three denominators):

Finally (step 3), divide the numerator by the denominator of the rational fraction to obtain the approximate value of the root:

  rounded to three digits of precision.

The actual value of is 1.41 to three significant digits. The relative error is 0.17%, so the rational fraction is good to almost three digits of precision. Taking more denominators gives successively better approximations: four denominators yields the fraction  , good to almost 4 digits of precision, etc.

In general, the larger the denominator of a rational fraction, the better the approximation. It can also be shown that truncating a continued fraction yields a rational fraction that is the best approximation to the root of any fraction with denominator less than or equal to the denominator of that fraction — e.g., no fraction with a denominator less than or equal to 70 is as good an approximation to as 99/70.

Assume that   is rational. That is, it can be reduced to a fraction  , where and are integers without a common factor.

This means that  .

Since   and  ,  .

This means that   and thus,  . This implies that   is an integer. Since x is not a perfect nth power, this is impossible. Thus   is irrational.

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.

Using Newton’s method

The th root of a number can be computed with Newton’s method, which starts with an initial guess and then iterates using the recurrence relation

until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten

This allows to have only one exponentiation, and to compute once for all the first factor of each term.

The approximation is accurate to 25 decimal places and is good for 51.

Newton’s method can be modified to produce various generalized continued fractions for the nth root. For example,

Digit-by-digit calculation of principal roots of decimal (base 10) numbers

Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number.

1. Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digits have been used, write «0» the number of times required to make a group) and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by   and add the digits from the next group. This will be the current value c.
2. Find p and x, as follows:
   • Let   be the part of the root found so far, ignoring any decimal point. (For the first step,  ).
   • Determine the greatest digit   such that  .
   • Place the digit   as the next digit of the root, i.e., above the group of digits you just brought down. Thus the next p will be the old p times 10 plus x.
3. Subtract   from   to form a new remainder.
4. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.

Find the square root of 152.2756.

  1 2. 3 4    / \/ 01 52.27 56

 01 100·1·00·12 + 101·2·01·11 ≤ 1 < 100·1·00·22 + 101·2·01·21 x = 1  01  y = 100·1·00·12 + 101·2·01·11 = 1 + 0 = 1 00 52 100·1·10·22 + 101·2·11·21 ≤ 52 < 100·1·10·32 + 101·2·11·31 x = 2  00 44  y = 100·1·10·22 + 101·2·11·21 = 4 + 40 = 44 08 27 100·1·120·32 + 101·2·121·31 ≤ 827 < 100·1·120·42 + 101·2·121·41 x = 3  07 29  y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = 729 98 56 100·1·1230·42 + 101·2·1231·41 ≤ 9856 < 100·1·1230·52 + 101·2·1231·51 x = 4  98 56  y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = 9856 00 00 Algorithm terminates: Answer is 12.34

Find the cube root of 4192 to the nearest hundredth.

  1 6. 1 2 4 3 / \/ 004 192.000 000 000

 004 100·1·00·13 + 101·3·01·12 + 102·3·02·11 ≤ 4 < 100·1·00·23 + 101·3·01·22 + 102·3·02·21 x = 1  001  y = 100·1·00·13 + 101·3·01·12 + 102·3·02·11 = 1 + 0 + 0 = 1 003 192 100·1·10·63 + 101·3·11·62 + 102·3·12·61 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71 x = 6  003 096  y = 100·1·10·63 + 101·3·11·62 + 102·3·12·61 = 216 + 1,080 + 1,800 = 3,096 096 000 100·1·160·13 + 101·3·161·12 + 102·3·162·11 ≤ 96000 < 100·1·160·23 + 101·3·161·22 + 102·3·162·21 x = 1  077 281  y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 + 480 + 76,800 = 77,281 018 719 000 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 ≤ 18719000 < 100·1·1610·33 + 101·3·1611·32 + 102·3·1612·31 x = 2  015 571 928  y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320 + 15,552,600 = 15,571,928 003 147 072 000 100·1·16120·43 + 101·3·16121·42 + 102·3·16122·41 ≤ 3147072000 < 100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51 x = 4 The desired precision is achieved: The cube root of 4192 is about 16.12

The principal nth root of a positive number can be computed using logarithms. Starting from the equation that defines r as an nth root of x, namely   with x positive and therefore its principal root r also positive, one takes logarithms of both sides (any base of the logarithm will do) to obtain

The root r is recovered from this by taking the antilog:

(Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.)

A number is represented in a floating point format as   which is also called scientific notation. Its square root is   and similar formulae would apply for cube roots and logarithms. On the face of it, this is no improvement in simplicity, but suppose that only an approximation is required: then just   is good to an order of magnitude. Next, recognise that some powers, , will be odd, thus for 3141.59 = 3.1415910 rather than deal with fractional powers of the base, multiply the mantissa by the base and subtract one from the power to make it even. The adjusted representation will become the equivalent of 31.415910 so that the square root will be 10.

A table with only three entries could be enlarged by incorporating additional bits of the mantissa. However, with computers, rather than calculate an interpolation into a table, it is often better to find some simpler calculation giving equivalent results. Everything now depends on the exact details of the format of the representation, plus what operations are available to access and manipulate the parts of the number. For example, Fortran offers an EXPONENT(x) function to obtain the power. Effort expended in devising a good initial approximation is to be recouped by thereby avoiding the additional iterations of the refinement process that would have been needed for a poor approximation. Since these are few (one iteration requires a divide, an add, and a halving) the constraint is severe.

So for a 32-bit single precision floating point number in IEEE format (where notably, the power has a bias of 127 added for the represented form) you can get the approximate logarithm by interpreting its binary representation as a 32-bit integer, scaling it by  , and removing a bias of 127, i.e.

For example, 1.0 is represented by a hexadecimal number 0x3F800000, which would represent   if taken as an integer. Using the formula above you get  , as expected from  . In a similar fashion you get 0.5 from 1.5 (0x3FC00000).

/* Assumes that float is in the IEEE 754 single precision floating point format */            /* Convert type, preserving bit pattern */ * ((((val.i / 2^m) - b) / 2) + b) * 2^m = ((val.i - 2^m) / 2) + ((b + 1) / 2) * 2^m) * b = exponent bias * m = number of mantissa bits    /* Subtract 2^m. */  /* Divide by 2. */    /* Add ((b + 1) / 2) * 2^m. */ /* Interpret again as float */

The three mathematical operations forming the core of the above function can be expressed in a single line. An additional adjustment can be added to reduce the maximum relative error. So, the three operations, not including the cast, can be rewritten as

Reciprocal of the square root

                            /* The next line can be repeated any number of times to increase accuracy */             

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.

An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:

Every positive real number x has a single positive nth root, called the principal nth root, which is written  . For n equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented using exponentiation as x^1/n.

For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root,   but −2 does not have any real 6th roots.

Every non-zero number x, real or complex, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.

The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are irrational. For example,

All nth roots of rational numbers are algebraic numbers, and all nth roots of integers are algebraic integers.

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:

Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5i and −5i, where i represents a number whose square is .

A cube root of a number x is a number r whose cube is x:

Every real number x has exactly one real cube root, written  . For example,

  and  

Every real number has two additional complex cube roots.