Square Root Calculator .Co


Decomposition of a   right triangle into five similar triangles, the basis for the aperiodic pinwheel tiling.

  • 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
, 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.

Notation for the (principal) square root of .

For example, = 5, since 25 = 5 ⋅ 5, or (5 squared).

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




Angle size and sector area are the same when the conic radius is . This diagram illustrates the circular and hyperbolic functions based on sector areas .

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 :


Babylonian clay tablet YBC 7289 with annotations. Besides showing the square root of 2 in sexagesimal (1 24 51 10), the tablet also gives an example where one side of the square is 30 and the diagonal then is 42 25 35. The sexagesimal digit 30 can also stand for =

, in which case 0 42 25 35 is approximately 0.7071065.


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 xn+1 = xn + , 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.

The A series of paper sizes

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.

Дополнительно:  Ноутбук стал сильно тормозить. Что делать в этом случае?


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 xn + 1 = (xn + a/xn) / 2, then each xn 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 .

  1. Dauben, Joseph W. (June 1983) Scientific American Georg Cantor and the origins of transfinite set theory. Volume 248; Page 122.
  2. Yee, Alexander. «Records Set by y-cruncher».
  3. Conrad, Keith. «Pell’s Equation II» . uconn.edu. Retrieved 2022.
  4. Browne, Malcolm W. (July 30, 1985) New York Times Puzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1. (Note: this is a widely cited article).
  5. Richard K. Guy: «The Strong Law of Small Numbers». American Mathematical Monthly, vol. 95, 1988, pp. 675–712
  6. Ivrissimtzis, Ioannis P.; Dodgson, Neil A.; Sabin, Malcolm (2005), « -subdivision», in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A. (eds.), Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003, Mathematics and Visualization, Berlin: Springer, pp. 285–299, doi:10.1007/3-540-26808-1_16, MR 2112357
  7. Kimberly Elam (2001), Geometry of Design: Studies in Proportion and Composition, New York: Princeton Architectural Press, ISBN 1-56898-249-6
  8. Jay Hambidge (1967), The Elements of Dynamic Symmetry, Courier Dover Publications, ISBN 0-486-21776-0
  9. Julian D. A. Wiseman, «Sin and cos in surds»
  10. LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR 0080682
  11. Khinchin, Aleksandr Yakovlevich (1964), Continued Fractions, University of Chicago Press, Chicago and London
  12. Ramanathan, K. G. (1984), «On the Rogers-Ramanujan continued fraction», Proceedings of the Indian Academy of Sciences, Section A, 93 (2): 67–77, doi:10.1007/BF02840651, ISSN 0253-4142, MR 0813071, S2CID 121808904
  13. Eric W. Weisstein, Ramanujan Continued Fractions at MathWorld

(fill in either box above to explore more about square roots)
Answers are rounded to seven decimal places.

  • positive numbers: 2
  • negative numbers: -3
  • imaginary numbers: -7i
  • complex numbers: 8+9i


  • The square root calculator provides the principal square root (the positive square root which is most commonly used).  
    Multiplying the principal square root by -1 will provide the negative square root if needed.
  • The square root of a negative number results in an imaginary number noted by the letter «i».

Square Root:

The square root of a number is a value that when multiplied by itself equals the original number.  
Example:   The square root of 9 = 3.   Thus: 3 x 3 = 9.

Where A and B are valid numbers:
A is the square root of B if:   A x A = B

  • Square Root Symbol / Square Root Sign:

    The square root of a number N is commonly represented by:  

    It can also be represented by:  

    In exponential notation, it can be represented by:   N1/2

    «Square root» can be abbreviated as «sqrt»   Example:   sqrt(2) = 1.4142136

    See: square root symbol if you need to add the symbol to a text document.

  • How are square roots used:

    Square roots can be used:

    • to solve for the distance between two points (Pythagorean Theorem)
    • to solve for the length of a side of a right triangle (Pythagorean Theorem)
    • to find the solutions to quadratic equations
    • to find normal distribution
    • to find standard deviation
    • basically to solve for a squared variable in an equation

    If you see an equation like N2 = 27 , you can solve for N by taking the square root of 27 which
    gives N = 5.1961524

How to Calculate the Square Root of:

  1. A Positive Number:

        The square root of a positive number can be calculated by:

    1. SquareRootCalculator.Co website
    2. —>

    3. square root calculator above
    4. handheld calculator
    5. spreadsheet
    6. square roots without a calculator

  2. A Negative Number:

    To calculate the square root of a negative number, find the square root of the same positive number and multiply by «i». ( where i represents an imaginary number and i = square root of -1)

    Example: square root of -5

    = (square root of 5) x (square root of -1)

    = (square root of 5) x (i)

    = 2.236068 x i

    = 2.236068i

  3. An Imaginary Number:

    To calculate the square root of an imaginary number, find the square root of the number as if it were a real number (without the i) and then multiply by the square root of i (where the square root of i = 0.7071068 + 0.7071068i)

    Example: square root of 5i

    = (square root of 5) x (square root of i)

    = (2.236068) x (0.7071068 + 0.7071068i)

    = 1.5811388 + 1.5811388i

  4. A Complex Number:

    To find the square root of a complex number, use the following equation:

    Square root (a + bi) = c + di

    c = (1/square root of 2) x square root of [ (square root of(a2+b2)) + a ]

    d = (sign of b/square root of 2) x square root of [ (square root of(a2+b2)) — a ]

    Example: the square root of 3-5i = c + di

    c = (1/square root of 2) x square root of [ (square root of(32+(-5)2)) + 3 ]

    c = 2.1013034
    d = (-1/square root of 2) x square root of [ (square root of(32+(-5)2)) — 3 ]
    d = -1.1897378
    Thus: square root of 3-5i = 2.1013034 — 1.1897378i

Need to calculate the square root of a number with a handheld calculator or looking to buy a new calculator?

Most handheld calculators have a square root function.
Below is a review of some that are recommended and how to use them.

Graphing Calculator:

  • Texas Instruments TI-84 Plus Graphing Calculator

    Square root on TI-84 Plus Calculator


    To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.

    (Note: This same method will also work with TI-83 and TI-81 calculators)

    (To see what the graph looks like on this calculator,
    click the show graph button below the calculator image on this page.)

    It can be used on many college entrance exams (check your exam criteria).

    It is a popular calculator. (If you need help, the likelihood of finding someone who knows how to use it is higher).

    The display shows seven lines of input/output. Long equations can be viewed and checked. (This is a nice benefit that graphing calculators have compared to scientific calculators that may only have a single line display.) Another benefit of having a large display is you can compare your current answer to past answers that are still on the screen. This can often help you discover an input error that may have otherwise gone unnoticed.

    It is bulkier than a scientific calculator.
    It costs around $85 (USD) more than a scientific calculator.

    Best price for this calculator as of 9-2-2014 is around $94.00 (USD).

  • Casio Graphing Calculator (fx-9750GII)

    Square root on Casio fx-9750GII Calculator


    Click to show y =


    <!— test —>

    To take the square root of a number, press [SHIFT] and then [] (the radical symbol is above the x2 key) and then the number that you want to find the square root of and then the [EXE] key.

    (To see this graph, click the show graph button below the calculator image on this page.)

    The cost is half the price of a TI-84 calculator.

    It is a little smaller than the TI-84 calculator.

    It can be used on many college entrance exams (check your exam criteria).

    The display shows seven lines of input/output.


    It is not as popular as the TI-84 calculator. (It may be harder to find someone to help if you have a question about how to use the calculator.)

    Best price as of 9-2-2014 is around $42.74 (USD).

Scientific Calculator:

  • Texas Instruments TI-36X Pro Scientific Calculator

    Square root on TI-36X-Pro Calculator

    To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.
    Note:  If your calculator is in «Math Print» mode, the display will show as the answer. To convert this answer to a decimal number, press the [<>~] key (the «Answer Toggle» key) which is the key above the [ENTER] key.

    It is good if a graphing calculator is not permitted.
    It is smaller and lighter than a graphing calculator.
    The display shows four lines of input/output. Some scientific calculators only show one line of input/output.

    The symbols shown on the function keys ( + , — , etc.) may be hard to see because they are the same silver color as the keys.

    Best price for this calculator on 9-2-2014 is around $19.99 (USD).

Math Help / Math Tutor

If you need help with math:

  • There are a lot of online resources that are free to use and are available when you need them. Some sites have online courses, instructional videos, and even math games to help you learn more about math.
  • Ask someone who is good at math to help. If you can’t find anyone who can help, a math teacher at a local school may be able to help or recommend a math tutor in your area.
  • A local library may be able to help not only with informational books but they may be able to recommend a math tutor also.
  • There are several websites that can help you find a local tutor if you would like in-person tutoring. Also, some tutors are able to help from a remote location if needed. Prices are typically between $10 to $50 (USD) per hour. Always research and try to find reviews, ratings, or feedback from others who have used a particular tutor before making a selection.

Square roots of positive integers

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.


In integral domains, including fields

Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2v2 = (uv)(u + v) is proved using the commutativity of multiplication. If and are square roots of the same element, then u2v2 = 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 = pe 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.

Properties and uses

The graph of the function f(x) = √x, made up of half a parabola with a vertical directrix

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| =
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.

Identities of Ramanujan

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


In rings in general

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.

Proofs of irrationality

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).

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.

Figure 1. Stanley Tennenbaum’s geometric proof of the irrationality of

Figure 2. Tom Apostol’s geometric proof of the irrationality of

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 2nm 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.

Relation to the golden ratio and Fibonacci numbers

The   diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.


(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  :


Square roots of matrices and operators

Square roots of negative and complex numbers

First leaf of the complex square root

Second leaf of the complex square root

Using the Riemann surface of the square root, it is shown how the two leaves fit together

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 √.

Nth roots and polynomial roots

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.

Geometric construction of the square root

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.

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