In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a square root of a number is a number such that ; in other words, a number whose ''
square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
Every
nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by
where the symbol
is called the ''
radical sign
In mathematics, the radical sign, radical symbol, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number x is written as
:\sqrt,
while the nth root of x is written as
:\sqrt
...
'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write
. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in
exponent notation, as .
Every
positive number has two square roots:
which is positive, and
which is negative. The two roots can be written more concisely using the
± sign as
. Although the principal square root of a positive number is only one of its two square roots, the designation "''the'' square root" is often used to refer to the principal square root.
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.
History
The
Yale Babylonian Collection YBC 7289
YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest ...
clay tablet was created between 1800 BC and 1600 BC, showing
and
respectively as 1;24,51,10 and 0;42,25,35
base 60 numbers on a square crossed by two diagonals. (1;24,51,10) base 60 corresponds to 1.41421296, which is a correct value to 5 decimal points (1.41421356...).
The
Rhind Mathematical Papyrus is a copy from 1650 BC of an earlier
Berlin Papyrus Berlin Papyrus may refer to several papyri kept in the Egyptian Museum of Berlin, including:
* Berlin Papyrus 3033 or the Westcar Papyrus, a storytelling papyrus
* Berlin Papyrus 3038 or the Brugsch Papyrus, a medical papyrus
* Berlin Papyrus 6619 ...
and other textspossibly the
Kahun Papyrus
The Kahun Papyri (KP; also Petrie Papyri or Lahun Papyri) are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the U ...
that shows how the Egyptians extracted square roots by an inverse proportion method.
In
Ancient India
According to consensus in modern genetics, anatomically modern humans first arrived on the Indian subcontinent from Africa between 73,000 and 55,000 years ago. Quote: "Y-Chromosome and Mt-DNA data support the colonization of South Asia by ...
, the knowledge of theoretical and applied aspects of square and square root was at least as old as the ''
Sulba Sutras'', dated around 800–500 BC (possibly much earlier). A method for finding very good approximations to the square roots of 2 and 3 are given in the ''
Baudhayana Sulba Sutra''.
Aryabhata, in the ''
Aryabhatiya'' (section 2.4), has given a method for finding the square root of numbers having many digits.
It was known to the ancient Greeks that square roots of
positive integers that are not
perfect squares are always
irrational numbers: numbers not expressible as a
ratio of two integers (that is, they cannot be written exactly as
, where ''m'' and ''n'' are integers). This is the theorem
''Euclid X, 9'', almost certainly due to
Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to:
* Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer
* ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer
* Theaetetus (crater)
Theaetetus ...
dating back to circa 380 BC.
The particular case of the
square root of 2 is assumed to date back earlier to the
Pythagoreans, and is traditionally attributed to
Hippasus. It is exactly the length of the
diagonal of a
square with side length 1.
In the Chinese mathematical work ''
Writings on Reckoning'', written between 202 BC and 186 BC during the early
Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."
A symbol for square roots, written as an elaborate R, was invented by
Regiomontanus (1436–1476). An R was also used for radix to indicate square roots in
Gerolamo Cardano's ''
Ars Magna''.
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.
According to Jeffrey A. Oaks, Arabs used the letter ''
jīm/ĝīm'' (), the first letter of the word "" (variously transliterated as ''jaḏr'', ''jiḏr'', ''ǧaḏr'' or ''ǧiḏr'', "root"), placed in its initial form () over a number to indicate its square root. The letter ''jīm'' resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician
Ibn al-Yasamin
Abu Muhammad 'Abdallah ibn Muhammad ibn Hajjaj ibn al-Yasmin al-Adrini al-Fessi () (died 1204) more commonly known as ibn al-Yasmin, was a Berber mathematician, born in Morocco and he received his education in Fez and Sevilla. Little is known of ...
.
The symbol "√" for the square root was first used in print in 1525, in
Christoph Rudolff's ''Coss''.
Properties and uses
The principal square root function
(usually just referred to as the "square root function") is a
function that maps the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of nonnegative real numbers onto itself. In
geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
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'',
:
and
:
The square root function is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
for all nonnegative ''x'', and
differentiable for all positive ''x''. If ''f'' denotes the square root function, whose derivative is given by:
:
The
Taylor series of
about ''x'' = 0 converges for ≤ 1, and 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
Physical may refer to:
* Physical examination, a regular overall check-up with a doctor
* ''Physical'' (Olivia Newton-John album), 1981
** "Physical" (Olivia Newton-John song)
* ''Physical'' (Gabe Gurnsey album)
* "Physical" (Alcazar song) (2004)
* ...
laws.
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
The square roots of the
perfect squares (e.g., 0, 1, 4, 9, 16) are
integers. In all other cases, the square roots of positive integers are
irrational numbers, and hence have non-
repeating decimals in their
decimal representations. Decimal approximations of the square roots of the first few natural numbers are given in the following table.
:
As 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
In cryptography, SHA-1 (Secure Hash Algorithm 1) is a cryptographically broken but still widely used hash function which takes an input and produces a 160- bit (20- byte) hash value known as a message digest – typically rendered as 40 hexa ...
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 1780. 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.
:
The
square bracket notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11,
; 3, 6, 3, 6, ... looks like this:
:
where the two-digit pattern repeats over and over again in the partial denominators. Since , the above is also identical to the following
generalized continued fractions:
:
Computation
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.
Most
pocket calculators have a square root key. Computer
spreadsheets and other
software are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as the
Newton's method (frequently with an initial guess of 1), to compute the square root of a positive real number. When computing square roots with
logarithm tables or
slide rules, one can exploit the identities
:
where and
10 are the
natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
and
base-10 logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
s.
By trial-and-error, one can square an estimate for
and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it is prudent to use the identity
:
as it allows one to adjust the estimate ''x'' by some amount ''c'' and measure the square of the adjustment in terms of the original estimate and its square. Furthermore, (''x'' + ''c'')
2 ≈ ''x''
2 + 2''xc'' when ''c'' is close to 0, because the
tangent line to the graph of ''x''
2 + 2''xc'' + ''c''
2 at ''c'' = 0, as a function of ''c'' alone, is ''y'' = 2''xc'' + ''x''
2. Thus, small adjustments to ''x'' can be planned out by setting 2''xc'' to ''a'', or ''c'' = ''a''/(2''x'').
The most common
iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
of square root calculation by hand is known as the "
Babylonian method" or "Heron's method" after the first-century Greek philosopher
Heron of Alexandria, who first described it.
The method uses the same iterative scheme as the
Newton–Raphson method yields when applied to the function y = ''f''(''x'') = ''x''
2 − ''a'', using the fact that its slope at any point is ''dy''/''dx'' = '(''x'') = 2''x'', but predates it by many centuries.
The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if ''x'' is an overestimate to the square root of a nonnegative real number ''a'' then ''a''/''x'' will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the
inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
), and so it can serve as a new overestimate with which to repeat the process, which
converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find ''x'':
# 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.
# Replace ''x'' by the average (''x'' + ''a''/''x'') / 2 between ''x'' and ''a''/''x''.
# Repeat from step 2, using this average as the new value of ''x''.
That is, if an arbitrary guess for
is ''x''
0, and , 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 computation of the square root of a positive number can be reduced to that of a number in the range . This simplifies finding a start value for the iterative method that is close to the square root, for which a
polynomial or
piecewise-linear approximation can be used.
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 .
The name of the square root
function varies from
programming language to programming language, with
sqrt
(often pronounced "squirt" ) being common, used in
C,
C++, and derived languages like
JavaScript,
PHP, and
Python.
Square roots of negative and complex numbers
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 written as ''j'', especially in the context of
electricity where "''i''" traditionally represents electric current) and called the
imaginary unit, which is ''defined'' such that . Using this notation, we can think of ''i'' as the square root of −1, but we also have 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 : 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 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
). The above Taylor series for
remains valid for complex numbers
with
The above can also be expressed in terms of
trigonometric functions:
Algebraic formula
When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root:
:
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:
:
Notes
In the following, the complex ''z'' and ''w'' may be expressed as:
*
*
where
and
.
Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general.
*
Counterexample for the principal square root: and
This equality is valid only when
*
Counterexample for the principal square root: and
This equality is valid only when
*
Counterexample for the principal square root: )
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 or which are not true in general.
Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that :
:
The third equality cannot be justified (see
invalid proof). It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains
The left-hand side becomes either
:
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 √.
''N''th roots and polynomial roots
The definition of a square root of
as a number
such that
has been generalized in the following way.
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 . 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.
Square roots of matrices and operators
If ''A'' is a
positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator ''B'' with ; we then define . In general matrices may have multiple square roots or even an infinitude of them. For example, the
identity matrix has an infinity of square roots,
[Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", ''Mathematical Gazette'' 87, November 2003, 499–500.] though only one of them is positive definite.
In integral domains, including fields
Each element of an
integral domain has no more than 2 square roots. The
difference of two squares identity is proved using the
commutativity of multiplication. If and are square roots of the same element, then . Because there are no
zero divisors this implies or , 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 . 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
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, let 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 quadratic residues and 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.
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.
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.
''Floruit'' (; abbreviated fl. or occasionally flor.; from Latin for "they flourished") denotes a date or period during which a person was known to have been alive or active. In English, the unabbreviated word may also be used as a noun indicatin ...
300 BC) gave the construction of the
geometric mean of two quantities in two different places
Proposition II.14an
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 o
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
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wi ...
. Let AHB be a line segment of length with and . 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. , from which we conclude by cross-multiplication that , and finally that
. When marking the midpoint O of the line segment AB and drawing the radius OC of length , 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.
See also
*
Apotome (mathematics)
*
Cube root
*
Functional square root
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying ...
*
Integer square root
In number theory, the integer square root (isqrt) of a non-negative integer ''n'' is the non-negative integer ''m'' which is the greatest integer less than or equal to the square root of ''n'',
: \mbox( n ) = \lfloor \sqrt n \rfloor.
For examp ...
*
Nested radical
*
Nth root
*
Root of unity
*
Solving quadratic equations with continued fractions
*
Square root principle
The Penrose method (or square-root method) is a method devised in 1946 by Professor Lionel Penrose for allocating the voting weights of delegations (possibly a single representative) in decision-making bodies proportional to the square root of th ...
*
Notes
References
*
*
*
*
* .
External links
Algorithms, implementations, and moreaul Hsieh's square roots webpage
AMS Featured Column, Galileo's Arithmetic by Tony Philipsncludes a section on how Galileo found square roots
{{DEFAULTSORT:Square Root
Elementary special functions
Elementary mathematics
Unary operations