Square root
In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x.[1] For example, 4 and −4 are square roots of 16 because .
Every
Every
Square roots of negative numbers can be discussed within the framework of
History
The Yale Babylonian Collection clay tablet YBC 7289 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.[5] (1;24,51,10) base 60 corresponds to 1.41421296, which is correct to 5 decimal places (1.41421356...).
The
In
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 dating back to c. 380 BC.[12] The discovery of irrational numbers, including the particular case of the square root of 2, is widely associated with the Pythagorean school.[13][14] Although some accounts attribute the discovery to Hippasus, the specific contributor remains uncertain due to the scarcity of primary sources and the secretive nature of the brotherhood.[15][16] It is exactly the length of the diagonal of a square with side length 1.
In the Chinese mathematical work
A symbol for square roots, written as an elaborate R, was invented by
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 (ج), the first letter of the word "جذر" (variously transliterated as jaḏr, jiḏr, ǧaḏr or ǧiḏr, "root"), placed in its initial form (ﺟ) 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.[19]
The symbol "√" for the square root was first used in print in 1525, in Christoph Rudolff's Coss.[20]
Properties and uses
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
For all real numbers x,
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 Taylor series of about x = 0 converges for |x| ≤ 1, and is given by
The square root of a nonnegative number is used in the definition of
Square roots of positive integers
A positive number has two square roots, one positive, and one negative, which are
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
n | truncated to 50 decimal places |
---|---|
0 | 0 |
1 | 1 |
2 | 1.41421356237309504880168872420969807856967187537694 |
3 | 1.73205080756887729352744634150587236694280525381038 |
4 | 2 |
5 | 2.23606797749978969640917366873127623544061835961152 |
6 | 2.44948974278317809819728407470589139196594748065667 |
7 | 2.64575131106459059050161575363926042571025918308245 |
8 | 2.82842712474619009760337744841939615713934375075389 |
9 | 3 |
10 | 3.16227766016837933199889354443271853371955513932521 |
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
As periodic continued fractions
One of the most intriguing results from the study of
= [1; 2, 2, ...] | |
= [1; 1, 2, 1, 2, ...] | |
= [2] | |
= [2; 4, 4, ...] | |
= [2; 2, 4, 2, 4, ...] | |
= [2; 1, 1, 1, 4, 1, 1, 1, 4, ...] | |
= [2; 1, 4, 1, 4, ...] | |
= [3] | |
= [3; 6, 6, ...] | |
= [3; 3, 6, 3, 6, ...] | |
= [3; 2, 6, 2, 6, ...] | |
= [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...] | |
= [3; 1, 2, 1, 6, 1, 2, 1, 6, ...] | |
= [3; 1, 6, 1, 6, ...] | |
= [4] | |
= [4; 8, 8, ...] | |
= [4; 4, 8, 4, 8, ...] | |
= [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...] | |
= [4; 2, 8, 2, 8, ...] |
The
where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since 11 = 32 + 2, 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
By trial-and-error,[23] 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
The most common
- 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 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 a = 0, the convergence is only linear; however, so in this case no iteration is needed.
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 n = 2.
The name of the square root
sqrt
[26] (often pronounced "squirt" [27]) being common, used in C and derived languages like C++, JavaScript, PHP, and PythonSquare 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 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
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
Algebraic formula
When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root:[28][29]
where sgn(y) = 1 if y ≥ 0 and sgn(y) = −1 otherwise.[30] 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 ±i 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: z = −1 and w = −1 This equality is valid only when
- Counterexample for the principal square root: w = 1 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.
Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that −1 = 1:
The third equality cannot be justified (see
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 n is an integer greater than two, a n-th root of is a number such that ; it is denoted
Given any
Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of n-th roots.
Square roots of matrices and operators
If A is a
In integral domains, including fields
Each element of an
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 p, let q = pe for some positive integer e. A non-zero element of the field Fq with q elements is a quadratic residue if it has a square root in Fq. 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.
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 ±i, ±j, and ±k. 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 n 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. 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 b = 1.
The construction is also given by
Euclid's second proof in Book VI depends on the theory of
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
- Integer square root
- Nested radical
- Nth root
- Root of unity
- Solving quadratic equations with continued fractions
- Square-root sum problem
- Square root principle
- Quantum gate § Square root of NOT gate (√NOT)
Notes
- ^ Gel'fand, p. 120 Archived 2016-09-02 at the Wayback Machine
- ^ "Squares and Square Roots". www.mathsisfun.com. Retrieved 2020-08-28.
- ISBN 978-0-7637-5772-4. Archived from the original on 2016-09-01. Extract of page 78 Archived 2016-09-01 at the Wayback Machine
- ^ Weisstein, Eric W. "Square Root". mathworld.wolfram.com. Retrieved 2020-08-28.
- ^ "Analysis of YBC 7289". ubc.ca. Retrieved 19 January 2015.
- ^ Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
- S2CID 119992603.
Seidenberg (pp. 501-505) proposes: "It is the distinction between use and origin." [By analogy] "KEPLER needed the ellipse to describe the paths of the planets around the sun; he did not, however invent the ellipse, but made use of a curve that had been lying around for nearly 2000 years". In this manner Seidenberg argues: "Although the date of a manuscript or text cannot give us the age of the practices it discloses, nonetheless the evidence is contained in manuscripts." Seidenberg quotes Thibaut from 1875: "Regarding the time in which the Sulvasutras may have been composed, it is impossible to give more accurate information than we are able to give about the date of the Kalpasutras. But whatever the period may have been during which Kalpasutras and Sulvasutras were composed in the form now before us, we must keep in view that they only give a systematically arranged description of sacrificial rites, which had been practiced during long preceding ages." Lastly, Seidenberg summarizes: "In 1899, THIBAUT ventured to assign the fourth or the third centuries B.C. as the latest possible date for the composition of the Sulvasutras (it being understood that this refers to a codification of far older material)."
- ^ Joseph, ch.8.
- doi:10.2307/2300909. Retrieved 30 March 2024.
- ^ Cynthia J. Huffman; Scott V. Thuong (2015). "Ancient Indian Rope Geometry in the Classroom - Approximating the Square Root of 2". www.maa.org. Retrieved 30 March 2024.
Increase the measure by its third and this third by its own fourth, less the thirty-fourth part of that fourth. This is the value with a special quantity in excess.
- ^ J J O'Connor; E F Robertson (November 2020). "Apastamba". www.mathshistory.st-andrews.ac.uk. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 30 March 2024.
- ^ Heath, Sir Thomas L. (1908). The Thirteen Books of The Elements, Vol. 3. Cambridge University Press. p. 3.
- ^ Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics (3rd ed.). Hoboken, NJ: John Wiley & Sons. pp. 51–53. ISBN 978-0470525487.
- ^ Stillwell, John (2010). Mathematics and Its History (3rd ed.). New York, NY: Springer. pp. 14–15. ISBN 978-1441960528.
- ^ Dauben (2007), p. 210.
- ^ "The Development of Algebra - 2". maths.org. Archived from the original on 24 November 2014. Retrieved 19 January 2015.
- ^ Oaks, Jeffrey A. (2012). Algebraic Symbolism in Medieval Arabic Algebra (PDF) (Thesis). Philosophica. p. 36. Archived (PDF) from the original on 2016-12-03.
- ISBN 84-86882-14-1.
- ISBN 9780387342283.
- ISBN 9780883850831.
- ISBN 978-1-136-84393-8. Archived from the original on 2017-03-01. Extract of page 41 Archived 2017-03-01 at the Wayback Machine
- ^ Heath, Sir Thomas L. (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324.
- ^ "Function sqrt". CPlusPlus.com. The C++ Resources Network. 2016. Archived from the original on November 22, 2012. Retrieved June 24, 2016.
- from the original on September 1, 2016. Retrieved June 24, 2016.
- ISBN 0-486-61272-4. Archived from the original on 2016-04-23., Section 3.7.27, p. 17 Archived 2009-09-10 at the Wayback Machine
- ISBN 978-0-470-25952-8. Archivedfrom the original on 2016-04-23.
- ^ This sign function differs from the usual sign function by its value at 0.
- ^ Maxwell, E. A. (1959). Fallacies in Mathematics. Cambridge University Press.
- ^ Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", Mathematical Gazette 87, November 2003, 499–500.
References
- ISBN 978-0-691-11485-9.
- ISBN 0-8176-3677-3.
- Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
- ISBN 978-0-486-20430-7.
- ISBN 978-1-4020-4559-2.
External links
- Algorithms, implementations, and more – Paul Hsieh's square roots webpage
- How to manually find a square root
- AMS Featured Column, Galileo's Arithmetic by Tony Philips – includes a section on how Galileo found square roots