Square root


Square root
Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of \sqrt{2h/g}, where h is the height and g is the acceleration of gravity.

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square (the result of multiplying the number by itself, or r × r) is x.[1] For example, 4 is a square root of 16 because 42 = 16.

Every non-negative real number x has a unique non-negative square root, called the principal square root, denoted by a radical sign as \scriptstyle \sqrt{x}. For positive x, the principal square root can also be written in exponent notation, as x1/2. For example, the principal square root of 9 is 3, denoted \scriptstyle \sqrt{9} \ = \ 3, because 32 = 3 × 3 = 9 and 3 is non-negative. 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.

Every positive number x has two square roots: \scriptstyle \sqrt{x}, which is positive, and \scriptstyle -\sqrt{x}, which is negative. Together, these two roots are denoted \scriptstyle \pm\sqrt{x} (see ± shorthand). 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 "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC.[2] The particular case \scriptstyle \sqrt{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.

The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign. For example, in the expression \scriptstyle \sqrt{ab+2}, ab + 2 is the radicand.

Contents

Properties

The graph of the function \scriptstyle f(x) = \sqrt{x}, made up of half a parabola with a vertical directrix.

The principal square root function \scriptstyle f(x) = \sqrt{x} (usually just referred to as the "square root function") is a function that maps the set of non-negative 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 (a superset of the rational numbers).

For all real numbers x


\sqrt{x^2} = \left|x\right| = 
\begin{cases} 
  x,  & \mbox{if }x \ge 0 \\
  -x, & \mbox{if }x < 0. 
\end{cases}
    (see absolute value)

For all non-negative real numbers x and y,

\sqrt{xy} = \sqrt x \sqrt y

and

\sqrt x = x^{1/2}.

The square root function is continuous for all non-negative x and differentiable for all positive x. If f denotes the square-root function, its derivative is given by:

f'(x) = \frac{1}{2\sqrt x}.

The Taylor series of √1 + x about x = 0 converges for |x| ≤ 1 and is given by

\sqrt{1 + x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + \textstyle \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + \dots,\!

which is a special case of a binomial series.

Computation

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 to compute the exponential function and the natural logarithm or common logarithm, and use them to compute the square root of a positive real number x using the identity

\sqrt{x} = e^{(\ln x)/2} or \sqrt{x} = 10^{(\log x)/2}.

The same identity is exploited when computing square roots with logarithm tables or slide rules.

The most common iterative method 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.[3] The method uses the same iterative scheme as the Newton-Raphson process yields when applied to the function \scriptstyle y=f(x)=x^2-a, using the fact that its slope at any point is \scriptstyle dy/dx=f'(x)=2x, but predates it by many centuries.[4] It involves a simple algorithm, which results in a number closer to the actual square root each time it is repeated. The basic idea is that if x is an overestimate to the square root of a non-negative real number a then \scriptstyle a/x\, will be an underestimate and so the average of these two numbers may reasonably be expected to provide a better approximation (though the formal proof of that assertion depends on the inequality of arithmetic and geometric means that shows this average is always an overestimate of the square root, as noted below, thus assuring convergence). To find x :

  1. Start with an arbitrary positive start value x (the closer to the square root of a, the fewer iterations will be needed to achieve the desired precision).
  2. Replace x by the average between x and a/x, that is: \scriptstyle (x + a/x) / 2\,, representing the Newton-Raphson scheme \scriptstyle x_{n+1} = x_n - f(x_n)/f'(x_n) resulting in \scriptstyle x_{n+1} = (x_n+a/x_n)/2,

(It is sufficient to take an approximate value of the average to ensure convergence)

  1. Repeat step 2 until x and a/x are as close as desired.

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.

Using the identity

\sqrt{a} = 2^{-n}\sqrt{4^n a},

the computation of the square root of a positive number can be reduced to that of a number in the range [1, 4). 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.

Square roots of negative and complex numbers

Complex square root
Second leaf of the complex square root
Using the Riemann surface of the square root, one can see 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 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 notice that 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 positive number, then the principal square root of –x is

\sqrt{-x} = i \sqrt x.

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

(i\sqrt x)^2 = i^2(\sqrt x)^2 = (-1)x = -x.

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.

Square root of an imaginary number

The square roots of i in the complex plane

The square root of i is given by

\sqrt{i} = \frac{1}{2}\sqrt{2} + i\frac{1}{2}\sqrt{2} = \frac{\sqrt{2}}{2}(1+i).

This result can be obtained algebraically by finding a and b such that

i = (a+bi)^2\,\!

or equivalently

i = a^2 + 2abi - b^2.\,\!

This gives the two simultaneous equations

\begin{cases}
2ab = 1\,\! \\
a^2 - b^2 = 0\,\!
\end{cases}

with solutions

a = b = \pm \frac{1}{\sqrt{2}}.

The choice of the principal root then gives

a = b = \frac{1}{\sqrt{2}}.

The result can also be obtained by using de Moivre's formula and setting

i = \cos\left (\frac{\pi}{2}\right ) + i\sin\left (\frac{\pi}{2}\right )

which produces

\begin{align}
 \sqrt{i} & = \left ( \cos\left ( \frac{\pi}{2} \right ) + i\sin \left (\frac{\pi}{2} \right ) \right )^{\frac{1}{2}}  \\
              & = \cos\left (\frac{\pi}{4} \right ) + i\sin\left ( \frac{\pi}{4} \right ) \\
              & = \frac{1}{\sqrt{2}} + i\left ( \frac{1}{\sqrt{2}} \right ) = \frac{1}{\sqrt{2}}(1+i) . \\
\end{align}

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 x + iy can be viewed as a point in the plane, (x, y), expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair (r, φ), where r ≥ 0 is the distance of the point from the origin, and φ is the angle that the line from the origin to the point makes with the positive real (x) axis. In complex analysis, this value is conventionally written re. If

 z=r e^{\varphi i} \text{ with } -\pi < \varphi \le \pi, \,

then we define the principal square root of z as follows:

\sqrt{z} = \sqrt{r} \, e^{i \varphi / 2}.

The principal square root function is thus defined using the nonpositive real axis as a branch cut. The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it isn't even continuous). The above Taylor series for √1 + x remains valid for complex numbers x with |x| < 1.

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

\sqrt{r \left(\cos \varphi + i \, \sin \varphi \right)]} = \sqrt{r}[\cos \frac{\varphi}{2} + i \sin \frac{\varphi}{2}]. \,

Algebraic formula

When the number is expressed using Cartesian coordinates the following formula can be used for the principal square root:[5][6]

\sqrt{x+iy} = \sqrt{\frac{r + x}{2}} \pm i \sqrt{\frac{r - x}{2}}

where the sign of the imaginary part of the root is taken to be same as the sign of the imaginary part of the original number, and

r = |x + iy| = \sqrt{x^2+ y^2}

is the absolute value or modulus of the original number. The real part of the principal value is always non-negative.

The other square root is simply –1 times the principal square root; in other words, the two square roots of a number sum to 0.

Notes

Because of the discontinuous nature of the square root function in the complex plane, the law √zw = √zw is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √zw at all, leading to the equality's failure. A similar problem appears with the complex logarithm and the relation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that –1 = 1:

-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1

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 (√–1)·(√–1). The left-hand side becomes either

\sqrt{-1} \cdot \sqrt{-1}=i \cdot i=-1

if the branch includes +i or

\sqrt{-1} \cdot \sqrt{-1}=(-i) \cdot (-i)=-1

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

\sqrt{-1 \cdot -1}=\sqrt{1}=-1,

where the last equality, √1 = –1, is a consequence of the choice of branch in the redefinition of √.

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 B2 = A; we then define A1/2 = √A = B. In general matrices may have multiple square roots or even an infinitude of them. For example the 2×2 identity matrix has an infinity of square roots.[7]

Uniqueness of square roots in general rings

In a ring we call an element b a square root of a iff b2 = a.

In an integral domain, suppose the element a has some square root b, so b2 = a. Then this square root is not necessarily unique, but it is "almost unique" in the following sense: If x too is a square root of a, then x2 = a = b2. So x2b2 = 0, or, by commutativity, (x + b)(xb) = 0. Because there are no zero divisors in the integral domain, we conclude that one factor is zero, and x = ±b. The square root of a, if it exists, is therefore unique up to a sign, in integral domains.

To see that the square root need not be unique up to sign in a general ring, consider the ring \mathbb{Z}/8\mathbb{Z} from modular arithmetic. Here, the element 1 has four distinct square roots, namely ±1 and ±3. On the other hand, the element 2 has no square root. See also the article quadratic residue for details.

Another example is provided by the quaternions \mathbb{H} in which the element −1 has an infinitude of square roots including ±i, ±j, and ±k. In fact, the set of square roots of -1 is exactly

 { ai + bj + ck  | a2 + b2 + c2 = 1 }. 

Hence this set is exactly the same size and shape as the (surface of the) unit sphere in 3-space.

Principal square roots of the positive integers

As decimal expansions

The square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots are irrational numbers, and therefore their decimal representations are non-repeating decimals.

\scriptstyle \sqrt {1} \scriptstyle =\, 1
\scriptstyle \sqrt {2} \scriptstyle \approx 1.414213562373095048801688724209698078569671875376948073176679737990732478462 (article) 1 million digits, 2 million, 5 million, 10 million
\scriptstyle \sqrt {3} \scriptstyle \approx 1.732050807568877293527446341505872366942805253810380628055806979451933016909 (article) 1 million digits
\scriptstyle \sqrt {4} \scriptstyle =\, 2
\scriptstyle \sqrt {5} \scriptstyle \approx 2.236067977499789696409173668731276235440618359611525724270897245410520925638 (article) 1 million digits
\scriptstyle \sqrt {6} \scriptstyle \approx 2.449489742783178098197284074705891391965947480656670128432692567250960377457 1 million digits
\scriptstyle \sqrt {7} \scriptstyle \approx 2.645751311064590590501615753639260425710259183082450180368334459201068823230 1 million digits
\scriptstyle \sqrt {8} \scriptstyle \approx 2.828427124746190097603377448419396157139343750753896146353359475981464956924 1 million digits
\scriptstyle \sqrt {9} \scriptstyle =\, 3
\scriptstyle \sqrt {10} \scriptstyle \approx 3.162277660168379331998893544432718533719555139325216826857504852792594438639 1 million digits
\scriptstyle \sqrt {11} \scriptstyle \approx 3.316624790355399849114932736670686683927088545589353597058682146116484642609
\scriptstyle \sqrt {12} \scriptstyle \approx 3.464101615137754587054892683011744733885610507620761256111613958903866033818
\scriptstyle \sqrt {13} \scriptstyle \approx 3.605551275463989293119221267470495946251296573845246212710453056227166948293
\scriptstyle \sqrt {14} \scriptstyle \approx 3.741657386773941385583748732316549301756019807778726946303745467320035156307
\scriptstyle \sqrt {15} \scriptstyle \approx 3.872983346207416885179265399782399610832921705291590826587573766113483091937
\scriptstyle \sqrt {16} \scriptstyle =\, 4
\scriptstyle \sqrt {17} \scriptstyle \approx 4.123105625617660549821409855974077025147199225373620434398633573094954346338
\scriptstyle \sqrt {18} \scriptstyle \approx 4.242640687119285146405066172629094235709015626130844219530039213972197435386
\scriptstyle \sqrt {19} \scriptstyle \approx 4.358898943540673552236981983859615659137003925232444936890344138159557328203
\scriptstyle \sqrt {20} \scriptstyle \approx 4.472135954999579392818347337462552470881236719223051448541794490821041851276

Note that if the radicand is not square-free one can simplify, for example \scriptstyle \sqrt{8} \ = \ \sqrt{4}\sqrt{2} \ = \ 2\sqrt{2}; \scriptstyle \sqrt{12} \ = \ \sqrt{4}\sqrt{3} \ = \ 2\sqrt{3}; \scriptstyle \sqrt{18} \ = \ \sqrt{9}\sqrt{2} \ = \ 3\sqrt{2} and \scriptstyle \sqrt{20} \ = \ \sqrt{4}\sqrt{5} \ = \ 2\sqrt{5}.

As expansions in other numeral systems

The square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots are irrational numbers, and therefore their representations in any standard positional notation system are non-repeating.

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 circa 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.

\scriptstyle \sqrt {2} \scriptstyle =\, [1; 2, 2, ...]
\scriptstyle \sqrt {3} \scriptstyle =\, [1; 1, 2, 1, 2, ...]
\scriptstyle \sqrt {4} \scriptstyle =\, [2]
\scriptstyle \sqrt {5} \scriptstyle =\, [2; 4, 4, ...]
\scriptstyle \sqrt {6} \scriptstyle =\, [2; 2, 4, 2, 4, ...]
\scriptstyle \sqrt {7} \scriptstyle =\, [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
\scriptstyle \sqrt {8} \scriptstyle =\, [2; 1, 4, 1, 4, ...]
\scriptstyle \sqrt {9} \scriptstyle =\, [3]
\scriptstyle \sqrt {10} \scriptstyle =\, [3; 6, 6, ...]
\scriptstyle \sqrt {11} \scriptstyle =\, [3; 3, 6, 3, 6, ...]
\scriptstyle \sqrt {12} \scriptstyle =\, [3; 2, 6, 2, 6, ...]
\scriptstyle \sqrt {13} \scriptstyle =\, [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
\scriptstyle \sqrt {14} \scriptstyle =\, [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
\scriptstyle \sqrt {15} \scriptstyle =\, [3; 1, 6, 1, 6, ...]
\scriptstyle \sqrt {16} \scriptstyle =\, [4]
\scriptstyle \sqrt {17} \scriptstyle =\, [4; 8, 8, ...]
\scriptstyle \sqrt {18} \scriptstyle =\, [4; 4, 8, 4, 8, ...]
\scriptstyle \sqrt {19} \scriptstyle =\, [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
\scriptstyle \sqrt {20} \scriptstyle =\, [4; 2, 8, 2, 8, ...]

The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in more traditional notation the simple continued fraction for the square root of 11 – [3; 3, 6, 3, 6, ...] – looks like this:


\sqrt{11} = 3 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \ddots}}}}}\,

where the two-digit pattern {3, 6} repeats over and over and over again in the partial denominators. Since 11 = 32+2, the above is also identical to the following generalized continued fractions:


\sqrt{11} = 3 + \cfrac{2}{6 + \cfrac{2}{6 + \cfrac{2}{6 + \cfrac{2}{6 + \cfrac{2}{6 + \ddots}}}}} = 3 + \cfrac{6\cdot 1}{20-1 - \cfrac{1}{20 - \cfrac{1}{20 - \cfrac{1}{20 - \ddots}}}}.

Geometric construction of the square root

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 \scriptstyle \sqrt {ab}, one can construct \scriptstyle \sqrt {a} simply by taking b = 1.

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 a + b 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. \scriptstyle \ a/h = h/b from which we conclude by cross-multiplication that \scriptstyle \ h^2 = ab and finally that \scriptstyle h = \sqrt {ab}. Note further that if you were to mark the midpoint O of the line segment AB and draw the radius OC of length \scriptstyle (a + b)/2 then clearly OC > CH i.e. \scriptstyle (a + b)/2 >= \sqrt {ab} (with equality when and only when a = b), 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: \scriptstyle \sqrt {1} can, of course, be constructed, and once \scriptstyle \sqrt {x} has been constructed, the right triangle with 1 and \scriptstyle \sqrt {x} for its legs has a hypotenuse of \scriptstyle \sqrt {x + 1}. The Spiral of Theodorus is constructed using successive square roots in this manner.

History

The Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 BC, showing \scriptstyle \sqrt{2} and \scriptstyle 30\sqrt{2} as 1;24,51,10 and 42;25,35 base 60 numbers on a square crossed by two diagonals.[8]

The Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows us how the Egyptians extracted square roots.[9]

In Ancient India, 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)[citation needed]. A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[10] Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.

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."[11]

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.[12]

The symbol √ for the square root was first used in print in 1525 in Christoph Rudolff's Coss, which was also the first to use the then-new signs '+' and '-'.[13]

See also

Notes

  1. ^ Gelfand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-817-63677-3. http://books.google.com/books?id=Z9z7iliyFD0C. , Extract of page 120
  2. ^ Heath, Sir Thomas L. (1908). The Thirteen Books of The Elements, Vol. 3. Cambridge University Press. p. 3. http://www.archive.org/stream/thirteenbookseu03heibgoog#page/n14/mode/1up. 
  3. ^ Heath, Sir Thomas L. (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324. http://books.google.com/?id=LOA5AAAAMAAJ&pg=PR323. 
  4. ^ Muller, Jean-Mic (2006). Elementary functions: algorithms and implementation. Springer. pp. 92–93. ISBN 0817643729. http://books.google.com/?id=g3AlWip4R38C , Chapter 5, p 92
  5. ^ Abramowitz, Miltonn; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Dover Publications. p. 17. ISBN 0-486-61272-4. http://books.google.com/books?id=MtU8uP7XMvoC. , Section 3.7.26, p. 17
  6. ^ Cooke, Roger (2008). Classical algebra: its nature, origins, and uses. John Wiley and Sons. p. 59. ISBN 0-470-25952-3. http://books.google.com/books?id=lUcTsYopfhkC. , Extract: page 59
  7. ^ Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", Mathematical Gazette 87, November 2003, 499-500.
  8. ^ Math.ubc.ca
  9. ^ Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
  10. ^ Joseph, ch.8.
  11. ^ Dauben, p. 210.
  12. ^ Smith, p. 148.
  13. ^ Manguel, Alberto (2006). "Done on paper: the dual nature of numbers and the page". The Life of Numbers. ISBN 8486882141. 

References

  • Imhausen, Annette (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. pp. 187–384. ISBN 0691114854. 
  • Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0691006598. 
  • Smith, David (1958). History of Mathematics. 2. New York: Dover Publications. ISBN 9780486204307. 

External links


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  • square root — noun count a number that you multiply by itself to produce a particular number. For example, the square root of 9 is 3 …   Usage of the words and phrases in modern English

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