Square roots often arise when solving Quadratic Equation s, or equations of the form , due to the variable being squared.
Every positive number ''x'' has two square roots. One of them is , which is positive, and the other is , which is negative. Together, these two roots are denoted . Square roots of negative numbers can be discussed within the framework of Complex Number s. Square roots of objects other than numbers can also be defined.
Square roots of of two integers. For example, cannot be written exactly as , where ''n'' and ''m'' are integers. Nonetheless, it is exactly the length of the Diagonal of a Square with side length 1. This has been known since ancient times, with the discovery that is irrational attributed to Hippasus , a disciple of Pythagoras . (''See Square Root Of 2 for proofs of the irrationality of this number.'')
with a vertical Directrix .]]
The principal square root function (usually just referred to as the "square root function") is a Function which maps the Set of non-negative real numbers onto itself, and, like all functions, always returns a unique value. The square root function also maps Rational Number s into Algebraic Number s (a superset of the rational numbers); is rational if and only if is a rational number which can be represented as a ratio of two Perfect Square s. In Geometrical terms, the square root function maps the Area of a Square to its side length.
For all positive real numbers and , and
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"http://wwwinformationdelightinfo/information/entry/Taylor_series" class="copylinks">Taylor Series of <math>\sqrt{x+1}</math> about <math>x=0</math> is <math>1 + rac{1}{2}x - rac{1}{8}x^2 + rac{1}{16} x^3 - rac{5}{128} x^4 + \dots</math> and converges for <math>\left x
ight < 1</math>
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and
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The usual definition of √''z'' is by introducing the following with −π < φ ≤ π, then we set the principal value to
:
where the two-digit pattern {3, 6} repeats over and over and over again in the partial denominators.
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
and
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.
Another method of geometric construction uses of
.
The
Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows us how the Egyptians extracted square roots.Anglin, W.S. (1994). ''Mathematics: A Concise History and Philosophy''. New York: Springer-Verlag.
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 B.C. (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 ''.Joseph, ch.8.
Aryabhata in the ''
Aryabhatiya '' (section 2.4), has given a method for finding the square root of numbers having many digits.
D.E. Smith in ''History of Mathematics'', says, about the existing situation in Europe: "In Europe these methods (for finding out the square and square root) did not appear before
Cataneo (1546). He gave the method of
Aryabhata for determining the square root". Smith, p. 148.
- Smith D.E., ''History of Mathematics'' (book 2)
- Joseph, George G., ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd ed. Penguin Books , London. (2000). ISBN 0-691-00659-8.
