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Square Root Of 7

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									                           Square Root Of 7
Square Root Of 7

In mathematics, a square root of a number a is a number y such that y2 = a, or, in other
words, a number y whose square (the result of multiplying the number by itself, or y × y) is
a.[1] For example, 4 is a square root of 16 because 42 = 16.

Every non-negative real number a has a unique non-negative square root, called the
principal square root, which is denoted by , where √ is called radical sign. For
example, the principal square root of 9 is 3, denoted , because 32 = 3 × 3 = 9 and 3 is
non-negative.

The term whose root is being considered is known as the radicand. The radicand is
the number or expression underneath the radical sign, in this example 9. Every
positive number a has two square roots: , which is positive, and , which is negative.

Together, these two roots are denoted (see ± shorthand). 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. For positive a, the
principal square root can also be written in exponent notation, as a1/2.
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Square roots of positive whole numbers 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.

The particular case 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 principal square root function (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).

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.


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 The method uses the same iterative scheme as the Newton-Raphson process yields
when applied to the function, using the fact that its slope at any point is , but predates
it by many centuries.

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 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 :

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.




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