what is [√]

In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. Every non-negative real number x has a unique non-negative square root, called the principal square root and denoted
圖片參考：http://upload.wikimedia.org/math/6/7/1/67160962b11ce4e32aa232acd43cf5e0.png
. The other square root of 9 (not the principal square root) is −3.
Square roots often arise when solving quadratic equations, or equations of the form ax2 + bx + c = 0, due to the variable x being squared.
Per the fundamental theorem of algebra, there are two solutions to the equation defining the square roots of any number (although these roots may not be distinct, as in the square root of zero). For a positive real number, the two square roots are the principal square root and the negative square root (denoted
圖片參考：http://upload.wikimedia.org/math/c/a/7/ca7cfa13728da069ec8c2a6ba90a8fcb.png
. For negative real numbers, the concept of imaginary and complex numbers has been developed to provide a mathematical framework to deal with the results. Square roots of objects other than numbers can also be defined.
Square roots of integers that are not perfect squares are always irrational numbers, i.e., numbers not expressible as a ratio of two integers. For example,
圖片參考：http://upload.wikimedia.org/math/c/4/7/c475af0fc6a341d865339933e251aba7.png
is irrational attributed to Hippasus, a disciple of Pythagoras. (See square root of 2 for proofs)
The square root symbol (
圖片參考：http://upload.wikimedia.org/math/1/6/0/160a5b4ac79375bd4c5e13c0f3a95f73.png
) was first used during the 16th century. It has been suggested that it originated as an altered form of lowercase r, representing the Latin radix (meaning "root").
Properties

The principal square root function
圖片參考：http://upload.wikimedia.org/math/4/c/3/4c3190daed02a84bc228e530395185c1.png
onto itself.

The principal square root function
圖片參考：http://upload.wikimedia.org/math/9/5/2/9522bb26ab3c9c2630ba6f2f72828378.png
always returns a unique value.

To obtain both roots of a positive number, take the value given by the principal square root function as the first root (root1) and obtain the second root (root2) by subtracting the first root from zero (ie root2 = 0 − root1).

The following important properties of the square root functions are valid for all positive real numbers x and y:


圖片參考：http://upload.wikimedia.org/math/8/a/e/8aeaa4179739d627ab535d12abd6b8cf.png


圖片參考：http://upload.wikimedia.org/math/5/d/7/5d77a845388eac74dd71c374aa174a68.png


圖片參考：http://upload.wikimedia.org/math/e/2/d/e2de293b0472dbb3319afb5d8fd9bd66.png


The square root function maps rational numbers to algebraic numbers; also,
圖片參考：http://upload.wikimedia.org/math/c/4/7/c475af0fc6a341d865339933e251aba7.png
is irrational.

In geometrical terms, the square root function maps the area of a square to its side length.

Contrary to popular belief,
圖片參考：http://upload.wikimedia.org/math/5/f/b/5fb2e730c094b72161aaadd42c3f8435.png
for real x (see absolute value).

Suppose that x and a are real numbers, and that x2 = a, and we want to find x. A common mistake is to "take the square root" and deduce that
圖片參考：http://upload.wikimedia.org/math/a/e/a/aea6eea8993dba7ac007fee164349bac.png
.

In calculus, for instance when proving that the square root function is continuous or differentiable, or when computing certain limits, the following identity often comes handy:


圖片參考：http://upload.wikimedia.org/math/5/6/9/5697d341960d5ab4a3e5cea0cebe6d01.png


valid for all non-negative numbers x and y which are not both zero.

The function
圖片參考：http://upload.wikimedia.org/math/c/f/c/cfcb940f8cfe449642fec6d14daea233.png
has the following graph, made up of half a parabola lying on its side:

圖片參考：http://upload.wikimedia.org/wikipedia/commons/7/72/Square_root.png


The function is continuous for all non-negative x, and differentiable for all positive x (it is not differentiable for x = 0 since the slope of the tangent there is ∞). Its derivative is given by


圖片參考：http://upload.wikimedia.org/math/5/d/d/5dd7f82c6f6a4fc1171d52d22bf5a371.png


The Taylor series of
圖片參考：http://upload.wikimedia.org/math/1/e/c/1eca0b4c44c8cd18bfa984d2bca2050b.png
about x = 0 can be found using the binomial theorem:






圖片參考：http://upload.wikimedia.org/math/8/1/8/818fa72d02e8ba8d5eb8a72779245e66.png


圖片參考：http://upload.wikimedia.org/math/4/6/a/46a53e7c4e61979d2ac4e81b3d685db8.png




圖片參考：http://upload.wikimedia.org/math/d/1/3/d138abe7ff6d26053a19c7b09c6fc987.png


for
圖片參考：http://upload.wikimedia.org/math/5/1/9/51952f374b1adf11cb320618b0208d74.png
.

2007-02-13 19:11:39 補充：
you are right ! gutvenn~

It can be a tick but it's most likely a square root in Math.

√=平方根(square root)

平方根:
一個合成數有一個數自"x"二次=那個合成數
那個數就是那個合成數的平方根(square root)

It means correct~