Square root
In mathematics, the square root of a non-negative real number x is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is x.For example, since .
This example suggests how square roots can arise when solving quadratic equations such as or, more generally, .
Extending the square root concept to negative real numbers gives rise to imaginary and complex numbers.
Square roots are often irrational numbers, requiring an infinite, non-repeating series of digits in their decimal representation. For example, cannot be written exactly in finite or repeating decimal form. Equivalently, it cannot be represented by a fraction whose numerator and denominator are integers. Nonetheless, it is exactly the length of the diagonal of a square with side length 1. The discovery that is irrational is attributed to the Pythagoreans.
The square root symbol (√) 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 square root function generally maps rational numbers to algebraic numbers; √x is rational if and only if x is a rational number which, after cancelling, is a fraction of two perfect squares. In particular, √2 is irrational.In geometrical terms, the square root function maps the area of a square to its side length.
Suppose that x and a are reals, and that x^{2}=a, and we want to find x. A common mistake is to "take the square root" and deduce that x = √a. This is incorrect, because the square root of x^{2} is not x, but the absolute value |x|, one of our above rules. Thus, all we can conclude is that |x| = √a, or equivalently x = ±√a.
In calculus, for instance when proving that the square root function is continuous or differentiable or when computing certain limitss, the following identity often comes handy:
The function f(x) = √x has the following graph, made up of half a parabola lying on its side:
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 &infin). Its derivative is given by
Computing square roots
Calculators
Pocket calculatorss typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of x using the identityBabylonian method
A commonly used algorithm for approximating √x is known as the "Babylonian method" and is based on Newton's method. It proceeds as follows:- start with an arbitrary positive start value r (the closer to the root the better)
- replace r by the average of r and x/r
- go to 2
This algorithm works equally well in the p-adic numbers, but cannot be used to identify real square roots with p-adic square roots; it is easy, for example, to construct a sequence of rational numbers by this method which converges to +3 in the reals, but to -3 in the 2-adics.
An exact "long-division like" algorithm
This method, while much slower than the Babylonian method, has the advantage that it is exact: if the given number has a square root whose decimal representation terminates, then the algorithm terminates and produces the correct square root after finitely many steps. It can thus be used to check whether a given integer is a square number.Write the number in decimal and divide it into pairs of digits starting from the decimal point. The numbers are laid out similar to the long division algorithm and the final square root will appear above the original number.
For each iteration:
- Bring down the most significant pair of digits not yet used and append them to any remainder. This is the current value referred to in steps 2 and 3.
- If denotes the part of the result found so far, determine the greatest digit that does not make exceed the current value. Place the new digit on the quotient line.
- Subtract from the current value to form a new remainder.
- If the remainder is zero and there are no more digits to bring down the algorithm has terminated. Otherwise continue with step 1.
____1__2._3__4_
| 01 52.27 56 1
x 01 1*1=1 1
____ __
00 52 22
2x 00 44 22*2=44 2
_______ ___
08 27 243
24x 07 29 243*3=729 3
_______ ____
98 56 2464
246x 98 56 2464*4=9856 4
_______
00 00 Algorithm terminates: answer is 12.34Although demonstrated here for base 10 numbers, the procedure works for any base, including base 2. In the description above, 20 means double the number base used, in the case of binary this would really be 100. The algorithm is in fact much easier to perform in base 2, as in every step only the two digits 0 and 1 have to be tested. See Shifting nth-root algorithm.
Pell's equation
Pell's equation yields a method for finding rational approximations of square roots of integers.Finding square roots in the head
Based on Pell's equation there is a methode to calculate the square root in the head, by simply subtraction of odd numbers.Ex: Square root of 27 is:
1) 27-1 = 26
2) 26-3 = 23
3) 23-5 = 18
4) 18-7 = 11
5) 11-9 = 2 First number is 5
2 x 100 = 200 and 5 x 20 + 1 = 101
1) 200-101 = 99 Next number is 1
99 x 100 = 9900 and 51 x 20 + 1 = 1021
1) 9900-1021 = 8879
2) 8879-1023 = 7856
3) 7856-1025 = 6831
4) 6831-1027 = 5804
5) 5804-1029 = 4775
6) 4775-1031 = 3744
7) 3744-1033 = 2711
8) 2711-1035 = 1676
9) 1676-1037 = 639 Next number is 9The result gives us 5.19 as the square root of 27
Continued fraction methods
Quadratic irrationals, that is numbers involving square roots in the form (a+√b)/c, have periodic continued fractions. This makes them easy to calculate recursively given the period. For example, to calculate √2, we make use of the fact that √2-1 = [0;2,2,2,2,2,...], and use the recurrence relation- a_{n+1}=1/(2+a_{n}) with a_{0}=0
Square roots of complex numbers
To every non-zero complex number z there exist precisely two numbers w such that w^{2} = z. The usual definition of √z is as follows: if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then we set √z = √r\ exp(iφ/2). Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous). The above Taylor series for √(1+x) remains valid for complex numbers x with |x| < 1.
When the number is in rectangular form the following formula can be used:
Note that because of the discontinuous nature of the square root function in the complex plane, the law √(zw) = √(z)√(w) is in general not true. Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that -1 = 1:
However the law can only be wrong up to a factor -1, √(zw) = ±√(z)√(w), is true for either ± as + or as - (but not both at the same time). Note that √(c^{2}) = ±c, therefore √(a^{2}b^{2}) = ±ab and therefore √(zw) = ±√(z)√(w), using a = √(z) and b = √(w).
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 B^{2} = A; we then define √A = B.
More generally, to every normal matrix or operator A there exist normal operators B such that B^{2} = A. In general, there are several such operators B for every A and the square root function cannot be defined for normal operators in a satisfactory manner. Positive definite operators are akin to positive real numbers, and normal operators are akin to complex numbers.