Square root Guide, Meaning , Facts, Information and Description
In mathematics, the square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is .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 of integers are often irrational numbers, i.e., numbers not expressible as one integer over another. (It is a misconception that mathematicians define irrational number to be one whose decimal expansion is infinite and non-repeating. That is equivalent, but nothing is sacred about base-10 numerals as opposed to other bases.) For example, cannot be written exactly as m/n, where n and m are integers (and so cannot be written in finite or repeating decimal form, although that is a fact of less interest to mathematicians.) Nonetheless, it is exactly the length of the diagonal of a square with side length 1.
The discovery that is irrational is attributed to Hippasus, a disciple of Pythagoras. After the number was revealed to be irrational, the Pythagoreans killed Hippasus, not wishing to believe this fundamental number could be infinitely long and nonrepeating. Other Greek philosophers celebrated the discovery with a sacrifice of 100 oxen (a hecatomb).
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").
The following important properties of the square root functions are valid for all positive real numbers and :
Properties
The square root function generally maps rational numbers to algebraic numbers; is rational if and only if is a rational number which, after cancelling, is a quotient of two perfect squares. In particular, is irrational.
In geometrical terms, the square root function maps the area of a square to its side length.
Suppose that and are reals, and that , and we want to find . A common mistake is to "take the square root" and deduce that . This is incorrect, because the square root of is not , but the absolute value , one of our above rules. Thus, all we can conclude is that , or equivalently .
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 has the following graph, made up of half a parabola lying on its side:
The function is continuous for all non-negative , and differentiable for all positive (it is not differentiable for since the slope of the tangent there is ∞). 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 using the identity
The same identity is exploited when computing square roots with logarithm tables or slide rules.
Babylonian method
A commonly used algorithm for approximating is known as the "Babylonian method" and is based on Newton's method. It proceeds as follows:
- start with an arbitrary positive start value (the closer to the root the better)
- replace by the average of and
- go to 2
This could be represented as where
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 in the reals, but to 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 12.34
Although 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.
Basic Newton iteration finds a single root of a function given a sufficently precise approximation to the root. The nature of which root will be given based on an approximation is dependent on the Newton fractal which we will not discuss here any further. The basic iteration is given by:
We have the two functions given as follows:
We first find the derivative of these two functions. We have:
We present the iteration for "g" as follows:
Ex: is:
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 9
To every non-zero complex number z there exist precisely two numbers w such that w2 = 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 √(c2) = ±c, therefore √(a2b2) = ±ab and therefore √(zw) = ±√(z)√(w), using a = √(z) and b = √(w).
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 √A = B.
More generally, to every normal matrix or operator A there exist normal operators B such that B2 = 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.Square roots using Newton iteration
There are two widely used functions used to find the square root of a number, say, "z". One finds the square root of "z" while the other finds the reciprocal of the square root of "z". The former gives sufficiently precise approximations to the square root of "z" with each iteration. The latter requires that one divide to obtain the square root of "z".
and
Note that for "f", we have . For "g", we have . If one multiplied the roots of "g" by "z", the result will be .
The iteration for "f" is derived here:
The iteration for "f" involves a division which is more time consuming than a multiplication in computer integer arithmetic. The iteration for "g" involves no division and is therefore recommended for large integers "z".
This iteration using "g" involves only a squaring and a two multiplications, as apposed to a division in the case of "f". In practical implementations of large integer square roots, the iteration involving "g" is faster for large integers "z" since division is at best , a constant times the time function of multiplication. The constant term is almost always 3 or more, meaning that a single division can almost never be faster than 3 multiplications.Pell's equation
Pell's equation yields a method for finding rational approximations of square roots of integers.Finding square roots using mental arithmetic
Based on Pell's equation there is a method to calculate square roots simply by subtracting odd numbers.2 x 100 = 200 and 5 x 20 + 1 = 101
1) 200-101 = 99 Next number is 1
The result gives us 5.19 as an approximation of the square root of 27Continued 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
to obtain √2 − 1 to some specific precision specified through n levels of recurrence, and add 1 to the result to obtain √2.Square roots of complex numbers
where the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number.
The third equality cannot be justified. (See invalid proof.)Square roots of matrices and operators