Square root

The square root ( colloquially root, square root english, short sqrt ) a non-negative number is that (uniquely determined ) non-negative number whose square is equal to the given number. The symbol for the square root is the root sign, the square root of the number is thus represented by. The number or the term under the root is called the radicand. Less common is the detailed spelling addition can be expressed as the square root of power: is equivalent to example is due and the square root of 9 is 3

Since the equation for two solutions, one usually defines the square root as the non-negative of the two solutions, ie it is always thus achieved that the concept of square root is unique. The two solutions of the equation, and thus are

• 3.1 Calculating the square root on graphic way

• 4.1 Definition
• 4.2 Example: Analysis of a complex square root

• 5.1 Calculation of square roots modulo a prime p 5.1.1 Calculation for the case of p mod 4 = 3
• 5.1.2 Calculation for the case p = 1 mod 4

Preface to the definitions

In the formal definition of the square root of two issues to consider:

• If we restrict ourselves to non-negative rational numbers, then the square root is not defined in many cases. Even in ancient times it was learned that about the number can not be a rational number (see Euclid's proof of the irrationality of the square root of 2).
• In general, there are two different numbers whose squares correspond to a predetermined number. For example, would be due to the number -3 a possible candidate for the square root of 9

The symbol for the square root was used for the first time during the 16th century. It is believed that the character is a modified form of small r, which is an abbreviation for the Latin word " radix " ( root ). Originally the symbol precedes the cube root; the horizontal extension was missing. Even Carl Friedrich Gauss therefore used parentheses for more complex root expressions and wrote, for example, instead of

In English, the square root is called the "square root", which is why the term is used, " sqrt " of the square root function, in many programming languages.

Square roots of real numbers

Definition: The square root of a nonnegative real number is the non-negative real number whose square is equal.

Equivalent to this, the real square root can be defined as a function like this: Be

The ( bijective ) function limitation of the square of the amount of non-negative real numbers. The inverse function of the function q is called the square root function

Comments

• Note that the square defined by said function for all real numbers, but is not reversible. It is neither injective nor surjective.
• The restriction q of the quadratic function is reversible and is reversed by the real root function. As only non-negative real numbers appear as images of q, the real root function is defined only for these numbers.
• By made ​​before inversion restriction of q to non-negative real numbers, the values ​​of the square root function are non-negative numbers. The restriction of the quadratic function in which different real numbers always have several squares on other subsets of would lead to other inverse functions, but they are not referred to as a real square root function.

Examples

Properties and calculation rules

The properties of the square root function will become apparent from the characteristics of the restricted to the amount of non-negative real numbers quadratic function:

• That is, the square root function is strictly increasing.
• Applies to the real amount for any real numbers a
• In contrast, only applies to non-negative a
• The square root function is differentiable on, there shall
• At the point 0, it is not differentiable, its graph has a vertical tangent there to the equation
• She's on every closed subinterval of its domain Riemann - integrable, one of its primitives is

Calculation of square roots of real numbers

Even if the square root is to be pulled out of a natural number, the result is often an irrational number whose decimal expansion so a non- periodic, non -terminating decimal fraction (namely, if and only if the result is not natural ). The calculation of a square root, which is not a rational number, that is to determine an approximation of sufficient accuracy. There are a number of ways:

• Written root Drag: This is an algorithm similar to the common methods of long division.
• Of intervals: This procedure is quite easy to understand if very tedious in the practical implementation.

• Babylonian square root method or Heron: This iterative method is often used when programming the root calculation for calculator, since it converges quickly. Is the Newton's method for finding zero points is applied to the function
• The Taylor series expansion of development with point 1 may be using the binomial series found. The series converges for pointwise to the function value of the root function.
• Calculation using CORDIC Algorithm: This method is mainly used in calculating units, FPUs and microcontrollers.

Determination of the square root on graphic way

One possibility is the Kathetensatz: The number whose square root is sought, is plotted on a number line from zero. About the distance between and is a semi-circle with radius drawn ( Thales circle). When a Lot is built to the baseline, which intersects the semicircle (height of a right triangle ). The distance between this intersection point to the zero point is the square root of ( cathetus ).

Square roots of complex numbers

Is a non-zero complex number, having the equation

Exactly two solutions, which are also known as roots or square roots of. These are in the Gaussian plane on the two intersections of the circle with the radius 0 and the bisector of the angle between the axis and positive. That of the two roots, which is in the right half-plane, called the principal value of the root.

If we write the complex number in the form

Where and are real with and, then for the principal value of the root:

The second root value ( value addition ) is given by point reflection (180 ° rotation ) at the zero point:

Definition

The complex function " squared- z", has just like the real quadratic function no inverse function, because it is not injective, but unlike the real numbers surjective, that is, every complex number is the square of a complex number. It can therefore be similar to the real ( non-negative ) square define complex square root functions by carrying out a restriction of the range of definition of a subset of D q of the complex numbers, on the q is injective and remains onto. Depending on which subset is selected for it, is obtained as the inverse square root function of the different branches.

The main branch of the complex square root function is obtained by using as the domain of q

Basis sets, this is the right half plane of the complex plane, starting from the edge include only the numbers with imaginary nichtnegativem to DH. The restriction of q on DH is a bijective mapping of DH on the complex numbers, so its inverse function, the main branch of the square root is defined on all of. The value of this inverse function is called the principal value of the square root of, If by a certain complex number, then it is this principal value.

Is in Cartesian coordinates given, ie with real numbers, and then results

For the main value of the square root, where the function for negative the value -1, and otherwise (that is, also, and thus, unlike the sign function ) has the value 1:

The only sub-branch of

Is given in polar coordinates, with then is the principal value of the square root by

Optionally, wherein the real ( non-negative ) is the square root of. The secondary value is again obtained as

The amount of the two roots therefore results than the square root of the sum of the complex number. At the main value, the argument is ( " the angle of z " s, u ) halved. The other solution is obtained geometrically by point reflection of this main value at the origin.

The argument of a complex number is the angle oriented in the complex plane, the points and in real coordinates. In the picture to the next example, the argument of z and the argument of w1 are color coded.

• Complex square root

Second branch

The Riemann surface of the square root can be seen how the two branches merge.

Example: Calculation of a complex square root

Wanted are the square roots of the radicand First, the amount is determined:

Thus, the principal value of the square root yields to

The other root is obtained by sign reversal:

Square roots modulo n

Also in the residue class ring to square roots can be defined. Quite analogous to the real and complex numbers is called a square root of, if:

However, one must use other methods than n when calculating real or complex square roots to compute square roots modulo. To determine the square roots of modulo, one can proceed as follows:

First, one determines the prime factorization

The modulus n, and then the solutions modulo prime powers of the individual, these solutions was finally recognized by applying the Chinese remainder theorem together to the sought solution.

Calculation of square roots modulo a prime p

For primes different from 2 is done calculating the square root of this:

To test whether any square has, it calculates the value of the Legendre symbol

Because it is:

In the first case does not have a square root in and in the second case, only the square root of 0 The interesting case is thus the third case, and therefore we assume in the following that apply.

Calculation for the case p mod 4 = 3

Is the Legendre symbol equal then

The two square roots of modulo

Calculation for the case p = 1 mod 4

Is the Legendre symbol equal then

The two square roots of modulo Here to Choose r so that

Applies. For this purpose, you can easily test different values ​​of r. The result is recursively

Defined.

Sample calculation for and:

According to the above formula are the square roots of by

Given. For we find by trial the value because it is:

The values ​​for and are thus obtained:

Substituting these values ​​gives

The means 15 and 22 are the two square roots of 3 modulo 37

Square roots of matrices

As a root of a square matrix is any matrix arising multiplied by itself:

As with the square root of a real or complex numbers, the square root of matrices is not necessarily unique. But if we consider only positive definite symmetric matrices, so rooting is clear: Every positive definite symmetric matrix has a unique positive definite symmetric root They are obtained by diagonalized using an orthogonal matrix (this is according to the spectral theorem always possible) and then the diagonal elements replaced by their roots; However, it is always to choose the positive root. See also Cholesky decomposition. The uniqueness follows from the fact that the exponential map is a diffeomorphism from the vector space of symmetric matrices on the subset of positive definite symmetric matrices.

Square root of an approximate integral operator

One can the definite integral function from 0 to and with a predetermined function that takes on the equidistant nodes the values ​​as matrix multiplication as follows numerical approach (used in):

It is intuitively clear that one can repeat this operation and preserves all the double integral:

Thus, the matrix is ​​regarded as numerically approximate the integral operator.

The matrix is ​​not diagonalizable and its Jordan canonical form is:

To draw a square root of it, you might as proceed as in the non- diagonalizable matrices described. However, there is in this case a more direct formal solution as follows:

With and

It denote the indices of the Subdiagonalen (0 is the diagonal ) and the exponent is equal to If, as a real and positive advance so is real and defined as positive.

This allows you to function as follows approaching a " half " definite integral from 0 numerically:

Searches all operators that multiplied by itself to give the approximate integral operator, so you must also use the negative sign, which means there are two solutions

To derive the formula can first invert, multiply the result by and finally invert again.