MD TANVIR ALAM: COMPLEX NUMBER BASIC

Elementary operations

Conjugation

Geometric representation of $z$ and its conjugate $\bar{z}$ in the complex plane

The complex conjugate of the complex number

z = x + yi

is defined to be x − yi. It is denoted $\bar{z}$ or $z^*\,$ . Geometrically, $\bar{z}$ is the "reflection" of z about the real axis. In particular, conjugating twice gives the original complex number: $\bar{\bar{z}}=z$ .
The real and imaginary parts of a complex number can be extracted using the conjugate:

$\operatorname{Re}\,(z) = \tfrac{1}{2}(z+\bar{z}), \,$

$\operatorname{Im}\,(z) = \tfrac{1}{2i}(z-\bar{z}). \,$

Moreover, a complex number is real if and only if it equals its conjugate.
Conjugation distributes over the standard arithmetic operations:

$\overline{z+w} = \bar{z} + \bar{w}, \,$

$\overline{z w} = \bar{z} \bar{w}, \,$

$\overline{(z/w)} = \bar{z}/\bar{w}. \,$

The reciprocal of a nonzero complex number

z = x + yi

is given by

$\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}=\frac{\bar{z}}{x^2+y^2}.$

This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. Inversive geometry, a branch of geometry studying more general reflections than ones about a line, can also be expressed in terms of complex numbers.

Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:

$(a+bi) + (c+di) = (a+c) + (b+d)i.\$

Similarly, subtraction is defined by

$(a+bi) - (c+di) = (a-c) + (b-d)i.\$

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.

Multiplication and division

The multiplication of two complex numbers is defined by the following formula:

$(a+bi) (c+di) = (ac-bd) + (bc+ad)i.\$

In particular, the square of the imaginary unit is −1:

$i^2 = i \times i = -1.\$

The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms.

$(a+bi) (c+di) = ac + bci + adi + bidi \$ (distributive law)

$= ac + bidi + bci + adi \$ (commutative law of addition—the order of the summands can be changed)

$= ac + bdi^2 + (bc+ad)i \$ (commutative law of multiplication—the order of the multiplicands can be changed)

$= (ac-bd) + (bc + ad)i \$ (fundamental property of the imaginary unit).

The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. Where at least one of $c$ and $d$ is non-zero:

$\,\frac{a + bi}{c + di} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i.$

Division can be defined in this way because of the following observation:

$\,\frac{a + bi}{c + di} = \frac{\left(a + bi\right) \cdot \left(c - di\right)}{\left (c + di\right) \cdot \left (c - di\right)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i.$

As shown earlier, $c-di$ is the complex conjugate of the denominator $c+di$ . The real part c and the imaginary part d of the denominator must not both be zero for division to be defined.

Square root

The square roots of a + bi (with b ≠ 0) are $\pm (\gamma + \delta i)$ , where

$\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}$

and

$\delta = \sgn (b) \sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}},$

where sgn is the signum function. This can be seen by squaring $\pm (\gamma + \delta i)$ to obtain a + bi.^[6]^[7] Here $\sqrt{a^2 + b^2}$ is called the modulus of a + bi, and the square root with non-negative real part is called the principal square root.

Polar form

Main article: Polar coordinate system

Figure 2: The argument φ and modulus r locate a point on an Argand diagram; $r(\cos \varphi + i \sin \varphi)$ or $r e^{i\varphi}$ are polar expressions of the point.

Absolute value and argument

An alternative way of defining points in the complex plane, other than using the x- and y-coordinates, is to use the distance of a point P from O, the point whose coordinates are (0, 0) (the origin), together with the angle between the line through P and O and the (horizontal) line which is the positive part of the real axis. This idea leads to the polar form of complex numbers.
The absolute value (or modulus or magnitude) of a complex number z = x + yi is

$\textstyle r=|z|=\sqrt{x^2+y^2}.\,$

If z is a real number (i.e., y = 0), then r = |x|. In general, by Pythagoras' theorem, r is the distance of the point P representing the complex number z to the origin.
The argument or phase of z is the angle of the radius OP with the positive real axis, and is written as $\arg(z)$ . As with the modulus, the argument can be found from the rectangular form $x+iy$ :^[8]

$\varphi = \arg(z) = \begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0 \\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\ \frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\ -\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\ \mbox{indeterminate } & \mbox{if } x = 0 \mbox{ and } y = 0. \end{cases}$

The value of φ must always be expressed in radians. It can change by any multiple of 2π and still give the same angle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principal value in the interval $(-\pi,\pi]$ is chosen. Values in the range $[0,2\pi)$ are obtained by adding $2\pi$ if the value is negative. The polar angle for the complex number 0 is undefined, but arbitrary choice of the angle 0 is common.
The value of φ equals the result of atan2: $\varphi = \mbox{atan2}(\mbox{imaginary}, \mbox{real})$ .
Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form

$z = r(\cos \varphi + i\sin \varphi ).\,$

Using Euler's formula this can be written as

$z = r e^{i \varphi}.\,$

Using the cis function, this is sometimes abbreviated to

$z = r \ \operatorname{cis} \ \varphi. \,$

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as^[9]

$z = r \ang \varphi . \,$

Multiplication, division and exponentiation in polar form

Multiplication of 2+i (blue triangle) and 3+i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers

z 1 = r 1 (cos φ 1 + i sin φ 1)

and

z 2 = r 2 (cos φ 2 + i sin φ 2)

the formula for multiplication is

$z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).\,$

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, which gives back i² = −1. The picture at the right illustrates the multiplication of

$(2+i)(3+i)=5+5i. \,$

Since the real and imaginary part of 5+5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangle are arctan(1/3) and arctan(1/2), respectively. Thus, the formula

$\frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3}$

holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-like formulas—are used for high-precision approximations of π.
Similarly, division is given by

$\frac{z_1}{ z_2} = \frac{r_1}{ r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right).$

This also implies de Moivre's formula for exponentiation of complex numbers with integer exponents:

$z^n = r^n\,(\cos n\varphi + i \sin n \varphi).$

The n-th roots of z are given by

$\sqrt[n]{z} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)$

for any integer

k

satisfying

0 \leq k \leq n - 1

. Here $\sqrt[n]{r}$ is the usual (positive) nth root of the positive real number r. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cⁿ = x there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z is considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely defined number. Formulas such as

$\sqrt[n]{z^n} = z$

(which holds for positive real numbers), do in general not hold for complex numbers

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Saturday, February 9, 2013

COMPLEX NUMBER BASIC