Complex Numbers

Complex Numbers-

Our IB Maths Tutors considers Complex numbers come into existence when the square of a number is negative because we know it very well that the square of a number will always be positive doesn’t matter whether the number is positive or negative.<img src="complex number world.jpg" alt="complex number world">

In cases like  {x^2} + 1 = 0 or  2{x^3} + 5x = 0 here, if we solve, we find the square of x=-1. We say that x is not real here. Generally, these types of cases are considered as Complex numbers. Complex numbers were first observed by mathematician Girolamo Cardano (1501-1575). In his book Ars Magna, he discussed the mechanics of complex numbers in details and thus he started Complex Algebra.

Standard form of Complex Numbers-

Complex numbers are defined as expressions of the form a + ib where a,b \in  R & i = \sqrt { - 1}

It is denoted by Z  i.e. z= a + ib.

‘a’  is called as real part of z= (Re z)

and ‘b’ is called as imaginary part of z =(Im z).

i or IOTA- iota is a unique symbol. it’s the ninth letter of Latin alphabet. It’s used to denote imaginary numbers whose square root is -1.
Click here to download the book “An Imaginary Tale The Story of i” a very interesting book on iota by Paul J. Nahin.

► Zero is both purely real as well as purely imaginary but not imaginary.

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i = \sqrt { - 1}   is called the imaginary unit. Also  i² =-l, {i^3} = -i, {i^4} = 1  etc.

► \sqrt a \sqrt b = \sqrt {ab}   only if at least one of either a or b is non-negative.

► Conjugate  Complex-  If  z=a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z ¯ or z*.  i.e.  z* = a – ib.

z + z*  =  2 Re(z) v

► z –  z* =  2i Im(z)

► zz* = a² + b² which is real If z lies in the 1st quadrant then lies z* in the 4th quadrant and -z* lies in the 2nd  quadrant.
 Algebraic Operations: The algebraic operations on complex numbers are similar to those on real numbers treating i as a  polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.

e.g.   z > 0,  4 + 2i < 2 + 4 i   are  meaningless .

However in real numbers, if {a^2} + {b^2} = 0   then a = 0 = b but in complex numbers,

{z_1}^2 + {\rm{ }}{z_2}^2 = 0{\rm{ }}   does not imply {z_{1{\rm{ }}}} = {z_{2{\rm{ }}}} = 0

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Equality In Complex Number: Two complex  numbers   {z_1} = {a_1} + i{b_{1}}\& {\rm{ }}{z_2} = {a_{2{\rm{ }}}} + i{b_{2{\rm{ }}}}

are equal if and only if their real  & imaginary parts coincide.

Representation  Of  A  Complex  Number  In  Various  Forms:

(a) Cartesian Form (Geometric Representation): Every complex number z = x + i y  can be represented by a point on the cartesian plane known as a complex plane (Argand diagram) by the ordered pair (x, y).

<img src="complex number.jpg" alt="complex number">

length OM is called modulus of the complex number denoted by  \left| z \right| & \theta   is called the argument or amplitude.

\left| z \right| = \sqrt {{a^2} + {b^2}}  and   \theta = {\tan ^{ - 1}}\frac{y}{x}

\left| z \right|  is always non-negative. Unlike real numbers   \left| z \right| = \left[ \begin{array}{l} z....if..z \ge 0\\ - z..if...z < 0 \end{array} \

Argument of a complex number is a many-valued function. If \theta   is the argument of a complex number then 2n\pi +\theta  where n \in  I will also be the argument of that complex number. Any two arguments of a complex number differ by 2n\pi

The unique value of  \theta  such that   - \pi < \theta \le \pi  is called the principal value of the argument. Unless otherwise stated, amp z  implies the principal value of the argument.

By specifying the modulus & argument a complex number is defined completely. For the complex number  0 + 0.i  the argument is not defined and this is the only complex number which is given by its modulus.
 There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.

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Few Basic Questions on complex numbers

Example-1 Compute real and imaginary part of  z{\rm{ }} = {\rm{ }}\frac{{i{\rm{ }} - {\rm{ }}4}}{{2i{\rm{ }} - {\rm{ }}3{\rm{ }}}}{\rm{ }}


                           z{\rm{ }} = \frac{{{\rm{ }}i{\rm{ }} - {\rm{ }}4{\rm{ }}}}{{2i{\rm{ }} - {\rm{ }}3}}

                              = \frac{{{\rm{ }}i{\rm{ }} - {\rm{ }}4{\rm{ }}}}{{2i{\rm{ }} - {\rm{ }}3}}.\frac{{{\rm{ 2}}i{\rm{ + 3 }}}}{{2i{\rm{ + }}3}}

=\frac{{{\rm{ 2}}{{\rm{i}}^2}{\rm{ + 3i - 8i - 12 }}}}{{{{(2i)}^2}{\rm{ }} - {\rm{ }}{3^2}}}

                             = \frac{{{\rm{ - 2 - 5i - 12 }}}}{{{\rm{ - 4 - 9}}}}

                             = \frac{{{\rm{ - 14 - 5i }}}}{{ - 13}}

                            = \frac{{{\rm{ - 14 }}}}{{ - 13}} - \frac{{{\rm{5i }}}}{{ - 13}}

so clearly Re(z)=14/13 and Im(z)=5/13

In the next article on complex numbers, we will learn about the Polar form of complex numbers and some other properties of complex numbers

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