Complex numbers learning | IB Elite Tutor

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.

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.

► 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 ${z_{1{\rm{ }}}} = {z_{2{\rm{ }}}} = 0$

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).

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 $\left| z \right| = \left[ \begin{array}{l} z....if..z \ge 0\\ - z..if...z < 0 \end{array} \right.not..correct..here$

► 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.