Complex Number System

Indian mathematician Mahavira (850 A.D.) was first to mention in his work 'Ganitasara sangraha'; 'As in nature of things a negative (quantity) is not a square (quantity), it has, therefore, no square root'. Hence, there is no real number x which satisfies the polynomial equation x²+1=0.

A symbol √(-1), denoted by letter i was introduced by Swiss Mathematician, Leonhard Euler to provide solution of equation x²+1=0.'i' was regarded as a fictitious or imaginary number which could be manipulated algebraically like an ordinary real number, except that its square was -1. The letter i was used to denote √(-1), possibly because i is the first letter of the Latin word, 'imaginarius'.

To permit solutions of such polynomial equations, the set of complex numbers is introduced. We can consider a complex number as having the form a + bi where a and b are real numbers. It is denoted by z, i.e., z = a+bi. 'a' is called as real part of j which is denoted by (Re z) and 'b' is called as imaginary part of z which is denoted by (Img). Here, a and b are real numbers.
 

Complex Plane

The point ‘z’ represented in red is the complex number z = a+ib which corresponds to an intercept ‘a’ on the x-axis while it corresponds to an intercept ‘b’ on y-axis. Complex number does not actually mean complicated. In fact, it just implies that a complex number is a combination of a real and imaginary number. The line in blue represents the position vector of the point z.     

Complex plane is also termed as the Argand plane. A complex number is actually a point or a position vector in a two dimensional Cartesian coordinate system. The real part of the complex number is graphed as the horizontal component while the imaginary part is plotted against the vertical component. When we talk of a position vector of a complex number, it always takes into consideration, the magnitude and the direction with respect to the origin. All this is also emphasized in the polar form.
 

 Some Key Points of Complex Numbers

Complex numbers is an important topic of IIT JEE. Some of the key points of IIT JEE complex numbers are listed below:

• As explained, complex numbers are generally expressed as z = a+ib. Hence z is:

1. Purely real, if b = 0

2. Imaginary, if b ≠ 0

3. Purely imaginary, if a = 0

• The set R of real numbers is a proper subset of the complex numbers.

• Zero is purely real as well as purely imaginary but not imaginary.

• i = √(-1), is called the imaginary unit. Also, i² = -1; i³ = i; i⁴=1, etc.

• √a√b√c  =  √(abc........)

iff at least one of a,b,c ........ is non-negative.

• If j = a + ib, then a-ib is called complex conjugate of j.

• Real numbers satisfy order relations whereas imaginary order relation, i.e.,

i > 0, 3+i < 2 are meaningless.   

Addition and Multiplication of Complex Numbers

Complex numbers are not very different from real numbers except for the fact that they contain imaginary numbers as well. The operations of addition and multiplication are defined in the same way as in the case of real numbers and commutative, distributive and associative properties also hold true. We discuss the addition and multiplication of complex numbers:

Addition of two complex numbers is the same as in real numbers. The real part is added to the real part, while the imaginary parts get added to each other.

(a + ib) + (c + id) = (a + c) + i(b + d)

Multipli9cation of complex numbers is also very simple. Here, every term gets multiplied to every other term and i² = -1. Using these facts, let us multiply the complex numbers.

 (a+ib)(c+id) = a.(c + id) + ib (c + id)

                      = [ac + aid] + [bic + bi.di]

                      = ac + adi + bci + bidi

                      = ac + adi + bci – bd

                      = (ac-bd) + (ad+bc)i

You may also view the video on complex numbers 

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