Complex Number

Complex Number

While solving the equation of the form ax2+bx+c=0, the roots of the equations can take three forms which are as follows:

Two Distinct Real Roots

Similar Root

No Real roots (Complex Roots)

The introduction of complex numbers in the 16th century made it possible to solve the equation 

x2+1=0. The roots of the equation are of the form x=±−1‾‾‾√ and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.
We denote −1‾‾‾√ with the symbol i, where i denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b.

Algebraic Operation on Complex numbers:

• Addition of two complex numbers
• Subtraction of two complex number
• Multiplication of two complex number
• Division of two complex number

Power of Iota (i) –

Depending upon the power of i, it can take the following values

i4k+1=i

i4k+2=−1

i4k+3=−i

i4k=1

where k can have any integral value (positive or negative).

Similarly we can find for the negative power of i, which are as follows

i−1=1i

Multiplying and dividing the above term with i, we have

i−1=1i×ii

i−1=ii2=i−1=−i

Note: −1‾‾‾√×−1‾‾‾√=−1×−1‾‾‾‾‾‾‾‾√=1‾√=1 contradicts to the fact that i2=−1.

Therefore for an imaginary number, a‾√×b‾√≠ab‾‾‾√.

Identities:

(i) (z1+z2)2=z21+z22+2z1z2

(ii) (z1−z2)2=z21+z22−2z1z2

(iii) z21−z22=(z1+z2)(z1–z2)

(iv) (z1+z2)3=z31+3z21z2+3z1z22+z32

(v) (z1−z2)3=z31−3z21z2+3z1z22−z32

Modulus and Conjugate of a Complex number:

Let z=a+ib be a complex number.

The Modulus of z is represented by |z|.

Mathematically, |z|=a2+b2‾‾‾‾‾‾‾√

The conjugate of z is denoted by z¯.

Mathematically, z¯=a–ib

Example- Express the following in a+ib form. 5+3√i1−3√i. And then find the Modulus and Conjugate of the complex number.] Solution- Given 5+3√i1−3√i z= 5+3√i1−3√i×1+3√i1+3√i=5−3+63√i1+3=12+32i Modulus, z¯=(12)2+(32)2‾‾‾‾‾‾‾‾‾‾‾‾√ z¯=(104)‾‾‾‾‾√ z¯=10√2 Conjugate, z¯=12–32i

Argand Plane & Polar Representation of Complex Number:

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis.