Complex Numbers -Model Problems - 1

To find iⁿ divide n by 4 to get 4m+r where m is the quotient and r is the remainder.

iⁿ will be equal to i^r

Prob: The value of i⁵³/i¹²¹ is

a. 2i
b. i
c. -2i
d. 2

53 = 13*4 + 1

So i⁵³ = i

121 = 30*4 + 1

So i¹²¹ = i

i⁵³/i¹²¹ is equal to i/i = 1

Answer (b)





Multiplication of complex numbers

(a1+ib1) (a2+ib2) by multiplying and simplifying we get

(a1a2 – b1b2) + i(a1b2+a2b1)


Prob: Multiply (2+i) by (2+i)

(2+i)(2+i) = 2*2 - 1*1 + i(2*1+2*1) = 3+4i


Division of complex numbers

z1/z2 = z1* Multiplicative inverse of z2

Multiplicative inverse of a+ib = a/(a² + b²) - ib/(a² + b²)

Prob: Find the result of (7+i)/(1+3i)

First step; Find multiplicative inverse of 1+3i = 1/(1²+3²) - i 3/(1²+3²)
= 1/10 - i 3/10

Therefore (7+i)/(1+3i) = (7+i)(1/10 - 3i/10)
= 7*1/10 - (1)(-3/10) + i(7*(-3/10)+1(1/10))
= 7/10 + 3/10 + i(-21/10+1/10)
= 10/10 + i(-20/10)
= 1 - 2i