Solved Examples on Complex Numbers
Example 1: Simplify:
a) (3i)(2 + 5i)
b) 8i + 15i β 7i
Solution:
a) (3i)(2 + 5i)
= (3i)(2) + (3i)(5i)
= 6i + 15i2
We know that i2 = β1
= 6i + 15(β1)
(3i)(2 + 5i) = 6i β 15
b) 8i + 15i β 7i = 23i β 7i = 16i
Example 2: Solve: (2+i)/(2-i).
Solution:
Given: (2+i)/(2-i)
Now multiply both numerator and denominator with (2+i)
(2+i)/(2-i) Γ (2+i)/(2+i)
(2+i)(2β i) = (2)2 β i2 {Since, i2 = β1}
= 4 β(β1) = 4+1 = 5
(2+i)/(2-i) Γ (2+i)/(2+i) = (2+i)2/5
= (22 + 2Γ2Γi + i2)/5
= (4 + 4i β1)/5
= (3+4i)/5
Hence, (2+i)/(2-i) = (3+4i)/5.
Example 3: Determine the sum and product of the complex numbers z1 = 2 + 3i and z2 = 1 β i as a complex number.
Solution:
Given: z1 = 2 + 3i
z2 = 1 β i
Sum:
z1 + z2 = 2 + 3i + 1 β i
= (2 + 1) + (3i β i)
z1 + z2 = 3 + 2i
Product:
z1 Γ z2 = (2 + 3i) Γ (1 β i)
= 2(1 β i) + 3i(1 β i)
= 2 β 2i + 3i β 3i2
= 2 + i β 3(β1) {Since, i2 = β1}
= 2 + i + 3
z1 Γ z2 = 5 + i
Example 4: Determine the difference and quotient of the complex numbers z1 = 5 β 4i and z2 = 3+ i as a complex number.
Solution:
Given: z1 = 5 β 4i
z2 = 3+ i
Difference:
z1 β z2 = 5 β 4i β (3+ i)
= (5 β 3) + (β4i β i)
z1 β z2 = 2 β 5i
Division:
z1/z2 = (5 β 4i)/(3+ i)
= (5 β 4i)/(3+ i) Γ (3 β i)/(3 β i)
(3 + i)(3 β i) = 32 β i2
= 9 β (β1) = 9 + = 10 {Since, i2 = β1}
(5 β 4i)/(3+ i) Γ (3 β i)/(3 β i) = [(5 β 4i)(3 β i)]/10
= (15 β 5i β 12i + 4i2)/10
= (15 β17i + 4(β1))/10 {Since, i2 = β1}
z1/z2 = (11 β 17i)/10
Example 5: Determine the modulus, conjugate, and reciprocal of the complex number 4 + 3i.
Solution:
Given: z = 4 + 3i
Modulus = β(42 + 32) = β(16 + 9) = β25 = 5
Conjugate of 4 + 3i is zΜ = 4 β 3i
Reciprocal = 1/(4 + 3i) = (4 β 3i)/(42+32)
=(4 β 3i)/(16 + 9) = (4 β 3i)/25
Reciprocal of 4 + 3i = (4 β 3i)/25
Is Every Real Number a Complex Number?
A complex number is referred to as the sum of a real number and an imaginary number. It is generally expressed as βzβ and is written in the form of a + ib, where a and b are real numbers and i = β(-1). Here, βaβ is a real part that is represented as Re(z) and βibβ is an imaginary part that is represented as Im(z). Some examples of complex numbers are 2 + 3i, 5β7i, 3 + iβ4, etc. The imaginary number is generally expressed either as βiβ or βjβ, whose value is equal to β(-1). Hence, the square of an imaginary number gives us a negative value. The square root of negative numbers can be calculated using complex numbers. Some applications of complex numbers are in signal processing, fluid dynamics, quantum mechanics, electromagnetism, vibration analysis, and also many scientific research areas.
Real numbers are referred to as the union of the set of rational numbers and the set of irrational numbers, i.e., positive numbers, whole numbers, integers, rational numbers, irrational numbers, etc. are real numbers. Some examples of real numbers are -4, -7/11, 0, 9, β6, 3.8, etc.
A number that gives a negative value when squared is called an imaginary number. It is the product of a non-zero real number and the imaginary unit βiβ, whose value is β(-1). An imaginary number can also be defined as the square root of negative numbers. Some examples of imaginary numbers are -2i, β5i, 3i, etc.