Question

Express the following complex numbers in the standard form a + ib:

(i) [(1 + i) (1 + √3i)]/(1 – i)

(ii) (2 + 3i)/(4 + 5i)

(iii) (1 – i)³/(1 – i³)

(iv) (1 + 2i)^-3

(v) (3 – 4i)/[(4 – 2i) (1 + i)]

Answer 1

(i) [(1 + i) (1 + √3i)]/(1 – i)

Now let us simplify and express in the standard form of (a + ib),

[(1 + i) (1 + √3i)]/(1 – i) = [1(1 + √3i) + i(1 + √3i)]/(1 - i)

= (1 + √3i + i + √3i²)/(1 – i)

= (1 + (√3 + 1)i + √3(-1))/(1 - i) [since, i² = -1]

= [(1 - √3) + (1 + √3)i]/(1 - i)

[by multiply and divide with (1 + i)]

= [(1 - √3) + (1 + √3)i]/(1 - i) × (1 + i)/(1 + i)

= [(1 - √3) (1 + i) + (1 + √3)i(1 + i)]/(1² – i²)

= [1 - √3 + (1 - √3)i + (1 + √3)i + (1 + √3)i²]/(1 - (-1)) [since, i² = -1]

= [(1 - √3) + (1 - √3 + 1 + √3)i + (1+ √3)(-1)]/2

= (-2√3 + 2i)/2

= -√3 + i

Thus the values of a, b are -√3, i

(ii) (2 + 3i)/(4 + 5i)

Now let us simplify and express in the standard form of (a + ib),

(2 + 3i)/(4 + 5i) = [multiply and divide with (4 - 5i)]

= (2 + 3i)/(4 + 5i) × (4 - 5i)/(4 - 5i)

= [2(4 - 5i) + 3i(4 - 5i)]/(4² – (5i)²)

= [8 – 10i + 12i – 15i²]/(16 – 25i²)

= [8 + 2i - 15(-1)]/(16 – 25(-1)) [since, i² = -1]

= (23 + 2i)/41

∴ The values of a, b are 23/41, 2i/41

(iii) (1 – i)³/(1 – i³)

Now let us simplify and express in the standard form of (a + ib),

(1 – i)³/(1 – i³) = [1³ – 3(1)²i + 3(1)(i)² – i³]/(1 - i².i)

= [1 – 3i + 3(-1)-i².i]/(1 – (-1)i) [since, i² = -1]

= [-2 – 3i – (-1)i]/(1 + i)

= [-2 - 4i]/(1 + i)

[By Multiply and divide with (1 - i)]

= [-2 - 4i]/(1 + i) × (1 - i)/(1 - i)

= [-2(1 - i) -4i(1 - i)]/(1² – i²)

= [-2 + 2i - 4i + 4i²]/(1 – (-1))

= [-2 - 2i + 4(-1)]/2

= (-6 - 2i)/2

= -3 – i

∴ The values of a, b are -3, -i

(iv) (1 + 2i)^-3

Now let us simplify and express in the standard form of (a + ib),

(1 + 2i)^-3 = 1/(1 + 2i)³

= 1/(1³ + 3(1)² (2i) + 2(1)(2i)² + (2i)³)

= 1/(1 + 6i + 4i² + 8i³)

= 1/(1 + 6i + 4(-1) + 8i².i) [since, i² = -1]

= 1/(-3 + 6i + 8(-1)i) [since, i² = -1]

= 1/(-3 - 2i)

= -1/(3 + 2i)

[By multiply and divide with (3 - 2i)]

= -1/(3 + 2i) × (3 - 2i)/(3 - 2i)

= (-3 + 2i)/(3² – (2i)²)

= (-3 + 2i)/(9-4i²)

= (-3 + 2i)/(9 - 4(-1))

= (-3 + 2i)/13

∴ The values of a, b are -3/13, 2i/13

(v) (3 – 4i)/[(4 – 2i) (1 + i)]

Now let us simplify and express in the standard form of (a + ib),

(3 – 4i)/[(4 – 2i) (1 + i)] = (3 - 4i)/[4(1 + i) - 2i(1 + i)]

= (3 - 4i)/[4 + 4i - 2i - 2i²]

= (3 - 4i)/[4 + 2i - 2(-1)] [since, i² = -1]

= (3 - 4i)/(6 + 2i)

[By multiply and divide with (6 - 2i)]

= (3 - 4i)/(6 + 2i) × (6 - 2i)/(6 - 2i)

= [3(6 - 2i) - 4i(6 - 2i)]/(6² – (2i)²)

= [18 – 6i – 24i + 8i²]/(36 – 4i²)

= [18 – 30i + 8 (-1)]/(36 – 4 (-1)) [since, i² = -1]

= [10 - 30i]/ 40

= (1 – 3i)/4

Thus the values of a, b are 1/4, -3i/4