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1Exercise 5.1
Express (3 + 2i) + (5 - 4i) and (3 + 2i)(5 - 4i) in standard form a + ib.
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Sum: (3 + 2i) + (5 - 4i) = 8 - 2i
Product: (3 + 2i)(5 - 4i) = 15 - 12i + 10i - 8i²
= 15 - 2i + 8
= 23 - 2i
2Exercise 5.1
Express 1/(2 + i) in the form a + ib.
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Multiply and divide by conjugate (2 - i):
1/(2 + i) × (2 - i)/(2 - i) = (2 - i)/(4 + 1)
= (2 - i)/5
= 2/5 - (1/5)i
3Exercise 5.1
Find the modulus and argument of z = 1 + √3 i.
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a = 1, b = √3
|z| = √(1 + 3) = 2
arg(z) = tan⁻¹(b/a) = tan⁻¹(√3) = π/3 (first quadrant)
z = 2(cos π/3 + i sin π/3)
4Exercise 5.1
Solve the quadratic equation x² + x + 1 = 0 using the quadratic formula.
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x = (-1 ± √(1 - 4))/2 = (-1 ± √-3)/2
= (-1 ± √3 i)/2
Roots: x = (-1 + √3 i)/2 and x = (-1 - √3 i)/2
(These are ω and ω², the cube roots of unity.)
5Exercise 5.1
If z = 3 + 4i, find z·z̄ and show that z·z̄ = |z|².
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z̄ = 3 - 4i
z·z̄ = (3 + 4i)(3 - 4i) = 9 - 12i + 12i - 16i² = 9 + 16 = 25
|z|² = (√(9+16))² = 25
Hence z·z̄ = |z|² = 25.
6Exercise 5.1
Find the real numbers x and y such that (x + iy)(2 + i) = 3 + 5i.
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Expand: (x + iy)(2 + i) = 2x + xi + 2yi + i²y = (2x - y) + (x + 2y)i
Equate real and imaginary parts:
2x - y = 3
x + 2y = 5
From first: y = 2x - 3. Substitute: x + 2(2x-3) = 5
x + 4x - 6 = 5 ⇒ 5x = 11 ⇒ x = 11/5
y = 2(11/5) - 3 = 22/5 - 15/5 = 7/5
x = 11/5, y = 7/5