Complex arithmetic, conjugates and equating parts

z = 3 + 2i, w = 1 − 4i

• Add: group the parts.
• Multiply: expand, then simplify.

1. z + w = (3 + 1) + (2 − 4)igroup the parts
2. = 4 − 2i
3. zw = (3 + 2i)(1 − 4i)
4. = 3 − 12i + 2i − 8i²expand
5. = 3 − 10i + 8i² = −1
6. = 11 − 10i

(7 + i) ÷ (1 − i)

• Multiply by the bottom's conjugate.
• Divide both parts.

1. (7 + i)/(1 − i) × (1 + i)/(1 + i)
2. bottom: (1 − i)(1 + i) = 1 + 1
3. = 2real
4. top: (7 + i)(1 + i)
5. = 7 + 7i + i + i²expand
6. = 6 + 8ii² = −1
7. (6 + 8i)/2 = 3 + 4idivide both parts

The 2025 error: the answer form

• A fraction over 17 is not a + bi.
• Split it into its two parts.

1. z/w = (3 + 2i)/(1 − 4i)
2. × (1 + 4i)/(1 + 4i)conjugate of the bottom
3. bottom: 1 + 16 = 17
4. top: 3 + 12i + 2i + 8i²
5. = −5 + 14ii² = −1
6. z/w = (−5 + 14i)/17not finished
7. = −5/17 + (14/17)ithe form asked

6 ÷ (3 − √3)

• The same conjugate trick.
• The bottom becomes rational.

1. 6/(3 − √3) × (3 + √3)/(3 + √3)
2. bottom: (3 − √3)(3 + √3)
3. = 9 − 3 = 6√3 × √3 = 3
4. top: 6(3 + √3) = 18 + 6√3
5. (18 + 6√3)/6 = 3 + √3divide both terms

(a + bi)(1 + 2i) = 7 + 4i

• Expand and group.
• Equate both parts.
• Solve the pair.

1. a + 2ai + bi + 2bi²expand
2. = (a − 2b) + (2a + b)ii² = −1, group
3. a − 2b = 7real parts
4. 2a + b = 4imaginary parts
5. 4a + 2b = 8double the second
6. 5a = 15, so a = 3add to the first
7. b = 4 − 6 = −2substitute back
8. a = 3, b = −2

The square roots of 3 + 4i

• Let the root be x + iy.
• Square it, then equate parts.

1. (x + iy)² = 3 + 4ix and y real
2. x² + 2xyi + i²y² = 3 + 4iexpand
3. (x² − y²) + 2xyi = 3 + 4ii² = −1
4. x² − y² = 3real parts
5. 2xy = 4imaginary parts
6. y = 2/x

Solving for x

• Substitute y = 2/x.
• A quadratic in x².

1. x² − y² = 3
2. y = 2/x
3. x² − 4/x² = 3substitute y = 2/x
4. x⁴ − 3x² − 4 = 0multiply by x²
5. (x² − 4)(x² + 1) = 0factorise
6. x² = 4 or x² = −1

Reject, keep both signs, check

• x is real: reject x² = −1, and say so.
• Keep both signs of the root.
• Square one root to check.

1. x² = −1x is real
2. x = 2: y = 2/2 = 1
3. x = −2: y = −1
4. roots 2 + i and −2 − i
5. (2 + i)² = 4 + 4i + i²check
6. = 3 + 4i

The answers: questions 1 and 2

1. 1. (2 + 3i)(4 − i)
2. = 8 − 2i + 12i − 3i²expand
3. = 8 + 10i + 3 = 11 + 10ii² = −1
4. 2. × (2 − i)/(2 − i)conjugate of the bottom
5. top: (5 + 5i)(2 − i) = 15 + 5i
6. bottom: 4 + 1 = 5
7. (15 + 5i)/5 = 3 + i

The answers: questions 3 and 4

1. 3. × (√5 + 1)/(√5 + 1)conjugate of the bottom
2. bottom: 5 − 1 = 4
3. 4(√5 + 1)/4 = √5 + 1the fours cancel
4. 4. (2a − b) + (a + 2b)i = 1 + 8iexpand, group
5. 2a − b = 1, a + 2b = 8equate parts
6. 5a = 10, so a = 2double the first, add
7. b = 2(2) − 1 = 3