Operations with Complex Numbers
Key Terms
- Addition/subtraction
- (a + bi) ± (c + di) = (a ± c) + (b ± d)i; combine real and imaginary parts separately.
- Multiplication
- Expand using FOIL and substitute i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
- Division
- Multiply numerator and denominator by the conjugate of the denominator to make the denominator real.
- Conjugate of denominator
- (c + di)(c − di) = c² + d² (a positive real number).
- Substituting i²
- The most commonly missed step: after expanding, replace every i² with −1 before collecting terms.
- Real denominator
- After dividing by (c² + d²), separate the result into its real part and imaginary part.
Operations with Complex Numbers
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
Multiplication: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i
Division: Multiply numerator and denominator by the conjugate of the denominator:
(a + bi)/(c + di) = (a + bi)(c − di) / (c + di)(c − di) = (a + bi)(c − di) / (c² + d²)
Worked Example 1 — Addition and Multiplication
Let z = 3 + 2i and w = 1 − 4i. Find: (a) z + w (b) z − w (c) zw
(a) z + w = (3 + 1) + (2 + (−4))i = 4 − 2i
(b) z − w = (3 − 1) + (2 − (−4))i = 2 + 6i
(c) zw = (3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i
Worked Example 2 — Division
Simplify (3 + 4i) / (1 − 2i)
Multiply top and bottom by the conjugate of the denominator, (1 + 2i):
(3 + 4i)(1 + 2i) / (1 − 2i)(1 + 2i) = (3 + 6i + 4i + 8i²) / (1 + 4) = (3 + 10i − 8) / 5 = (−5 + 10i) / 5 = −1 + 2i
Addition and Subtraction — Component-wise
Adding and subtracting complex numbers works component-by-component, just like adding vectors. Real parts combine with real parts, imaginary parts with imaginary parts. This is straightforward and mirrors how we handle algebraic expressions.
Think of it this way: i is just a symbol (for now), and adding 3i and 5i gives 8i in exactly the same way that adding 3x and 5x gives 8x. Complex addition simply extends this to two components simultaneously.
Multiplication — Expand and Replace i²
Multiplying complex numbers is just like expanding two brackets using FOIL or the distributive law, with one extra step: replace every i² with −1.
(a + bi)(c + di) = ac + adi + bci + bdi²
Now substitute i² = −1:
= ac + adi + bci + bd(−1) = (ac − bd) + (ad + bc)i
The result is a complex number in standard form. You do not need to memorise the formula — just expand and collect.
Division — The Conjugate Technique
Division by a complex number requires removing the imaginary part from the denominator. The strategy: multiply the fraction by (denominator’s conjugate)/(denominator’s conjugate) = 1, which does not change the value but converts the denominator to a real number.
Why does this work? Because (c + di)(c − di) = c² + d², which is always a positive real number when (c + di) ≠ 0. Once the denominator is real, you can split the fraction into real and imaginary parts.
Always write the final answer in standard form a + bi with both parts explicitly stated. For example, write −1 + 2i, not just −1 + 2i/1.
Algebraic Laws That Do (and Don’t) Apply
Complex numbers satisfy all the usual algebraic laws: commutativity (z + w = w + z, zw = wz), associativity, and distributivity. These are inherited from the algebraic structure.
What does NOT apply: inequalities. There is no ordering on ℂ. You cannot say z > w for complex numbers (unless they happen to be real). Never write z > 0 for a complex number.
Exam Strategy: Checking Your Answer
After computing z/w, verify by multiplying your answer by w and confirming you get z back. For example, if you claim (3 + 4i)/(1 − 2i) = −1 + 2i, check: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i² = −1 + 4i + 4 = 3 + 4i. ✓
Mastery Practice
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Fluency
Q1 — Addition and Subtraction
Let z = 4 + 3i and w = 2 − 5i. Calculate: (a) z + w (b) z − w (c) 2z + w
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Fluency
Q2 — Multiplying Complex Numbers
Expand and simplify: (a) (2 + 3i)(1 + i) (b) (3 − i)(3 + i) (c) i(4 − 2i)
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Fluency
Q3 — Dividing Complex Numbers
Express each in the form a + bi: (a) 1/(1 + i) (b) (2 + i)/(3 − i)
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Fluency
Q4 — Scalar Multiplication
If z = −2 + 5i, find: (a) 3z (b) −2z (c) (1/2)z
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Understanding
Q5 — Combined Operations
Given z = 1 + 2i and w = 3 − i, find: (a) z² (b) zw − z̅ (c) (z + w)(z − w)
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Understanding
Q6 — Division with Result Verification
Find (5 + 3i) / (2 + i) in Cartesian form, then verify your answer.
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Understanding
Q7 — Solving for Unknown Complex Number
Find z if 2z + 3i = (1 + i)z + (4 − i).
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Understanding
Q8 — Real and Imaginary Parts After Operations
If z = (2 + ai)/(1 + bi) is real (Im(z) = 0), find a relationship between a and b.
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Problem Solving
Q9 — Simplifying a Complex Fraction
Simplify ((1 + i) / (1 − i))² and express in the form a + bi.
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Problem Solving
Q10 — Complex Equation with Two Unknowns
Find real numbers p and q such that (p + qi)² = 5 + 12i.