The complex numbers extend the reals with i, a new number whose square is −1.
A negative has no real square root, but with i every quadratic equation gains a solution.
Written as a + bi, it carries a real part and an imaginary part, and times its conjugate it turns real again.
Here you plot them on the complex plane and watch addition and multiplication in action.
Birth of the Imaginary Unit i
We needed a solution to x² = -1, so we defined i² = -1. i steps off the real axis and opens a new dimension.
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Powers of i cycle through four values
💡 Cycle of i
①i⁰ = 1, i¹ = i, i² = -1, i³ = -i
②i⁴ returns to 1 → 4-cycle
③i^n: determined by n mod 4
Complex Plane — Numbers as Points
A complex number z = a + bi is plotted as a point (a, b) on the complex plane. Horizontal axis = real part, vertical = imaginary.
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Complex plane: z = a + bi and conjugate z̄ = a - bi
🔑 Key Ideas
①z = a + bi → point (a, b)
②z̄ = a - bi → reflection across real axis
③|z| = distance to origin = √(a² + b²)
Operations on Complex Numbers
Addition
(a+bi) + (c+di) = (a+c) + (b+d)i
Add real and imaginary parts separately
Multiplication
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
Distribute, then apply i² = -1
Product with Conjugate
z · z̄ = (a+bi)(a-bi) = a² + b² = |z|²
Always real (imaginary part vanishes)
Equality of Complex Numbers
Equality
a + bi = c + di ⟺ a = c, b = d
Real and imaginary parts must each be equal
📐 Using Equality
①Compare real & imaginary parts to solve equations
②z = z̄ ⟺ b = 0 (purely real)
③z = -z̄ ⟺ a = 0 (purely imaginary)
Work It Out
Example 1
Express (2 + 3i)(1 − 2i) in the form a + bi. (i is the imaginary unit)
1
Expand using the distributive law.
2 − 4i + 3i − 6i²
2
Substitute i² = −1 and simplify.
2 − i + 6 = 8 − i
▸ 8 − i
Substituting i² = −1 is the heart of complex multiplication.
Example 2
Find the value of i + i² + i³ + … + i¹⁰.
1
Powers of i cycle through i, −1, −i, 1 with period 4, and any four consecutive terms sum to 0.
i + i² + i³ + i⁴ = 0
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The sum from i¹ to i⁸ is 0, leaving i⁹ + i¹⁰.
i⁹ + i¹⁰ = i + (−1) = −1 + i
▸ −1 + i
Judge a power by the remainder of the exponent divided by 4.
Wrap-up
Imaginary Unit
i² = -1; powers cycle every 4
i⁰=1, i¹=i, i²=-1, i³=-i
Modulus
|z| = |a + bi| = √(a² + b²)
Distance from origin in the complex plane
Grade-10 school exam type
For the complex number z = 3 − 4i, find z·z̄. (z̄ is the conjugate of z)
①5
②7
③25
④−7
⑤−25
▸ ③ 25
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The conjugate is z̄ = 3 + 4i, so multiply the two.
z·z̄ = (3 − 4i)(3 + 4i)
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Use the product of a sum and a difference.
= 3² − (4i)² = 9 + 16 = 25
🎯 Exam Points
①Powers of i: take the exponent mod 4
②Division: multiply by conjugate to rationalize
③Memorize z · z̄ = |z|²
④Equality: compare real & imaginary parts separately