Algebra of complex numbers — A/L Combined Mathematics

What must I remember about Algebra of complex numbers?

• Addition and Subtraction of complex numbers: (x1 + iy1) ± (x2 + iy2) = (x1 ± x2) + i(y1 ± y2); add/subtract real and imaginary parts independently.
• Multiplication of complex numbers: (x1 + iy1)(x2 + iy2) = (x1*x2 - y1*y2) + i(x1*y2 + x2*y1), using i² = -1.
• Division of complex numbers: Multiply numerator and denominator by the complex conjugate of the denominator: (x1 + iy1)/(x2 + iy2) = [(x1 + iy1)(x2 - iy2)] / (x2² + y2²) = [(x1*x2 + y1*y2) + i(x2*y1 - x1*y2)] / (x2² + y2²).
• Square root of a complex number: To find √(a + ib), set (x + iy)² = a + ib => x² - y² = a and 2xy = b; solve simultaneously for real x and y.
• Algebraic properties: ℂ forms a Field satisfying commutativity, associativity, and distributivity of addition and multiplication.

Test yourself (1 minute)

1. Denominator rationalization

   Never leave an imaginary unit i in the denominator; multiply by complex conjugate z* to make denominator |z|^2.

2. Sign rule for square roots

   If b > 0 in √(a + ib), then x and y have the SAME sign; if b < 0, x and y have OPPOSITE signs.

3. Multiplicative inverse

   z^-1 = 1/z = z* / |z|^2 = (x - iy) / (x² + y²).

Practise Algebra of complex numbers questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 21

What should I study first?

What comes next?

Next in the syllabus

Where does Algebra of complex numbers fit in the syllabus?

Year 2 · Term 1