The complex number system — A/L Combined Mathematics

What must I remember about The complex number system?

• Imaginary unit i: Defined such that i² = -1; powers of i cycle every 4 steps: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and i^4k+r = i^r.
• Complex Number definition: A number z of the form z = x + iy where x, y ∈ ℝ; x is the Real Part (Re(z)) and y is the Imaginary Part (Im(z)).
• Purely real and purely imaginary numbers: z is purely real if Im(z) = y = 0; z is purely imaginary if Re(z) = x = 0 with y ≠ 0.
• Equality of Complex Numbers: Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if their real parts are equal AND their imaginary parts are equal: x1 = x2 and y1 = y2.
• Set inclusion hierarchy: Natural numbers ℕ ⊂ Integers ℤ ⊂ Rationals ℚ ⊂ Reals ℝ ⊂ Complex numbers ℂ.

Test yourself (1 minute)

1. Complex numbers CANNOT be ordered

   Expressions like z1 > z2 or z1 < z2 have NO mathematical meaning in ℂ unless both numbers are strictly real.

2. Square root of -a (for a > 0)

   √(-a) = i√a; be careful: √(-a) * √(-b) = (i√a)(i√b) = i² √(ab) = -√(ab), NOT +√(ab).

3. Solving quadratic with negative discriminant

   For ax² + bx + c = 0 with Δ = b² - 4ac < 0, roots are complex conjugates: x = [-b ± i√(4ac - b²)] / (2a).

Practise The complex number system questions today

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

What comes next?

Next in the syllabus

Where does The complex number system fit in the syllabus?

Year 2 · Term 1