Complex Numbers and Quadratic Equations

Every real number, when squared, gives a value that is zero or positive. So the equation x² + 1 = 0, that is x² = −1, has no solution among real numbers. To solve such equations, a new number called the imaginary unit, denoted i, is introduced and defined by i² = −1 (so i = √−1).

Using i, the square root of any negative real number can be written directly: for a > 0, √(−a) = i√a. For example, √−9 = i√9 = 3i, and √−16 = i√16 = 4i.

A complex number is a number of the form z = a + ib, where a and b are real numbers. Here a is the real part, written Re(z) = a, and b is the imaginary part, written Im(z) = b. The set of all complex numbers is denoted C. Every real number a can be written as a + i0, so every real number is also a complex number.

Two complex numbers a + ib and c + id are equal if and only if a = c and b = d — their real parts are equal and their imaginary parts are equal.

Powers of i repeat in a cycle of 4: i¹ = i, i² = −1, i³ = i²·i = −i, i⁴ = (i²)² = 1, then i⁵ = i again. In general, for any integer k: i^4k = 1, i^4k+1 = i, i^4k+2 = −1, i^4k+3 = −i.

Worked Example: Simplify i³⁷ + 1/i⁶⁷.

Since 37 = 4(9) + 1, i³⁷ = i^4×9+1 = i. Since 67 = 4(16) + 3, i⁶⁷ = i^4×16+3 = i³ = −i, so 1/i⁶⁷ = 1/(−i) = i (because i × (−i) = −i² = 1, so 1/(−i) = i). Hence i³⁷ + 1/i⁶⁷ = i + i = 2i.

Let z₁ = a + ib and z₂ = c + id be two complex numbers.

Addition: z₁ + z₂ = (a + c) + i(b + d) — add the real parts and add the imaginary parts.

Subtraction: z₁ − z₂ = (a − c) + i(b − d).

Multiplication: treat i like a variable and expand, then replace i² by −1: z₁ z₂ = (a + ib)(c + id) = ac + iad + ibc + i²bd = (ac − bd) + i(ad + bc).

Addition and multiplication of complex numbers are both commutative and associative, and multiplication distributes over addition, exactly as for real numbers. The additive identity is 0 = 0 + i0, and the additive inverse of z = a + ib is −z = −a − ib. The multiplicative identity is 1 = 1 + i0.

Worked Example: If z₁ = 3 + 2i and z₂ = 1 − 4i, find z₁ + z₂ and z₁ z₂.

z₁ + z₂ = (3 + 1) + i(2 − 4) = 4 − 2i.

z₁ z₂ = (3)(1) + (3)(−4i) + (2i)(1) + (2i)(−4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i.

For z = a + ib, the conjugate is z̄ = a − ib (only the sign of the imaginary part changes). A key fact is that z z̄ = (a + ib)(a − ib) = a² − (ib)² = a² + b², which is always a non-negative real number.

For a nonzero complex number z = a + ib, its multiplicative inverse is the complex number z⁻¹ such that z · z⁻¹ = 1. Multiplying numerator and denominator of 1/z by z̄ gives z⁻¹ = z̄/(a² + b²) = a/(a² + b²) − i·b/(a² + b²).

To divide z₁ by a nonzero z₂, multiply numerator and denominator by the conjugate of z₂: z₁/z₂ = (z₁ z̄₂)/(z₂ z̄₂), which turns the denominator into a real number.

Worked Example: Divide (2 + 3i) by (1 − 2i).

Multiply top and bottom by the conjugate (1 + 2i): numerator = (2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i² = 2 + 7i − 6 = −4 + 7i. Denominator = (1)² + (2)² = 5. So (2 + 3i)/(1 − 2i) = −4/5 + (7/5)i.

Also, the multiplicative inverse of 3 − 4i is: |3 − 4i|² = 9 + 16 = 25, so (3 − 4i)⁻¹ = (3 + 4i)/25 = 3/25 + (4/25)i.

For z = a + ib, the modulus is |z| = √(a² + b²), a non-negative real number that represents the distance of the point (a, b) from the origin when z is plotted in a plane.

Properties of modulus: for complex numbers z₁, z₂, we have |z₁ z₂| = |z₁||z₂|, and |z₁/z₂| = |z₁|/|z₂| whenever z₂ ≠ 0.

Properties of conjugate: the conjugate of a sum is the sum of the conjugates, (z₁ + z₂)‾ = z̄₁ + z̄₂; the conjugate of a product is the product of the conjugates, (z₁ z₂)‾ = z̄₁ z̄₂. Also z + z̄ = 2a = 2Re(z) and z − z̄ = 2ib = 2i·Im(z); in particular, z is real if and only if z = z̄.

Worked Example: Find |z| for z = −3 + 4i, and verify |z₁ z₂| = |z₁||z₂| for z₁ = 1 + i, z₂ = 2 − i.

|z| = √((−3)² + 4²) = √(9 + 16) = √25 = 5.

|z₁| = √2, |z₂| = √5, so |z₁||z₂| = √10. Also z₁z₂ = (1 + i)(2 − i) = 2 − i + 2i − i² = 2 + i + 1 = 3 + i, so |z₁z₂| = √(9 + 1) = √10. Both sides equal √10, confirming the property.

A complex number z = a + ib can be represented as the point P(a, b) in a plane, where the horizontal axis is called the real axis and the vertical axis the imaginary axis. This plane is called the Argand plane (or complex plane), and |z| equals the distance OP from the origin.

In polar form, z is written as z = r(cosθ + i sinθ), where r = |z| = √(a² + b²), and θ, called the argument of z (written arg z), satisfies cosθ = a/r and sinθ = b/r, so tanθ = b/a. The principal argument is the value of θ chosen in the interval (−π, π].

To find θ: locate the quadrant of the point (a, b) from the signs of a and b, find the reference (acute) angle using tanθ = b/a, and then adjust it to the correct quadrant to get the principal value.

Worked Example: Express z = 1 + i√3 in polar form.

r = √(1² + (√3)²) = √(1 + 3) = 2. cosθ = 1/2, sinθ = √3/2. Since both a and b are positive, θ lies in the first quadrant, so θ = π/3. Hence z = 2(cos(π/3) + i sin(π/3)).

Another Example: For z = −1 − i, r = √(1 + 1) = √2. Here a and b are both negative, so the point lies in the third quadrant; the reference angle is π/4, giving the principal argument θ = −(π − π/4) = −3π/4. So z = √2(cos(−3π/4) + i sin(−3π/4)).

For a quadratic equation ax² + bx + c = 0 with real coefficients a, b, c (a ≠ 0), the discriminant is D = b² − 4ac. The roots are given by the quadratic formula x = (−b ± √D)/(2a). If D > 0, the roots are real and distinct; if D = 0, the roots are real and equal; if D < 0, there is no real root, and the equation has two roots that are complex conjugates of each other.

When D < 0, write D = −(4ac − b²), where 4ac − b² > 0, so √D = √(−(4ac−b²)) = i√(4ac − b²). The roots become x = (−b ± i√(4ac − b²)) / (2a) — a conjugate pair, since real coefficients can never produce just one non-real root on its own.

As with real roots, the sum and product of the roots follow the same relations for complex roots α and β: sum of roots α + β = −b/a, and product of roots αβ = c/a.

Worked Example: Solve x² + 2x + 5 = 0.

Here a = 1, b = 2, c = 5, so D = 2² − 4(1)(5) = 4 − 20 = −16 < 0. So √D = √−16 = 4i. Roots: x = (−2 ± 4i)/2 = −1 ± 2i.

Check: Sum of roots = −2 = −b/a ✓. Product of roots = (−1 + 2i)(−1 − 2i) = (−1)² − (2i)² = 1 − 4i² = 1 + 4 = 5 = c/a ✓.