Flash Cards: Complex Numbers and Quadratic Equations

A complex number is of the form a + ib, where a and b are real numbers and i is the imaginary unit, defined as √(-1).

'i' is defined as √(-1). Thus, i² = -1.

Two complex numbers z₁ = a + ib and z₂ = c + id are equal if a = c and b = d.

The modulus of a complex number z = a + ib is |z| = √(a² + b²).

The conjugate of z = a + ib is denoted as ¯z = a - ib.

For z₁ = a + ib and z₂ = c + id, the sum is z₁ + z₂ = (a + c) + i(b + d).

The sum of two complex numbers is also a complex number.

The difference z₁ - z₂ is given by z₁ + (-z₂).

For z₁ = a + ib and z₂ = c + id, the product is z₁z₂ = (ac - bd) + i(ad + bc).

The quotient z₁/z₂ (where z₂ ≠ 0) is defined by z₁ · (1/z₂).

i² = -1; i³ = -i; i⁴ = 1, with the pattern repeating every four powers.

For a positive a, √(-a) = i√a.

(z₁ + z₂)² = z₁² + z₂² + 2z₁z₂.

For any two complex numbers z₁ and z₂, z₁ + z₂ = z₂ + z₁.

The additive identity is the complex number 0 + 0i (or simply 0), satisfying z + 0 = z.

The multiplicative identity for complex numbers is 1 + 0i (or simply 1), such that z · 1 = z.

In the Argand plane, a complex number z = x + iy is represented as the point (x, y).

If z = x + iy, then its conjugate ¯z = x - iy is the reflection of z across the real axis.