Chapter 1 - Complex Numbers

What is the definition of i and what is the value of i²?

i = √-1 and i² = -1

What is an imaginary number?

A number of the form bi, where b ∈ ℝ.

What is a complex number?

A number written in the form a + bi, where a, b ∈ ℝ.

How are complex numbers added or subtracted?

By adding or subtracting their real parts and imaginary parts separately.

How do you multiply a real number by a complex number?

Multiply as usual by expanding brackets; normal algebraic rules apply.

What happens when b² - 4ac < 0 in a quadratic equation ax² + bx + c = 0?

The equation has two distinct complex roots, neither of which is real.

What is the complex conjugate of z = a + bi?

z* = a - bi

How do complex roots occur for real coefficients a, b, c in a quadratic equation?

As a conjugate pair — if one root is a + bi, the other is a - bi.

If the roots of a quadratic are α and β, how can the equation be expressed?

(x - α)(x - β) = 0 or x² - (α + β)x + αβ = 0

If f(z) is a polynomial with real coefficients and z₁ is a root, what other root must exist?

The conjugate z₁* must also be a root.

What are the possible root combinations for a cubic equation with real coefficients?

Either all three roots are real, or one root is real and the other two form a complex conjugate pair.

What are the possible root combinations for a quartic equation with real coefficients?

All four roots are real, or two roots are real and the other two form a complex conjugate pair, or two pairs of complex conjugate roots.