Complex Numbers in Cartesian Form

Why Do We Need Complex Numbers?

The real number system has a fundamental gap: you cannot take the square root of a negative number. Consider the equation x² + 1 = 0. Rearranging gives x² = −1, but no real number satisfies this — squaring any real number always gives a non-negative result. For centuries, mathematicians dismissed such equations as “impossible.”

The solution was to define a new symbol i such that i² = −1. This is not a trick or a sleight of hand — it is a legitimate extension of the number system, just as fractions extended whole numbers and irrationals extended rationals. The resulting system of complex numbers is complete in a profound sense: every polynomial equation has a solution in ℂ (the Fundamental Theorem of Algebra).

The Cartesian Form a + bi

Every complex number can be written as z = a + bi where a and b are real numbers. We call this Cartesian form (or rectangular form). The number a is the real part Re(z), and b is the imaginary part Im(z) — note that Im(z) is the real number b, not the expression bi.

Special cases: when b = 0, z = a is a real number; when a = 0, z = bi is a pure imaginary number. This shows that the real numbers are a subset of the complex numbers.

The Imaginary Unit and Powers of i

Since i² = −1, we can simplify any power of i by reducing the exponent modulo 4:

Power	Value	Reasoning
i¹	i	definition
i²	−1	definition
i³	−i	i² × i = −1 × i
i&sup4;	1	i² × i² = (−1)(−1)
i&sup5;	i	i&sup4; × i = 1 × i

To simplify iⁿ: divide n by 4 and check the remainder. Remainder 0 → 1, remainder 1 → i, remainder 2 → −1, remainder 3 → −i.

The Conjugate and its Key Property

The conjugate of z = a + bi is z̅ = a − bi. Geometrically, taking the conjugate reflects the number across the real axis on the Argand diagram (covered in a later lesson).

The most important algebraic property is: z · z̅ = (a + bi)(a − bi) = a² + b². This is a real number, and it is always ≥ 0. This property is the engine behind dividing complex numbers — multiplying numerator and denominator by the conjugate of the denominator converts the denominator to a real number.

Equality of Complex Numbers

Two complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal. This gives us a powerful technique: if we know a + bi = c + di, we can equate real parts (a = c) and imaginary parts (b = d) to solve for unknowns. This technique appears constantly throughout Unit 2.

Common error: Students sometimes write i² = +1 instead of −1. Always remember: i² = −1 by definition.