Fundamentals of Complex Numbers

The foundation of complex numbers is the imaginary number i, defined as √(-1), which means that i² = -1. A complex number takes the form a + bi, where a is the real part and b is the imaginary part. For example, 3 + 4i is a complex number with 3 as its real component and 4i as its imaginary component.

When working with powers of i, remember these patterns: i³ = -i and i⁴ = 1. Generally, odd powers result in either i or -i, while even powers give 1 or -1.

When solving quadratics that yield complex solutions, you can either factorise, complete the square, or use the quadratic formula. For example, to solve x² + 6x + 25 = 0 by completing the square: rewrite as $x + 3$² - 9 + 25 = 0, which simplifies to $x + 3$² + 16 = 0. Since the square equals -16, we get x = -3 ± 4i.

💡 Complex solutions to quadratic equations always come in conjugate pairs—if a + bi is a solution, then a - bi will also be a solution.

Complex Conjugates and Argand Diagrams

The complex conjugate of z = a + bi is written as z̄ = a - bi. Conjugates are incredibly useful because when you multiply a complex number by its conjugate, you get a real number: $a + bi$$a - bi$ = a² + b².

This property helps simplify complex division problems by multiplying both numerator and denominator by the conjugate of the denominator. Also remember that $a + b$ + $a - b$ = 2a, which can help isolate real components.

Argand diagrams allow us to represent complex numbers visually on a coordinate plane. The real part corresponds to the x-axis, and the imaginary part to the y-axis. For instance, z₁ = 6 + 8i would be plotted at the point (6, 8), while z₂ = -2 + 4i would be at (-2, 4).

By plotting complex numbers this way, we transform algebraic expressions into geometric relationships, making many complex number operations easier to understand and visualize.

Complex Numbers as Vectors

Complex numbers can be treated as vectors on the Argand diagram, with vector addition following the same rules as complex addition. When you add z₁ + z₂, you're effectively connecting the corresponding vectors head-to-tail.

The modulus of a complex number represents its length or magnitude as a vector. For z = a + bi, the modulus is calculated as |z| = √$a² + b²$, using the Pythagorean theorem. For example, if z = 7 + 24i, then |z| = √(7² + 24²) = 25.

Radian measure becomes important when we start looking at complex numbers in polar form. One radian is defined as the angle where the arc length equals the radius of the circle. This measurement system helps express complex numbers in terms of their angle and distance from the origin.

🔍 The vector interpretation of complex numbers creates a beautiful bridge between algebra and geometry, allowing us to solve problems in either domain.

Arguments and Polar Form

The argument of a complex number is the angle measured anticlockwise from the positive x-axis to the vector representing the complex number. For quick reference: 360° = 2π, 180° = π, and 90° = π/2 radians.

To find the argument θ of a complex number z = x + iy, we use the formula tanθ = y/x (the ratio of imaginary to real parts). However, be careful with this approach as you need to consider which quadrant the complex number lies in.

A complex number can be written in two equivalent forms: the Cartesian form z = x + iy and the polar form z = r$cosθ + isinθ$, where r is the modulus. This polar representation becomes extremely useful when multiplying, dividing, or raising complex numbers to powers.

This connection between trigonometric functions and complex numbers reveals deep mathematical relationships that will become essential when you study advanced topics like Euler's formula.

Operations in Polar Form

Converting between Cartesian and polar forms is straightforward. For example, with z = 3 + 4i, we find r = |z| = √(3² + 4²) = 5 and θ = tan⁻¹(4/3) ≈ 0.927 rad. Therefore, z = 5$cos0.927 + isin0.927$.

When multiplying complex numbers in polar form, you multiply the moduli and add the arguments. If z₁ = r₁$cosA + isinA$ and z₂ = r₂$cosB + isinB$, then z₁ × z₂ = r₁r₂$cos(A+B) + isin(A+B)$.

For division, divide the moduli and subtract the arguments: z₁/z₂ = $r₁/r₂$$cos(A-B) + isin(A-B)$. Remember that cos(-θ) = cosθ and sin(-θ) = -sinθ when working with negative angles.

🌟 The polar form transforms complex multiplication and division from complicated algebraic operations into simple arithmetic with moduli and addition/subtraction of angles!

Solving Complex Equations

To divide complex numbers, multiply both numerator and denominator by the conjugate of the denominator. For example, to find $3+5i$/$1+i$, multiply by $1-i$/$1-i$: $3+5i$/$1+i$ × $1-i$/$1-i$ = $3-3i+5i-5i²$/$1-i²$ = $3-3i+5i+5$/2 = $8+2i$/2 = 4+i

Finding the square root of a complex number involves solving a system of equations. For √$3+4i$, we set $a+bi$² = 3+4i and expand: a²+2abi-b² = 3+4i. Comparing real and imaginary parts gives us:

• a²-b² = 3
• 2ab = 4

From the second equation, a = 2/b. Substituting into the first: $2/b$²-b² = 3, which leads to b⁴+3b²-4 = 0. Factoring gives $b²+4$$b²-1$ = 0, so b = ±1.

When b = 1, we get a = 2, leading to one solution: 2+i.

💡 Always check your solutions by squaring them to verify they equal the original complex number!

Completing Complex Solutions

When b = -1 from our equation in the previous page, we find a = -2, giving us the second solution: -2-i.

Therefore, the complete solution for √$3+4i$ = ±$2+i$. This means both 2+i and -$2+i$ are square roots of 3+4i. You can verify this by squaring each answer and checking that you get 3+4i.

This example demonstrates the process for finding square roots of complex numbers, which involves setting up and solving simultaneous equations. The technique requires careful algebraic manipulation and a good understanding of complex number properties.

Remember that complex numbers typically have two square roots, just as real numbers do. The only exception is zero, which has just one square root (zero itself).