Complex numbers

Habari Mhandisi Mtarajiwa! Welcome to the World Beyond Real Numbers!

Ever been told in secondary school that you can't find the square root of a negative number? Well, get ready to bend the rules! In engineering, we face problems that seem impossible with ordinary numbers. That's where Complex Numbers come in. Think of them as a secret level in the game of mathematics, unlocking solutions for everything from electrical circuits to signal processing. So, grab your calculator, and let's dive into this new, exciting dimension!

So, What's the "i"dea? (Or in our case, "j"!)

The whole journey starts with one "impossible" problem: what is the square root of -1? Mathematicians created a solution and called it 'i' (for imaginary). But as engineers, we already use 'i' for electric current. To avoid confusion when you're dealing with KPLC's power lines, we use the letter 'j' instead.

• The Imaginary Unit: j
• Definition:

  j = √(-1)

• The most important rule:

  j² = -1

This simple rule is the foundation of everything we are about to learn. Whenever you see j², just replace it with -1. Easy, right?

Anatomy of a Complex Number

A complex number is a single number that has two parts, a real part and an imaginary part. It's like Githeri – you have maize (the real part) and beans (the imaginary part) mixed together to make one dish.

The standard form is:

z = a + bj

• z is the complex number.
• a is the Real Part, Re(z).
• b is the Imaginary Part, Im(z).

Example: For the complex number z = 3 + 4j:

• The real part is 3.
• The imaginary part is 4.

Visualizing the Impossible: The Argand Diagram

How do you draw a number that's "imaginary"? We use a special graph called an Argand Diagram. It looks just like the Cartesian (x-y) plane you know, but with a twist:

• The horizontal axis (x-axis) is the Real Axis.
• The vertical axis (y-axis) is the Imaginary Axis.

So, the complex number z = a + bj is plotted as the point (a, b).

      Imaginary Axis
            ^
            |
          4j+             ● z = 3 + 4j
            |
          3j+
            |
          2j+
            |
           j+
            |
 ---+---+---+---+---+---> Real Axis
    0   1   2   3   4

> **Image Suggestion:** A vibrant, clear diagram of the Argand Plane. The horizontal axis is labeled "Real Axis (Re)" and the vertical axis is labeled "Imaginary Axis (Im)". A point 'z = 3 + 4j' is plotted at coordinates (3, 4). A vector (an arrow) is drawn from the origin (0,0) to this point. The style should be a modern, clean educational graphic.

Let's Do Some Maths! Operations with Complex Numbers

1. Addition and Subtraction

This is the easy part. Just add or subtract the "like terms" – reals with reals, and imaginaries with imaginaries. It's like sorting your coins: you put the shillings with the shillings and the cents with the cents.

Example: Let z₁ = 4 + 5j and z₂ = 2 - 3j.

Addition:

z₁ + z₂ = (4 + 5j) + (2 - 3j)
        = (4 + 2) + (5j - 3j)
        = 6 + 2j

Subtraction:

z₁ - z₂ = (4 + 5j) - (2 - 3j)
        = 4 + 5j - 2 + 3j  (Watch the signs!)
        = (4 - 2) + (5j + 3j)
        = 2 + 8j

2. Multiplication

Here, we use the FOIL method (First, Outer, Inner, Last), just like in algebra. But remember our golden rule: j² = -1.

Example: Let's multiply z₁ = (3 + 2j) and z₂ = (1 + 4j).

z₁ * z₂ = (3 + 2j)(1 + 4j)

// F - First: (3 * 1) = 3
// O - Outer: (3 * 4j) = 12j
// I - Inner: (2j * 1) = 2j
// L - Last:  (2j * 4j) = 8j²

        = 3 + 12j + 2j + 8j²

// Now, combine like terms and substitute j² = -1
        = 3 + (12j + 2j) + 8(-1)
        = 3 + 14j - 8
        = (3 - 8) + 14j
        = -5 + 14j

3. The Conjugate: The Mirror Image

The complex conjugate is a simple but powerful tool. To find the conjugate of a complex number, you just flip the sign of the imaginary part. The conjugate of z = a + bj is written as z* or z̄ and is equal to a - bj.

On the Argand diagram, it's the reflection of the number across the real axis. We need this for division!

4. Division

You can't just divide by a complex number directly. The trick is to turn the denominator into a real number. We do this by multiplying both the top and bottom of the fraction by the conjugate of the denominator.

Example: Let's calculate (2 + 5j) / (3 - j).

Step 1: Find the conjugate of the denominator (3 - j). It is (3 + j).

Step 2: Multiply the top and bottom by the conjugate.

  (2 + 5j)   (3 + j)
= ---------- * ---------
  (3 - j)    (3 + j)

Step 3: Multiply out the top (numerator) and the bottom (denominator).

// Top (Numerator):
(2 + 5j)(3 + j) = 6 + 2j + 15j + 5j²
                = 6 + 17j + 5(-1)
                = 6 - 5 + 17j
                = 1 + 17j

// Bottom (Denominator):
(3 - j)(3 + j) = 9 + 3j - 3j - j²
               = 9 - (-1)
               = 10

Step 4: Combine them and write in standard form a + bj.

  1 + 17j     1     17
= --------- = -- + --- j
     10      10    10

= 0.1 + 1.7j

A Different Perspective: Polar Form

Sometimes, describing our point (a, b) with a distance and an angle is more useful, especially in AC circuits. Think of a piki-piki rider at the city centre (origin). Instead of telling them to go 3 blocks east and 4 blocks north (Cartesian form: 3 + 4j), you can tell them to ride 5 kilometres in a specific direction (Polar form).

• Modulus (r): The direct distance from the origin. We find it using Pythagoras' theorem. r = |z| = √(a² + b²).
• Argument (θ): The angle from the positive Real axis. We find it using trigonometry. θ = tan⁻¹(b/a).

The Polar Form is written as: z = r(cosθ + jsinθ).

Image Suggestion:** An Argand Diagram showing the complex number `z = a + bj`. The vector from the origin to the point (a, b) is labeled 'r' (the modulus). The angle between this vector and the positive Real axis is labeled 'θ' (the argument). The sides 'a' and 'b' of the right-angled triangle are also shown. This visually connects the Cartesian and Polar forms.

Mhandisi at Work!

Imagine you are an electrical engineer in Nairobi, working on the power grid. You're analyzing an AC circuit. The voltage and current are not in sync; they are out of phase. Using real numbers alone would be a nightmare! But with complex numbers (specifically, their polar form, called phasors), you can easily represent these phase differences. The modulus (r) represents the magnitude (e.g., Volts or Amps), and the argument (θ) represents the phase angle. This makes calculating impedance (the total opposition to current flow, Z = R + jX) and power much, much easier. You're literally using imaginary numbers to keep the lights on in the real world!

You've Done It!

Congratulations! You've taken your first steps into the powerful world of complex numbers. It might seem strange at first, but this is a fundamental tool in an engineer's toolkit. Keep practicing the operations, and you'll soon see how they make "impossible" problems completely solvable. Keep up the great work, and see you in the next lesson!