Let's start with a simple equation: x squared equals 4. This has real solutions: positive 2 and negative 2, as we can see on the number line. But what happens when we try to solve x squared equals negative 1? We need a real number that when squared gives negative 1. However, any real number squared is always non-negative. This creates a mathematical gap that real numbers cannot fill.
To solve our problem, we make a bold mathematical step: we define a new number called i, the imaginary unit, where i squared equals negative 1. Let's explore the powers of i. i to the first power is simply i. i squared is negative 1 by definition. i cubed equals i squared times i, which is negative 1 times i, giving us negative i. i to the fourth power equals i squared times i squared, which is negative 1 times negative 1, giving us 1. Notice the cyclic pattern - the powers of i repeat every four steps!
Now we can define complex numbers. A complex number z has the form a plus b i, where a and b are real numbers. Here, a is called the real part and b is the imaginary part. Let's look at some examples. 3 plus 2i has real part 3 and imaginary part 2. Negative 1 plus 4i has real part negative 1 and imaginary part 4. The number 5 is a pure real number where the imaginary part is zero. And 3i is a pure imaginary number where the real part is zero. We can add complex numbers by adding their real parts and imaginary parts separately.
We can visualize complex numbers geometrically using the complex plane. The horizontal axis represents real numbers, and the vertical axis represents imaginary numbers. Each complex number a plus b i corresponds to the point with coordinates a comma b. For example, 3 plus 2i is plotted at point 3 comma 2. We can also represent complex numbers as vectors from the origin. When we add complex numbers, it's like vector addition - we add the corresponding components.
Here's the most beautiful insight about complex numbers: multiplying by i represents a 90-degree counterclockwise rotation in the complex plane. Let's see this in action. Start with the number 2 on the real axis. When we multiply by i, we get 2i, which is on the imaginary axis - a perfect 90-degree rotation! Let's try another example: 1 plus i multiplied by i gives us negative 1 plus i. Again, we see a 90-degree counterclockwise rotation. If we multiply by i four times, we complete a full 360-degree rotation and return to the original number. This geometric interpretation makes complex multiplication intuitive and powerful.