Introduction to Complex Numbers

One of the most fascinating properties of $i$ is that its powers follow a repeating cycle of four. This cycle allows us to simplify $i$ raised to any integer power, no matter how large. The pattern is: - $i^1 = i$ - $i^2 = -1$ - $i^3 = i^2 \cdot i = -i$ - $i^4 = i^2 \cdot i^2 = (-1)(-1) = 1$

After $i^4$, the pattern repeats ($i^5 = i$, $i^6 = -1$, etc.). To simplify $i^n$, divide the exponent $n$ by 4 and look at the remainder. The remainder tells you which of the four values in the cycle is the answer.

A complex number is a combination of a real number and an imaginary number, written in the standard form $a + bi$. Here, $a$ is the real part and $bi$ is the imaginary part. When adding or subtracting complex numbers, we treat $i$ like a variable (like $x$). We combine like terms: add the real parts together and add the imaginary parts together. For example, $(2 + 3i) + (4 + 5i) = (2+4) + (3i+5i) = 6 + 8i$. Always ensure your final answer is in $a + bi$ format.