Table of Contents
In Algebra II, complex numbers are usually introduced as numbers that extend the real number system so that certain equations, like $x^2 + 1 = 0$, can have solutions. The key new idea that makes this possible is the imaginary unit.
This chapter focuses on what the imaginary unit is, how it is defined, and how to work with its basic powers. General operations with complex numbers (addition, multiplication, etc.) are treated in the separate chapter on operations with complex numbers, so we will not go into full detail about them here.
The definition of the imaginary unit
The imaginary unit is a special symbol, written as $i$, defined by the property
$$
i^2 = -1.
$$
This is not something you can do with real numbers: there is no real number whose square is $-1$, because for any real $x$,
$$
x^2 \ge 0.
$$
So $i$ is introduced as a new kind of number with the defining rule $i^2 = -1$.
From this definition, we can solve equations like
$$
x^2 + 1 = 0.
$$
Rewriting,
$$
x^2 = -1.
$$
Using $i^2 = -1$, any solution must satisfy $x^2 = i^2$, so
$$
x = i \quad \text{or} \quad x = -i.
$$
Thus, $i$ and $-i$ are the (complex) solutions of $x^2 + 1 = 0$.
The square root of negative numbers and $i$
Using the imaginary unit, we represent square roots of negative real numbers. For any positive real number $a$:
$$
\sqrt{-a} = i\sqrt{a}.
$$
Examples:
- $\sqrt{-9} = i\sqrt{9} = 3i$,
- $\sqrt{-5} = i\sqrt{5}$.
It is important to remember:
- $\sqrt{-1} = i$ by definition (since $i^2 = -1$),
- For $a > 0$, $\sqrt{-a}$ is defined as $i\sqrt{a}$.
Be careful: rules about square roots that work for positive real numbers do not always carry over unchanged when negatives and $i$ are involved. For example, for real positives, $\sqrt{ab} = \sqrt{a}\sqrt{b}$, but blindly doing this with negatives leads to contradictions:
$$
\sqrt{-1}\sqrt{-1} = i \cdot i = -1,
$$
but if you treat it as $\sqrt{(-1)(-1)} = \sqrt{1} = 1$, you would get $1$ instead of $-1$. This shows that the real-number rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ cannot be used without care once negative numbers and $i$ enter.
Basic algebraic behavior of $i$
Although $i$ is a new kind of number, it can be manipulated algebraically using its defining property $i^2 = -1$.
Some simple consequences:
- $i^1 = i$ (just $i$ itself),
- $i^2 = -1$ (by definition),
- $i^3 = i^2 \cdot i = (-1)\cdot i = -i$,
- $i^4 = i^2 \cdot i^2 = (-1)\cdot(-1) = 1$.
So the powers of $i$ repeat in a cycle of length $4$:
$$
\begin{aligned}
i^1 &= i,\\
i^2 &= -1,\\
i^3 &= -i,\\
i^4 &= 1,\\
i^5 &= i,\\
i^6 &= -1, \text{ and so on.}
\end{aligned}
$$
More generally, any integer power $i^n$ can be simplified by noticing that only the remainder when $n$ is divided by $4$ matters.
- If $n \equiv 0 \pmod{4}$, then $i^n = 1$.
- If $n \equiv 1 \pmod{4}$, then $i^n = i$.
- If $n \equiv 2 \pmod{4}$, then $i^n = -1$.
- If $n \equiv 3 \pmod{4}$, then $i^n = -i$.
For example:
- $i^{10}$: divide $10$ by $4$, getting remainder $2$, so $i^{10} = i^2 = -1$.
- $i^{17}$: $17$ divided by $4$ leaves remainder $1$, so $i^{17} = i$.
- $i^{23}$: $23$ divided by $4$ leaves remainder $3$, so $i^{23} = -i$.
This pattern makes working with high powers of $i$ straightforward.
Imaginary numbers versus the imaginary unit
The imaginary unit $i$ is just one specific complex number satisfying $i^2 = -1$. Using $i$, we can build many related numbers:
- Any number of the form $bi$, where $b$ is a real number (positive, negative, or zero), is called a purely imaginary number.
So:
- $3i$, $-2i$, and $\frac{1}{2}i$ are purely imaginary numbers.
- $0i = 0$ is actually just the real number $0$.
In later chapters on complex numbers, a general complex number will be written in the form
$$
a + bi,
$$
where $a$ and $b$ are real numbers. Here, $a$ is the real part and $b$ is the imaginary part. In this chapter, we only need to note that $i$ is the basic building block of the imaginary part of such numbers.
Simplifying expressions with $i$
When simplifying expressions involving $i$, the key idea is to repeatedly use $i^2 = -1$ and the power cycle of $i$.
Some typical simplifications:
- Simplify powers of $i$:
- $i^7 = i^{4+3} = i^4 \cdot i^3 = 1 \cdot (-i) = -i$.
- $i^{12} = (i^4)^3 = 1^3 = 1$.
- Simplify products involving $i$:
- $3i \cdot 4i = 12i^2 = 12(-1) = -12$.
- $(-5i)(-2i) = 10i^2 = 10(-1) = -10$.
- Express square roots of negative numbers using $i$:
- $\sqrt{-27} = \sqrt{-1 \cdot 27} = \sqrt{-1}\sqrt{27} = i\sqrt{27} = 3i\sqrt{3}$.
In more advanced work with complex numbers, expressions will be simplified until they are written in the standard form $a + bi$. At this stage, learning to reduce powers of $i$ and to rewrite $\sqrt{-a}$ as $i\sqrt{a}$ are the main skills related to the imaginary unit.
Summary of key facts about the imaginary unit
- The imaginary unit $i$ is defined by $i^2 = -1$.
- $\sqrt{-1} = i$ and, for $a > 0$, $\sqrt{-a} = i\sqrt{a}$.
- Powers of $i$ repeat every four steps:
$$
i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1,
$$
and then the pattern repeats. - Any number of the form $bi$ with real $b$ is a purely imaginary number.
- The imaginary unit is the fundamental building block that allows us to extend the real numbers to the complex numbers, which will be described more fully in the following chapter on operations with complex numbers.