Introduction to Complex Numbers

In earlier classes, you worked only with real numbers. But some equations do not have real solutions. For example:

Consider the equation \(x^2 + 1 = 0\). If you try to solve it using real numbers, you get

\(x^2 = -1\)

There is no real number whose square is \(-1\). So, to handle such equations, we extend our number system and introduce a new kind of number called a complex number.

Complex numbers help in solving many problems in algebra, trigonometry, and even in physics and engineering. So this chapter is like opening a new door in the number system.

To deal with square roots of negative numbers, we first define a special number called the imaginary unit.

Definition: The imaginary unit \(i\) is defined by

\(i^2 = -1\).

Using this, we can write the square root of negative numbers in a simple way. For any positive real number \(a > 0\), we define

\(\sqrt{-a} = \sqrt{a} \cdot i\).

Some quick examples:

• \(\sqrt{-1} = i\)
• \(\sqrt{-4} = 2i\)
• \(\sqrt{-9} = 3i\)

So, instead of saying “no real solution”, we now express answers using \(i\).

Because \(i^2 = -1\), higher powers of \(i\) repeat in a simple pattern. It is very useful to remember this as a note.

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = i^2 \cdot i = -1 \cdot i = -i\)
• \(i^4 = i^2 \cdot i^2 = (-1)(-1) = 1\)

After this, the pattern repeats every 4 powers:

• \(i^5 = i\)
• \(i^6 = -1\)
• \(i^7 = -i\)
• \(i^8 = 1\), and so on.

In general, to simplify \(i^n\), you can divide \(n\) by 4 and use the remainder to match one of \(i, -1, -i, 1\).

Now we are ready to define a complex number.

Definition: A complex number is any number that can be written in the form

\(z = a + ib\)

where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\).

This way of writing \(z\) is called the standard form or Cartesian form of a complex number.

Examples:

• \(3 + 2i\) is a complex number with \(a = 3\) and \(b = 2\).
• \(-5 + 0i\) is just the real number \(-5\).
• \(0 + 4i\) is a purely imaginary number, usually written as \(4i\).

In a complex number \(z = a + ib\), the two components \(a\) and \(b\) have special names.

Definition:

• The real part of \(z\) is \(a\), written as \(\Re(z) = a\).
• The imaginary part of \(z\) is \(b\), written as \(\Im(z) = b\).

Note that the imaginary part is the number \(b\) itself, not \(ib\).

Examples:

• If \(z = 3 + 2i\), then \(\Re(z) = 3\) and \(\Im(z) = 2\).
• If \(z = -5 + 0i\), then \(\Re(z) = -5\) and \(\Im(z) = 0\).
• If \(z = 0 - 7i\), then \(\Re(z) = 0\) and \(\Im(z) = -7\).

Two complex numbers are equal only when both their real parts and imaginary parts are equal.

Definition: Let \(z_1 = a + ib\) and \(z_2 = c + id\). Then

\(z_1 = z_2 \iff a = c \text{ and } b = d\).

So, if any one part is different, the complex numbers are not equal.

Examples:

• \(3 + 2i = 3 + 2i\) because real parts: \(3 = 3\) and imaginary parts: \(2 = 2\).
• \(4 + 5i \neq 4 + 3i\) because imaginary parts: \(5 \neq 3\).
• \(2 - i \neq 1 - i\) because real parts: \(2 \neq 1\).

This idea is very useful when we compare complex expressions and form equations in terms of their real and imaginary parts.