Understanding Complex Numbers

The Need for Complex Numbers

Consider the quadratic equation $x^2 - 1 = 0$. Factoring gives $(x-1)(x+1) = 0$, so its two solutions, $x=1$ and $x=-1$, are real numbers.

The equation $x^2 + 1 = 0$ is different. It requires $x^2 = -1$, but the square of every real number is nonnegative. The equation therefore has no real solution.

Complex numbers extend the real number system so that equations such as this one can be solved.

Imaginary Numbers

The imaginary unit, denoted by $i$, is defined by the equation

With the principal square root convention, this is also written as

For a nonnegative real number $a$, the principal square root of $-a$ is $i\sqrt{a}$. For example:

Numbers like $2i$ and $3i$ are called purely imaginary numbers.

General Form

Complex numbers are generally written in the form $z = a + bi$, where:

• $a=\operatorname{Re}(z)$ is the real part.
• $b=\operatorname{Im}(z)$ is the imaginary part.
• $i$ is the imaginary unit ($\sqrt{-1}$ ).

Both $a$ and $b$ are real numbers. The expression $bi$ is the imaginary term, while its coefficient $b$ is the imaginary part.

Identifying Real and Imaginary Parts

Take each example in turn and name its real part and its imaginary part.

1. $2 + 3i$

   • Real part ($a$): $2$
   • Imaginary part ($b$): $3$

2. $5 - 4i$ This is the same as $5 + (-4)i$.

   • Real part ($a$): $5$
   • Imaginary part ($b$): $-4$

3. $\sqrt{2}$ This is an ordinary real number, but it can also be considered a complex number with an imaginary part of $0$. Its form is $\sqrt{2} + 0i$.

   • Real part ($a$): $\sqrt{2}$
   • Imaginary part ($b$): $0$

4. $-7i$ This is a purely imaginary number. Its form is $0 + (-7)i$.

   • Real part ($a$): $0$
   • Imaginary part ($b$): $-7$

Exercise

Determine the real and imaginary parts of each complex number below. Two of these expressions hide their value inside a square root or a power of $i$, so simplify each one first and only then name the two parts.

1. $2 + \sqrt{(-2)^2}$
2. $2 + i^2$
3. $1 + \sqrt{-9}$
4. $1 + 2i$

Worked Solutions

Each solution below separates the real and the imaginary part before simplifying, so you can follow the same order on your own expression.

1. $2 + \sqrt{(-2)^2} = 2 + \sqrt{4} = 2 + 2 = 4$.

   This can be written as $4 + 0i$.

   • Real part: $4$
   • Imaginary part: $0$

2. $2 + i^2 = 2 + (-1) = 1$.

   This can be written as $1 + 0i$.

   • Real part: $1$
   • Imaginary part: $0$

3. $1 + \sqrt{-9} = 1 + 3i$.

   This can be written as $1 + 3i$.

   • Real part: $1$
   • Imaginary part: $3$

4. $1 + 2i$.

   This can be written as $1 + 2i$.

   • Real part: $1$
   • Imaginary part: $2$