Q.Let the population of the world in $t$ years after 2010 be given by the formula $P = 4.7(1.02)^{t}$ billions. Calculate the total population of the world in the year 2029 to the nearest million.

Complex Numbers: A First Look

You already know that some equations have no solution in the real numbers. For example, $x^2 + 1 = 0$ asks for a number whose square is $-1$. No real number works — the square of any real number is zero or positive. So we are stuck.

But what if we simply invent a number whose square is $-1$? That is exactly what mathematicians did. They defined a new number, called $i$ (for "imaginary"), with the property:

This one step opens up an entirely new number system.

What is a complex number?

A complex number is any number of the form

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit defined above.

• $a$ is called the real part.
• $b$ is called the imaginary part (note: the imaginary part is $b$, not $bi$).

For example:

• $3 + 4i$ has real part $3$, imaginary part $4$.
• $-2 + 0i$ is just the real number $-2$.
• $0 + 5i$ is $5i$, a purely imaginary number.

Every real number is also a complex number (with $b = 0$). So complex numbers are a superset of the real numbers.

Why "imaginary"?

The name is unfortunate. These numbers are no more imaginary than negative numbers or fractions. They are a perfectly consistent extension of the real numbers. The term "imaginary" stuck because when they were first proposed, many mathematicians found the idea of a number whose square is negative to be absurd.

Today, complex numbers are essential in engineering, physics, signal processing, and pure mathematics.

Arithmetic with complex numbers

You treat $i$ like any other algebraic symbol, with the rule $i^2 = -1$.

Addition and subtraction: Add real parts and imaginary parts separately.

Multiplication: Expand like binomials, then replace $i^2$ with $-1$.

Example: $(2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5 + i$.

The complex conjugate

For a complex number $z = a + bi$, its conjugate is $\overline{z} = a - bi$. The conjugate is useful because:

which is always a non-negative real number. This lets you divide complex numbers:

The big picture

Complex numbers let you solve every polynomial equation. For instance, $x^2 + 1 = 0$ has solutions $x = i$ and $x = -i$. This completeness is one reason they are so powerful.

For now, just remember: a complex number is $a + bi$, with $i^2 = -1$, and you do algebra with it like any other expression — just remember to replace $i^2$ with $-1$ when it appears.