# Every imaginary number is also a complex number?

How is it possible that every imaginary number (multiple of i ) is also a complex number?

• What's a complex number?
– user63181
Apr 5, 2014 at 13:51
• Every natural number is also a real number. The same concept. The imaginary numbers form a subset of the complex numbers. Apr 5, 2014 at 13:52
• $\mathbb{C}:=\{ a+bi|a,b\in\mathbb{R} \}$ now choose $a=0$ Apr 5, 2014 at 13:52
• Is it the root of some polynomial with real coefficients? Then it's a complex number. (One can further weaken that to only needing complex coefficients.) Aug 2, 2014 at 16:19

Any real multiple of $i$, say $ai$, is also a complex number $0+ai$. Complex number is a number of the form $a+bi$, where $a, b\in \mathbb {R}$. Zero is obviously a real number, so everything works out nicely.

First of all, lets see the definition of a complex number:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit,

For an imaginaty number it is: 0+bi

Note that 0 is a real number, so it didn't break the rule.

Similarly any real number is: a+0i which is a complex number too.

The standard definition of a complex number is any number that can be written as: $a+b\,i$ where $a,b\in\Bbb R$ and $i=\sqrt{-1}$. So even an imaginary number, that is a number of the form $b\,i$, is a complex number since it can be written as: $0+b\,i$.