Question

I know that real numbers consist of the natural or counting numbers, whole numbers, integers, rational numbers and irrational numbers. But what about imaginary number i. And is it possible that i can use an imaginary number for a real number.

Answer 1

The imaginary unit $i$ belongs to the set of complex numbers, denoted by $\mathbb C$. These numbers take the form $a+bi$, where $a,b$ are any real numbers.

The set of real numbers, $\mathbb R$, is a subset of $\mathbb C$, where each number in $\mathbb R$ can be obtained by taking $b=0$ and letting $a$ be any real number.

But any number in $\mathbb C$ with non-zero imaginary part is not a real number. This includes $i$.

• "is it possible that i can use an imaginary number for a real number"

I'm not sure what you mean by this part of your question. It is possible to represent any real number as a complex number, but not a purely imaginary one. All real numbers are complex, but not all complex numbers are real. For example, 2 is real and complex because $2=2+0i$.

There are some operations that you can carry out on purely imaginary numbers to get a purely real number. A famous example is raising $i$ to the $i$-th power. Since $i=e^{i\pi/2}$, we have

$i^i=\left(e^{i\pi/2}\right)^i=e^{i^2\pi/2}=e^{-\pi/2}\approx0.2079$