Question

What is 5i equal to?

Answer 1

We explain the concept of complex number and find its general form. Then we use the general form to convert the given form of $5i$ in full complex number. We also state the relation between complex number $i$ where ${{i}^{2}}=-1,{{i}^{3}}=-i,{{i}^{4}}=1$.

Complete step-by-step solution:
Combination of both the real number and imaginary number is a complex number.
Complex numbers are often represented on the complex plane, sometimes known as the Argand plane or Argand diagram. In the complex plane, there is a real axis and a perpendicular, imaginary axis. All the complex numbers can be represented in the form of a circle with a centre point at the origin.
Complex numbers are the numbers that are expressed in the form of $a+ib$ where $a,b$ are real numbers and $i$ is an imaginary number called ‘iota’.
The given expression is a representation of the complex number.
Here the complex number is the term $i=\sqrt{-1}$. 5 is the constant multiple to the term $i$.
The relation and conditions for the complex number $i$ is that ${{i}^{2}}=-1,{{i}^{3}}=-i,{{i}^{4}}=1$.
Now we try to express the $5i$ in full complex form.
We know that $a=\sqrt{{{a}^{2}}}$. We use that to express that in the form of
$5i=5\sqrt{-1}=\sqrt{25}\sqrt{-1}=\sqrt{-25}$.

Note: We need to remember that real numbers are actually a part of imaginary numbers. In the general form of $a+ib$, if we take the value of $b=0$, we get any real number. Therefore, we can say that the real number set is a subset of a complex number.