What are Complex Numbers? Operations on Complex Numbers

What are Complex Numbers?

As we already know that the square of a real number is positive, therefore, the solution of the below-mentioned equation is not available in Real numbers.

\displaystyle {{x}^{2}}+1=0\,\,\Rightarrow \,\,{{x}^{2}}=-1

Therefore, in order to remove this deficiency of real numbers, we require a number whose square must be -1.

Hence, the mathematicians introduced a larger set of number which is called the set of complex numbers that includes real numbers and every number whose square is not positive.

They designed another number i.e. – 1 which is called an imaginary unit. Complex Number is denoted by the a letter \displaystyle i which contain the property that \displaystyle {{i}^{2}}=-1.

Leonard Euler was a Swiss Mathematician who first used the symbol \displaystyle i for the number \displaystyle \sqrt{{-1}}.

It is pertinent to mention here that, the \displaystyle i is not a real number.

Complex numbers are represented by Z and the set of all complex numbers is represented by C.

Pure Imaginary Number

It is a square root of a negative real number. For example, \displaystyle \sqrt{{-1}},\,\sqrt{{-11}},\,\,\sqrt{{-23}}.

Basic Arithmetic Operations on Complex Numbers

1. Addition

Consider, \displaystyle {{Z}_{1}}=a+ib\,\,\,and\,\,\,\,{{Z}_{2}}=c+id are two complex numbers, where a, b, c and d belong to the real numbers then the sum of these two complex numbers will be

\displaystyle {{Z}_{1}}+{{Z}_{2}}=a+ib\,+c+id=a+c+i(b+d)

Examples:

a – \displaystyle a.\,(4-i)+(12-20i)

Solution:

\displaystyle 4-i+12-20i=16-21i

b – \displaystyle (14-i)+(10-6i)

Solution:

\displaystyle 14-i+10-6i=24-7i

c – \displaystyle (2-i)+(8-3i)

Solution:

\displaystyle 2-i+8-3i=10-4i

2. Subtraction

Consider, \displaystyle {{Z}_{1}}=a+ib\,\,\,and\,\,\,\,{{Z}_{2}}=c+id are two complex numbers, where a, b, c and d belong to the real numbers then the difference between these two complex numbers will be

\displaystyle {{Z}_{1}}-{{Z}_{2}}=(a+ib\,)-(c+id)=a+ib-c-id=a-c+i(b-d)

Examples:

a – \displaystyle (4-i)-(12-20i)

Solution:

\displaystyle 4-i-12+20i=-8+19i

b – \displaystyle (14-i)-(10-6i)

Solution:

\displaystyle 14-i-10+6i=4+5i

c – \displaystyle (2-i)-(8-3i)

Solution: \displaystyle 2-i-8+3i=-6+2i

3. Multiplication:

Consider, \displaystyle {{Z}_{1}}=a+ib\,\,\,and\,\,\,\,{{Z}_{2}}=c+id are two complex numbers, where a, b, c and d belong to the real numbers then the product of these two complex numbers will be

\displaystyle \begin{array}{l}{{Z}_{1}}{{Z}_{2}}=(a+ib\,)(c+id)=ac+aid+ibc+{{i}^{2}}bd\\=ac+aid+ibc+(-1)bd=ac+aid+ibc-bd=(ac-bd)+(ad+bc)i\end{array}

Some multiplication examples of real numbers are given here.

  1. \displaystyle \begin{array}{l}=16-6i-8i+3{{i}^{2}}=16-14i+3(-1)\\=16-14i-3=13-14i\end{array}
  2. \displaystyle \begin{array}{l}(3-2i)(4+5i)\\=12+15i-8i-10{{i}^{2}}=12+7i-10(-1)\\=12+10+7i=22+7i\end{array}

4. Division

Consider, \displaystyle {{Z}_{1}}=a+ib\,\,\,and\,\,\,\,{{Z}_{2}}=c+id are two complex numbers, such that, \displaystyle {{Z}_{2}}\ne 0, the division of these complex numbers is

\displaystyle \frac{{{{Z}_{1}}}}{{{{Z}_{2}}}}=\frac{{a+bi}}{{c+di}}

Few division examples of real numbers are given below: –

complex number division
Complex number division

Read also: What is a Real Number? Definition and Properties