Question

How do you simplify \[\left( 1+i \right)\left( 1-i \right)\]?

Answer 1

Hint: To solve the given question, we need to know the expansion of the expression \[\left( a+b \right)\left( c+d \right)\]. The expression \[\left( a+b \right)\left( c+d \right)\] is expanded by multiplying each term of the first bracket with each term of the second bracket and then adding their products. Algebraically it is expressed as, \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\]. We will simplify the given expression using this expansion.

Complete step-by-step solution:
We are asked to simplify the expression \[\left( 1+i \right)\left( 1-i \right)\]. This expression of the form \[\left( a+b \right)\left( c+d \right)\]. We know that this expression is expanded by multiplying each term of the first bracket with each term of the second bracket and then adding their products. Algebraically it is expressed as, \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\].
Here, we have \[a=c=1,b=i\And d=-i\]
Substituting these values in the expansion of the general expression, we get
\[\begin{align}
  & \Rightarrow \left( 1+i \right)\left( 1-i \right) \\
 & \Rightarrow 1-i+i-{{i}^{2}} \\
\end{align}\]
Simplifying the above expression, we get
\[\Rightarrow 1-{{i}^{2}}\]
Here \[i\] is a complex number, and \[i=\sqrt{-1}\]. Substituting its value in the above expression, we get
\[\begin{align}
  & \Rightarrow 1-{{\left( \sqrt{-1} \right)}^{2}} \\
 & \Rightarrow 1-\left( -1 \right)=1+1=2 \\
\end{align}\]
Hence, the simplification of the expression \[\left( 1+i \right)\left( 1-i \right)\] is 2.

Note: We can also use a different expansion of an algebraic expression to solve the given question. As we can see that here the first terms in the two brackets are the same, and the second terms in the brackets are of opposite signs. So, it is of the form \[\left( a-b \right)\left( a+b \right)\]. The expansion of this expression is \[{{a}^{2}}-{{b}^{2}}\].
For the given question, we have \[a=1\And b=i\]. substituting the values in the expansion of the expression, we get
\[\begin{align}
  & \Rightarrow \left( 1+i \right)\left( 1-i \right) \\
 & \Rightarrow 1-{{i}^{2}} \\
\end{align}\]
Here \[i\] is a complex number, and \[i=\sqrt{-1}\]. Substituting its value in the above expression, we get
\[\begin{align}
  & \Rightarrow 1-{{\left( \sqrt{-1} \right)}^{2}} \\
 & \Rightarrow 1-\left( -1 \right)=1+1=2 \\
\end{align}\]
Thus, we get the same answer from both methods. To solve these types of questions, we should know the expansions of different expressions.