Question

How do you simplify $(7 - 4i) + (2 - 3i)$?

Answer 1

In this question, we need to simplify the given complex number to obtain the solution. The complex number has two parts namely, real part and imaginary part. Firstly, we need to combine the like terms together. So combine the real part in one parenthesis and the real part in another parenthesis. Then simplify the terms and obtain the solution. Here $i$ represents the imaginary unit.

Complete step by step solution:
Given the complex number $(7 - 4i) + (2 - 3i)$
We are asked to simplify the complex number and obtain the solution.
A complex number is a number that can be expressed in the form of $a + ib$ …… (1)
where a and b are real numbers and $i$ represents the imaginary unit, which satisfies the equation ${i^2} = - 1$.
Because no real number satisfies this equation, $i$ is called an imaginary number.
A complex number contains two parts which are the real part and imaginary part.
In the equation (1), a is the real part and b is the imaginary part of the complex number $a + ib$.
Now consider $(7 - 4i) + (2 - 3i)$.
Firstly, we make rearrangements in the above expression in such a way that combine the real part in one parenthesis and imaginary part in the other parenthesis and simplify it.
Now combining real and imaginary part, we get,
$\Rightarrow (7 + 2) + ( - 4i - 3i)$
Now simplifying the like terms we get,
$\Rightarrow 9 - 7i$

So the simplification of the given expression $(7 - 4i) + (2 - 3i)$ is equal to $9 - 7i$.

Note: Complex numbers are expressions in the form $x + iy$, where $x$ is the real part and $y$ is the imaginary part. These numbers cannot be marked on the real number line.
(Here note that the imaginary part is $y$, and not $iy$)
Students note that the backbone of this new number system is the number $i$, also known as the imaginary unit. So, from here we can conclude that any imaginary number is also a complex number and any real number is also a complex number.
In the above problem students get confused when combining the terms. They must be careful in combining real and imaginary parts.