Question

How do you simplify $(5 + 2i) - (2 + 3i)$ ?

Answer 1

Here we have to subtract a complex number from another complex number. We simplify or solve the given subtraction by opening the brackets and subtracting real part from real part and imaginary part from imaginary part of the two complex numbers in the brackets.

• In a complex number $z = x + iy$, the real part is x and the imaginary part is y. Only real parts of two or more complex numbers can be added or subtracted to each other. Similarly only imaginary parts of two or more complex numbers can be added or subtracted from each other.
• If $z = x + iy$ is a complex number, then conjugate of this complex number is $\overline z = x - iy$
• Subtraction is a process of deducting the values, if we have to subtract ‘b’ from ‘a’, we write $a - b$ which means we are deducting ‘b’ values from ‘a’.

Complete step by step solution:
We are given the subtraction of two complex numbers $(5 + 2i)$ and $(2 + 3i)$
Open the brackets of both the complex numbers
$\Rightarrow (5 + 2i) - (2 + 3i) = 5 + 2i - 2 - 3i$
Pair the real terms separately and imaginary terms separately
$\Rightarrow (5 + 2i) - (2 + 3i) = (5 - 2) + (2i - 3i)$
Solve the subtraction of real terms separately and imaginary terms separately
$\Rightarrow (5 + 2i) - (2 + 3i) = 3 + ( - i)$
Multiply the sign outside the bracket and sign inside the bracket
$\Rightarrow (5 + 2i) - (2 + 3i) = 3 - i$
$\therefore$Solution of $(5 + 2i) - (2 + 3i)$ is $3 - i$

Note: Many students make the mistake of not multiplying the sign outside the bracket to the sign inside the bracket in the end. Keep in mind multiplication of a negative and a positive sign gives negative sign. Also, many students write the value of $i = \sqrt { - 1}$ in the end which is also an answer, but always choose the values of the answer according to the question.