Question

How do you find the sum of the complex number: \[\left( {4 - 2i} \right) + \left( {12 + 7i} \right)\]?

Answer 1

Hint: A complex number, is made of a real number and some multiple of i and is of the from \[a + bi\]. Complex numbers, as any other numbers, are added, subtracted, multiplied or divided, and then those expressions can be simplified. And here to find the sum of the given complex number; we need to add and subtract the real terms together and then simplify the imaginary terms together.

Complete step by step solution:
Let us write the given data:
\[\left( {4 - 2i} \right) + \left( {12 + 7i} \right)\]
If \[\left( {a + bi} \right)\] and \[\left( {c + di} \right)\] are two complex numbers:
To add or subtract complex numbers: We need to add or subtract the real terms together and then add or subtract the imaginary terms together i.e.,
\[ \Rightarrow a + bi + c + di = \left( {a + c} \right) + \left( {b + d} \right)i\] ……………. 1
Hence, the imaginary terms are written combining both the numbers.
Now, as given here we have:
 \[\left( {4 - 2i} \right) + \left( {12 + 7i} \right)\]
As per the equation 1 we get:
\[ \Rightarrow 4 - 2i + 12 + 7i = \left( {4 + 12} \right) + \left( { - 2 + 7} \right)i\]
Simplifying the numbers, we get:
\[ \Rightarrow 4 - 2i + 12 + 7i = 16 + 5i\]

Therefore, we get:
\[\left( {4 - 2i} \right) + \left( {12 + 7i} \right) = 16 + 5i\]

Note: A complex number is a number that can be written in the form \[a + bi\], where a and b are real numbers and i is the imaginary unit. Multiplication of two complex numbers is also a complex number.
Although real numbers are subsets of complex numbers and hence the sum of two complex numbers is always a complex number. To add or subtract, combine like terms and to multiply monomials, multiply the coefficients and then multiply the imaginary numbers i and to multiply complex numbers that are binomials, use the Distributive Property of Multiplication.