Question

Express the complex number $3\left( 7+i7 \right)+i\left( 7+i7 \right)$ in the form a + ib

Answer 1

Hint: Use the same method to approach when approached with simplification: having a real number only thing is to just separate terms with ‘i'. Also use the fact ${{i}^{2}}=-1$.

Complete Step-by-Step solution:
In the question we have to express by simplifying $3\left( 7+7i \right)+i\left( 7+7i \right)$ and writing in the form a + ib.
Before doing so, we will learn what complex numbers are.
A complex number is a number that can be written in form of a + bi, where a, b are real numbers and i is a solution of the equation ${{x}^{2}}=-1$ .This is because no real value satisfies for equation ${{x}^{2}}+1=0$ or ${{x}^{2}}=-1$ , hence i is called imaginary number. For the complex number a + ib, a is considered as real part and b as imaginary part. Despite the historical nomenclature “imaginary” complex numbers are regarded in the mathematical sciences as just as “real” as real numbers and are fundamental in any aspects of scientific description of the natural world
Now it is given that,
$3\left( 7+7i \right)+i\left( 7+7i \right)$
which can be further written as,
$21+21i+7i+7{{i}^{2}}$
Now using the fact ${{i}^{2}}=-1$ and separating constants and terms related to i we get,
$21-7+28i=14+28i$
So, the answer after simplifying the expression we get is, 14 + 28i which can be written in the form of a + bi where a = 14 and b = 28.
Hence the answer is 14 + 28i

Note: While doing simplification of complex numbers keep in mind that all rules are the same when it is considered with real numbers only difference is terms containing ‘i' to be always separated with that of constant.