Question

How do you simplify \[\dfrac{{1 + 2i}}{{3 - 4i}}\]?

Answer 1

Hint: To simplify this question , we need to solve it step by step . . Here we are going to solve this given question by performing some calculations with the concepts of imaginary numbers and conjugates . In order to obtain the simplified form of the given complex number, obtain the conjugate value of the denominator which ensures that the denominator is real . Now the question comes: what is the complex conjugate of a complex number . It is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign . We decide the sign by comparing the denominator . If there is a negative sign in the denominator then the conjugate will be of same magnitude but of opposite sign and vice versa to obtain the required result .

Complete step by step solution:
We are Given a complex number \[\dfrac{{1 + 2i}}{{3 - 4i}}\] let it be
Here i is the imaginary number which is commonly known as the iota.
The form which we are given is called the rectangular form of complex numbers.
Conjugate is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign and we have to multiply the numerator and denominator by the conjugate of the denominator . We decide the sign by comparing the denominator . If there is a negative sign in the denominator then the conjugate will be of the same magnitude but of opposite sign . As per our given question the conjugate of our denominator will be \[3 + 4i\]
Now multiplying the conjugate value of the denominator which ensures that the denominator is real .
\[
  \dfrac{{1 + 2i}}{{3 - 4i}} \times \dfrac{{3 + 4i}}{{3 + 4i}} \\
  \dfrac{{(1 + 2i)(3 + 4i)}}{{(3 - 4i)(3 + 4i)}} \;
 \]
Now expand the above to simplify –
\[
  \dfrac{{3 + 10i + 8{i^2}}}{{9 - 16{i^2}}} \\
  \dfrac{{ - 5 + 10i}}{{25}} \\
   - \dfrac{5}{{25}} + \dfrac{{10}}{{25}}i \\
   - \dfrac{1}{5} + \dfrac{2}{5}i \;
 \]
Therefore, the required simplified answer is \[ - \dfrac{1}{5} + \dfrac{2}{5}i\]
So, the correct answer is “ \[ - \dfrac{1}{5} + \dfrac{2}{5}i\]”.

Note: 1. Real Number: Any number which is available in a number system, for example, positive, negative, zero, whole number, discerning, unreasonable, parts, and so forth are Real numbers. For instance: 12, - 45, 0, 1/7, 2.8, √5, and so forth, are all the real numbers.
2. A Complex number is a number which are expressed in the form $a + ib$ where $ib$ is the imaginary part and $a$ is the real number .i is generally known by the name iota. \[\]
or in simple words complex numbers are the combination of a real number and an imaginary number .
3. The Addition or multiplication of any 2-conjugate complex number always gives an answer which is a real number.