Question

Write $\dfrac{{a + bi}}{{c + di}}$ in standard form?

Answer 1

Hint: For solving this equation, we must have the knowledge about Complex numbers and about imaginary numbers. Complex numbers are those numbers which are written in the form of $a + bi$ where “I” is an imaginary number which is generally known as Iota and “a and b” are known as real numbers.

Complete step by step answer:
More about complex number:-
The combination of both the Real and Imaginary number are termed to as a Complex number. When we perform any arithmetic operations of complex numbers such as addition, multiplication, division and subtraction, we generally combine the similar terms together. This means that we have to combine the real number with the real number and imaginary number with the imaginary number.

Difference between Real and Imaginary numbers:-
1. Real Numbers are those values which is generally present in number system such as positive, negative, zero, fractions, rational and irrational numbers
 For example $1, - 5, \dfrac{3}{2}, \sqrt 8$ etc.

2. Imaginary Numbers are those values which are not real. Whenever we square an imaginary number it always gives a negative result.
 For example $\sqrt { - 5} , \sqrt { - 13 }$ etc.

Now we will solve the given equation $\dfrac{{a + bi}}{{c + di}}$ in its standard form
Multiplying the numerator and denominator by the conjugate of the denominator we get
$ \Rightarrow \left( {\dfrac{{a + bi}}{{c + di}}} \right) \times \left( {\dfrac{{c - di}}{{c - di}}} \right)$
\[ \Rightarrow \dfrac{{ac - adi + cbi + bd{i^2}}}{{{c^2} - cdi + cdi - {d^2}{i^2}}} \:\: or \:\: \dfrac{{ac + \left( {cb - ad} \right)i - bd{i^2}}}{{{c^2} - {d^2}{i^2}}}\]

Now putting the value of ${i^2} = 1$ we get:-
$ \Rightarrow \dfrac{{ac + \left( {cb + ad} \right)i - bd\left( { - 1} \right)}}{{{c^2} - {d^2}\left( { - 1} \right)}} \:\: or \:\: \dfrac{{ac + \left( {cb - ad} \right)i + bd}}{{{c^2} + {d^2}}} $
$ \Rightarrow \dfrac{{ac + bd + \left( {cb - ad} \right)i}}{{{c^2} + {d^2}}} \:\: or \:\: \dfrac{{ac + bd}}{{{c^2} + {d^2}}} + \dfrac{{\left( {cb - ad} \right)i}}{{{c^2} - {d^2}}}$
$ \Rightarrow \dfrac{{ac + bd}}{{{c^2} + {d^2}}} + \dfrac{{cb - ad}}{{{c^2} + {d^2}}}i$

Thus, the standard form of $\dfrac{{a + bi}}{{c + di}}$ is $ \dfrac{{ac + bd}}{{{c^2} + {d^2}}} + \dfrac{{cb - ad}}{{{c^2} + {d^2}}}i$.

Note: Imaginary numbers which are generally called complex numbers are used in our daily life. These are used to solve real-life applications such as electricity and also used for solving quadratic equations. These are also used in signal processing, which is very useful in wireless technologies and cellular technology as well as in radar and also in biology.