
How do you multiply $(a - bi)(a + bi)$?
Answer
445.2k+ views
Hint: Solve the question by multiplying $(a - bi)(a + bi)$ and then simplify the equation to get the answer. You should also take care of the fact that $i \times i = {i^2} = - 1$.
Complete step by step solution:
Let $y = (a - bi)(a + bi)$
Therefore on solving the equation we get,
$y = a.a + a.bi - a.bi - bi.bi$
$y = {a^2} - {b^2}{(i)^2}$
As we know that $i \times i = {i^2} = - 1$
Therefore after substituting the value of iota in the above expression we get
$y = {a^2} + {b^2}$
which is the required answer.
Note:
The square root of a negative real number is called an imaginary quantity or imaginary number.
The quantity $\sqrt { - 1} $ is an imaginary number , denoted by ‘i’ and is called iota.
$i = \sqrt { - 1} $
${i^2} = - 1$
${i^3} = i$
${i^4} = 1$
In other words,
${i^n} = {( - 1)^{n/2}}$ , if n is an even integer
${i^n} = {( - 1)^{(n - 1)/2}}$ , if n is an odd number
A number in the form of $z = x + iy$, where $x,y \in R$ is called a complex number.
The numbers x and y are called the real part and imaginary part of the complex number z respectively.
i.e $x = \operatorname{Re} (z)$ and $y = \operatorname{Im} g(z)$
MULTIPLICATION OF COMPLEX NUMBERS
Let ${z_1} = {x_1} + i{y_1}$ and be any two complex numbers , then there multiplication is defined as
${z_1}{z_2} = ({x_1} + i{y_1})({x_2} + i{y_2})$
$ = ({x_1}{x_2} - {y_1}{y_2}) + i({x_1}{y_2} + {x_2}{y_1})$
Properties of multiplication
1. Commutative : ${z_1}{z_2} = {z_2}{z_1}$
2. Associative: $({z_1}{z_2}){z_3} = {z_1}({z_2}{z_3})$
3. Multiplicative Identity: $z.1 = 1.z = z$
Here, 1 is the multiplicative identity of z.
4. Multiplicative Inverse: Every non-zero complex number z there exists a complex number ${z_1}$ such that ${z_1}z = 1 = z{z_1}$.
5. Distribution Law:
(a) ${z_1}({z_2} + {z_3}) = {z_1}{z_2} + {z_1}{z_3}$ (left distribution).
(b) $({z_1} + {z_2}){z_3} = {z_1}{z_3} + {z_2}{z_3}$(right distribution).
Complete step by step solution:
Let $y = (a - bi)(a + bi)$
Therefore on solving the equation we get,
$y = a.a + a.bi - a.bi - bi.bi$
$y = {a^2} - {b^2}{(i)^2}$
As we know that $i \times i = {i^2} = - 1$
Therefore after substituting the value of iota in the above expression we get
$y = {a^2} + {b^2}$
which is the required answer.
Note:
The square root of a negative real number is called an imaginary quantity or imaginary number.
The quantity $\sqrt { - 1} $ is an imaginary number , denoted by ‘i’ and is called iota.
$i = \sqrt { - 1} $
${i^2} = - 1$
${i^3} = i$
${i^4} = 1$
In other words,
${i^n} = {( - 1)^{n/2}}$ , if n is an even integer
${i^n} = {( - 1)^{(n - 1)/2}}$ , if n is an odd number
A number in the form of $z = x + iy$, where $x,y \in R$ is called a complex number.
The numbers x and y are called the real part and imaginary part of the complex number z respectively.
i.e $x = \operatorname{Re} (z)$ and $y = \operatorname{Im} g(z)$
MULTIPLICATION OF COMPLEX NUMBERS
Let ${z_1} = {x_1} + i{y_1}$ and be any two complex numbers , then there multiplication is defined as
${z_1}{z_2} = ({x_1} + i{y_1})({x_2} + i{y_2})$
$ = ({x_1}{x_2} - {y_1}{y_2}) + i({x_1}{y_2} + {x_2}{y_1})$
Properties of multiplication
1. Commutative : ${z_1}{z_2} = {z_2}{z_1}$
2. Associative: $({z_1}{z_2}){z_3} = {z_1}({z_2}{z_3})$
3. Multiplicative Identity: $z.1 = 1.z = z$
Here, 1 is the multiplicative identity of z.
4. Multiplicative Inverse: Every non-zero complex number z there exists a complex number ${z_1}$ such that ${z_1}z = 1 = z{z_1}$.
5. Distribution Law:
(a) ${z_1}({z_2} + {z_3}) = {z_1}{z_2} + {z_1}{z_3}$ (left distribution).
(b) $({z_1} + {z_2}){z_3} = {z_1}{z_3} + {z_2}{z_3}$(right distribution).
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

How do you convert 295 meters per second to kilometers class 7 maths CBSE

Trending doubts
10 examples of friction in our daily life

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Prokaryotic Cells and Eukaryotic Cells

State and prove Bernoullis theorem class 11 physics CBSE

What organs are located on the left side of your body class 11 biology CBSE

Write down 5 differences between Ntype and Ptype s class 11 physics CBSE
