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Multiplication of Two Complex Numbers

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Last updated date: 19th Apr 2024
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An Overview of Complex Numbers

Complex numbers are very helpful in finding the square root of the negative numbers. Complex numbers have applications in many scientific research, fluid dynamics, vibration analysis, quantum mechanics, and electromagnetism. In this chapter, we are going to learn what a complex number is. How to multiply complex numbers and the Complex number differs from the Real number and Imaginary numbers. We will also learn to use methods for solving complex numbers.


What is a Complex Number?

A complex number is the summation of a real number and an imaginary number. A complex number is the form of a+ib, and it is usually represented by z. Here both a and b are real numbers. The value ‘a’ is called the real part, which is denoted by Re(z), and ‘b’ is called an imaginary part Im(z). Also, ib is also an imaginary number.


A combination of both the real and imaginary numbers is a complex number.

We know $i^2=-1$, let us calculate the value of 'i ' raised to the power.


$i^3$

$i^{2}\times i$

-i

$i^4$

$i^{2}\times i^2=-1 \times -1$

1

$i^5$

$i^{2} \times i^{2}\times i$

i

$i^6$

$i^{2}\times i^{2} \times i^2$

-1


Difference Between Complex Number, Real Number and Imaginary Number

To find out that what the difference amongst the three numbers refer to the table mentioned below:-


COMPLEX NUMBER

REAL NUMBER

IMAGINARY NUMBER

-1+2i

-1

2i

7-9i

7

-9i

-6i

0

-6i (Purely imaginary)

6

6

0i (Purely Real)


Multiplication of Complex Number

Multiplication of complex numbers or product of complex numbers is a procedure of the multiplication of two or more complex numbers by applying the distributive property. Mathematically, if we have two complex numbers, z = a + ib and w = c + id, then the multiplication of complex numbers z and w is written as zw = (a + ib) (c + id). Multiplying complex numbers is similar to multiplying polynomials. Polynomial identity is used to solve the multiplication of complex numbers: (a+b) (c+d) = ac + ad + bc + bd.


The formula for multiplying complex number is given as:

$(a+i b)(c+i d)=a c+i a d+i b c+i^2 b d,$ is

$(a+i b)(c+i d)=(a c-b d)+i(a d+b c)$.


Multiplying complex number


Multiplying complex number


How to Multiply Complex Numbers?

When multiplying complex numbers, the most effective method is to use either FOIL method or the Distributive property to simplify the expressions. It also works when it is in need to multiply imaginary numbers. Following are the steps or stages which is used to multiply complex numbers by using the distributive property:-


Use the distributive property to multiply the real part of the first factor by the second factor.


Repeat step 1, but with the imaginary part of the first factor.


Combine like terms, as alike, with the algebraic expressions.


Simplify the complex number and express the final answer in the form $a+bi$ or $a-bi$


Example: $(-3+2i)(5-4i)$

Ans: $(-3+2i)(5-4i)$

$=-3(5-4i)+2i(5-4i)$

$=-15+12i+10i-8i^2$

$=22i-8i^2-15= -15+22i+8$

$=-7+22i$


Product of Two Complex Number

The product of two complex numbers can be expressed in the standard form A + iB where A and B are real.


Let $Z_1=p+i q$ and $Z_2=r+is$ is to be two complex numbers $(p, q, r$ and $s$ are real), then the product $Z_1 Z_2$ is defined as


$Z_1 Z_2=(p r-q s)+i(p s+q r)$

SUM of Complex Numbers

$(5-4 i)+(-2+6 i)=5-2-4 i+6 i$

$=3+2 i$


PRODUCT of Complex Numbers

$(5-4 i)+(-2+6 i)=-10+30 i+8 i-24 i^2$

$=-10+38 i-24(-1)$

$=-10+38 i+24$

$=14+38 i$


Multiplicative Inverse of Complex Numbers

The multiplicative inverse of a complex number on multiplying with the given complex number results in the multiplicative identity of 1 . The multiplicative inverse of the complex number $z=a+i b$ is $z^{-1}=\dfrac{\bar{z}}{|z|^2}$. Here $\bar{z}=a-i b$, and $|z|=\sqrt{a^2+b^2}$.

$z . z^{-1}=1$

$z^{-1}=\dfrac{\bar{z}}{|z|^2}$


For finding the multiplicative inverse of a complex number, we need the complex number's conjugate and the complex number's modulus. The conjugate of the complex number is $\bar{z}=a-i b$, and the modulus of the complex number is $|z|=\sqrt{a^2+b^2}$.


Let us take a simple example of finding the multiplicative inverse of a complex number $z=3+4 i$. For this complex number, the conjugate complex number is $\bar{z}=3-4 i$, and the modulus of the complex number is $|z|=\sqrt{3^2+4^2}=5$. And the multiplicative inverse is $z^{-1}=\dfrac{3-4 i}{5}=\dfrac{3}{5}- \dfrac{4 \mathrm{i}}{5}$.

Solved Examples

Q 1. Find the product of two complex numbers $(-2+\sqrt{3} i)$ and $(-3+2 \sqrt{3} i)$ and express the result in standard form $A+i B$.

Ans: $(-2+\sqrt{3 i})(-3+2 \sqrt{3 i})$

$=-2(-3+2 \sqrt{3} i)+\sqrt{3} i(-3+2 \sqrt{3} i)$

$=6-4 \sqrt{3} i-3 \sqrt{3} i+2(\sqrt{3} i)^2$

$=6-7 \sqrt{3} i-6$

$=6-6-7 \sqrt{3} i$

$=0-7 \sqrt{3} i$, which is the required from $A+iB$ where $A=0$ and $B=-7 \sqrt{3}$


Q 2. Multiply complex numbers $z=3-2 i$ and $w=-4+3 i$.

Ans: For multiplying complex numbers $z$ and $w$, we will use the formula $(a+i b)(c+i d)=(a c-b d)+i(a d+b c)$. Here $a=3, b$ $=-2, c=-4, d=3$

$(3-2 i)(-4+3 i)=[3 \times(-4)-(-2) \times 3)+i(3 \times 3+(-2) \times(-4))$ $=(-12+6)+i(9+8)$ $=-6+17 i$

$(3-2 \mathrm{i})(-4+3 \mathrm{i})=-6+17 \mathrm{i}$


Q 3. Find the square of the complex number $(-4+6 \mathrm{i})$.

Ans: To find the square of a complex number, we will use the formula $(a+i b)^2=\left(a^2-b^2\right)+i$ 2ab. Here, $a=-4$ and $b=6$

$(-4+6 i)^2=\left((-4)^2-6^2\right)+i 2 \times(-4) \times 6$

$=(16-36)-48 i$

$=-20-48 i$

$(-4+6 i)^2=-20-48 i$


Practice Questions

Q 1. The formula for multiplying complex numbers is:

  1. $(a+i b)(c+i d)=(a c+b d)+i(a d+b c)$

  2. $(a+i b)(c+i d)=(a c-b d)+i(a d+b c)$

  3. $(a+i b)(c+i d)=(a c+b d)+i(b c+a d)$

  4. $(a+i b)(c+i d)=(a b-c d)+i(a b+c d)$

Ans: (b)

$(a+i b)(c+i d)=(a c-b d)+i(a d+b c)$


Q 2. Find the multiplicative inverse of $\sqrt{2}+7 i$

Ans: $=\dfrac{\sqrt{2}}{51}-\dfrac{7}{51} i$


Q 3. Find the product of two complex numbers $(-2+\sqrt{3i})$ and $(-3+2 \sqrt{3i})$ and express the result in standard from $\mathrm{A}+\mathrm{iB}$.

Ans: $\mathrm{A}+\mathrm{iB}$, where $\mathrm{A}=0$ and $\mathrm{B}=-7 \sqrt{3}$


Summary

In contrast with real numbers, there is no natural ordering of the complex numbers. Specifically, there is no linear ordering on the complex numbers that is harmonious with addition and multiplication. Hence, the complex numbers do not carry the formation of an ordered field. Mathematically, a complex number is an element of the number system which extends to the real numbers with a specific element denoted 'i', which is known as the imaginary unit. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers.

FAQs on Multiplication of Two Complex Numbers

1. How to Graph Complex Numbers?

The complex number of the form z = a + ib can be represented in the argand plane. The complex number  z = a + ib can be presented as the coordinates of a point as (Re(z), Im(z) ) = (a, ib). Here the real part is presented with reference to the x-axis, and the imaginary part is presented with reference to the y-axis.

2. What is the Multiplication of Complex Numbers with a Real Number?

Multiplying complex numbers with real numbers is evaluated as $a (c + id) = ac + i ad$, which is a complex number.

3. What Properties apply to Multiplication with Complex Numbers?

The following properties can be defined for the multiplication of complex numbers:

  • Closure law,

  • Commutative law,

  • Associative law,

  • Distributive law.