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Operations with Complex Numbers

When a number is in the form c+id then it is called as complex numbers.

The following are the types of operations on complex numbers: Equality of Complex numbers, Sum of two Complex numbers, Negative of a Complex number, Additive identity of the Complex number, Additive inverse of a Complex number, Product of two Complex numbers, Multiplicative identity of Complex numbers, Conjugate complex numbers, Quotient of two non-zero Complex numbers, Reciprocal of a non-zero complex number or multiplicative inverse of a non-zero complex number. We will see these operations below.

Square root of a negative number is known as an imaginary number. `sqrt(-a)`  a > 0 is an imaginary number.

or

A number whose square is negative is known as an imaginary number.

`sqrt(-1)` , `sqrt(-2)`, `sqrt(-3)`

The symbol i,

i = `sqrt(-1)`

We write,

`sqrt(-3)` = `sqrt(3)` `sqrt(-1)` = `sqrt(3)` i = i `sqrt(3)`

`sqrt(-4)` = 2i = i2

Powers of i,

i = `sqrt(-1)`

i2 = -1

Let p > 0 be a positive integer such that p > 4. Let q be the quotient and r be the remainder, when p is divided by 4.

i.e., p = 4q + r where 0 `<=` r < 4.

Hence iP = i 4q+r = i4q.ir = ( i4 )q . ir = 1q .ir = ir ( $\therefore$ i4 = 1)

Square root of a negative number is known as an imaginary number .

If x and y are real numbers, then x + iy is called a complex number. x is called the real part and y is called the imaginary part.

Related Calculators
Adding Complex Numbers Calculator Complex Number Calculator
Dividing Complex Numbers Calculator Calculator for Order of Operations
 

Adding Complex Numbers

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The sum of two complex numbers (a + bi) and (c + di) was the complex number [(a + b) (c + d) i ]. In other words, the sum of two complex numbers is another complex number. To find the sum, we add the real parts together and then add imaginary parts together.

Here is the examples for adding complex numbers -

Solved Examples

Question 1: Adding the following complex numbers (3a + 5i) and (-2 +2i)
Solution:
Step 1: Add the like terms of the two complex numbers, we get 3a + (-2a) = 1a 

Step 2: Add the imaginary numbers, we get 5i + 2i = 7i 

Step 3: combine the terms, (3a + 5i) + (-2a +2i)


Correct answer is a + 7i
Question 2: Adding the following complex numbers (4 + 5) and (-3 -2)
Solution:
We know that the square root of -1 ( ) is denoted as i
(4 + 5i) + (-3 - 2i)

Step 1: Add the real numbers of the two complex numbers, we get 4 + (-3) = 1

Step 2: Add the imaginary numbers, we get 5i + (- 2i) = 3i

Step 3: combine the terms, 1 + 3i 


Correct answer is 1 + 3

Subtracting Complex Numbers

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The subtraction of the complex numbers means that the process of subtracting the real part and the imaginary part of the other complex numbers real and also the imaginary part respectively.

Here is the examples for adding complex numbers -

Solved Examples

Question 1: The given complex number is (13+9i) - (23+91i)
Solution:

First subtract the real part separately and subtract the imaginary part separately.

(13+9i) - (23+91i)

= (13 - 23) + (9-91)i

= -10 + (-82) i



Correct answer is -10 - 82 i
Question 2: Perform subtraction for the complex numbers (82+65i) - (20+5i)
Solution:

First subtract the real part separately and subtract the imaginary part separately 

(82+65i) - (20+5i)

= (82- 20) + (65-5)i

 = 62 + (60) i



Correct answer is -62 + 60 i

Multiplying Complex Numbers

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Multiplying complex numbers like binomials, i2 = -1, suppose if we have two complex numbers (a + bi) and (c + di). We need to multiply them together.

Multiply the complex number by using Z = x + yi by the real number p. Then the equation is given as

(x + yi)p = xp + ypi.

When the complex number is multiplied by using the real number p then there is a transformation by stretching the vector by a factor p without rotation.
When the multiplication of p > 0 then the transformation streches the complex plane C by using the factor p from origin.
When multiplication of p < 0 the transformation diverts toward 0

Here is the examples for adding complex numbers -

Solved Examples

Question 1: (2 + 2i)(5 + 3i)
Solution:

The multiplication of the real numbers in the arithmetic form is given as,

  (2 + 2i)(5 + 3i)
= 10 + 6i + 10i + 6i2
 = 10 + 16i - 6  (where i2 = -1)



Question 2: (3 + 7i)(4 + 3i)
Solution:

Multiplication of the complex number in algebric form is given as,

(3 + 7i)(4 + 3i) 

= 12 + 9i + 28i + 21i2

= 12 + 37i - 21



Correct answer is -9+37

Dividing Complex Numbers

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Dividing complex numbers is much like rationalizing the denominator of a readical expression. For the complex number z = a + bi we define its complex conjugate as $\hat{z}$ = a - bi

z x $\hat{z}$ = (a + bi)(a - bi) = a2 + b2
The product of a complex number and its conjugate is always be a non negative real number. Thus we use this property to divide a complex numbers.

To simplify the quotient $\frac{a+bi}{c+di}$, we multiply the numerator and the denominator by the complex conjugate of the denominator:

$\frac{a+bi}{c+di}$ = ($\frac{a+bi}{c+di}$) ($\frac{c-di}{c-di}$) = $\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$

Here is the examples for adding complex numbers-

Solved Examples

Question 1: $\frac{8+4i}{4+2i}$
Solution:

Here we can see 8+4i =2(4+2i)

So we can change it as $\frac{2(4+2i)}{4+2i}$



Correct answer is 2
Question 2: $\frac{7+3i}{2+2i}$
Solution:
Here we have to use conjugates.
The conjugate of the complex number 2+2i is 2-2i
Now we are going to multiply the numerator and denomnator by the conjugate.
So, $\frac{(7+3i)(2-2i)}{(2+2i)(2-2i)}$

Now we can expand and simplify

= $\frac{14 -14i + 6i - 6i^2}{4+4}$

$\frac{14 -14i + 6i + 6}{4+4}$

= $\frac{20 - 8i}{8}$


Correct answer is 2.5 - i

If x and y are real numbers, then x + iy is called a complex number.

x is called the real part and y is called the imaginary part.

The complex number x + iy is also written as an ordered pair(x, y) and is denoted by z.

i.e., z = x + iy

The positive value $\sqrt{x^2 + y^2}$ is called the modulus of Z and is denoted by |Z|.

i.e., Z = x + iy, then | Z | = $\sqrt{x^2 + y^2}$

In Z = x + iy, if y = 0, then Z is purely real and if x = 0, then Z is imaginary.

Examples:

2 + 4i, 6 - 5i, 3 + $\sqrt{2i}$, $\sqrt{3}$ - 2i are complex numbers.

Set of real numbers is a proper subset of the set of Complex numbers.

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Equality of Complex Numbers

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Two complex numbers are equal if their corresponding real parts and imaginary parts are separately equal.

i.e., a + ib = x + iy ⇔ a = x and b = y

Product of Two Complex Numbers

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Let Z1 = a + ib and Z2 = c + id, then

Z1Z2 = (a + ib)(c + id)

= ac + iad + ibc + i2 db

= ac + i(ad + bc) - bd ( i2 = -1)

= (ac - bd) + i(ad + bc) which is a complex number.

Conjugate Complex Numbers

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If Z = a + ib is a complex number, then a - ib is called the complex conjugate of a + ib and the conjugate is denoted by $\overline{Z}$.

Remark:

The sum and product of two conjugate numbers is always real.

For Z = a + ib, $\overline{Z}$ = a - ib

Z + $\overline{Z}$ = (a + ib) + (a - ib)

= (a + a) + i(b - b)

= 2a + i(0)

= 2a

Z . $\overline{Z}$ = (a + ib)(a - ib)

= a2 - (ib)2

= a2 - (-1)b2

= a2 + b2

Hence Z + $\overline{Z}$ and Z . $\overline{Z}$ are both real.

Reciprocal of a Complex Number

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It is also called as multiplicative inverse of a non-zero complex number

Let Z = a + ib, then $\frac{1}{Z}$ = $\frac{1}{a + ib}$

$\frac{1}{Z}$ = $\frac{1}{a + ib}$ $\frac{(a-ib)}{(a-ib)}$

= $\frac{a - ib}{a^2 + b^2}$

= $( \frac{a}{a^a + b^a} - \frac{ib}{a^a + b^a} )$

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