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Algebraic operations on complex numbers are always expressed by four fundamental arithmetic operations that are addition, subtraction, multiplication, and division. A complex number comprises two numbers: a real number and an imaginary number. Algebraic approaches are the only way to express algebraic operations on complex numbers. The relationship between the number of operations is explained using simple algebraic laws such as associative, commutative, and distributive law.
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Keyterms: Algebraic operations, complex numbers, fundamental arithmetic operations, addition, subtraction, multiplication, division, associative, commutative, distributive law
Complex Numbers
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A complex number is a number that comprises both real and imaginary variables. The variables we use to perform mathematical calculations are called real numbers. However, except in the case of complex numbers, imaginary numbers are rarely applied in computation.
Representation of a complex number
Equality of Complex Numbers
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Imagine that Y1 and Y2 are the two complex numbers.
Here Y1 = a1+i b1 and Y2 = a2+i b2
If both the complex numbers (Y1 and Y2) are equal that is
Y1=Y2
The real part of the first complex number is equal to the real part of the second complex number and the imaginary part of the first complex number is equal to the imaginary part of the second complex number.
That is Re (Y1) = Re (Y2) and Im (Y1) = Im (Y2)
As a result, the equality of complex numbers indicates. If a1+ib1 = a2+ib2, then a1=a2 and b1=b2.
Power of i
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The alphabet i is referred to as the iota and is helpful to represent the imaginary part of the complex number.
Further the iota(i) is very helpful to find the square root of negative numbers. We have the value of i2 = -1, and this is used to find the value of √-4 = √i24 = +i.
The value of i2 = 1 is the fundamental aspect of a complex number. Let us try and understand more about the increasing powers of i.
i = √-1
i2 = -1
i3 = i.i2 = i(-1) = -i
i4 = (i2)2 = (-1)2 = 1
i4N = 1
i4N+1 = i
i4N+2 = -1
i4N+3 = -i
Operations on Complex Numbers
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The following are the basic algebraic operations on complex numbers:
- Addition of Complex Numbers
- Subtraction of Complex Numbers
- Multiplication of Complex Numbers
- Division of Complex Numbers
Addition of Complex Numbers
A complex number is defined as z=a+ib, where a and b both are real numbers.
Consider the two complex numbers, z1 = a1 + ib1 and z2 = a2 + ib2.
Then the complex numbers z1 and z2 are added as follows:
z1+z2 = (a1+a2) +i (b1+b2)
The real portion of the resulting complex number is equal to the sum of the real parts of each complex number, and the imaginary part is equal to the sum of the imaginary parts of each complex number, that is
Re (z1+z2) = Re (z1) +Re (z1) and Im(z1+z2 )=Im( z1)+Im(z2)
Properties of Addition of Complex Numbers
| Name of the Property | Description | Expression |
|---|---|---|
| Closure property | A complex number is formed by adding two complex numbers. | z1 + z2 = z |
| Commutative property | The result is unchanged by the order in which two complex numbers are added. | z1 + z2 = z2 + z1 |
| Associative property | Rearranging three complex numbers while adding them has no effect on the outcome. | (z1+z2) +z3 = z1+(z2+z3) |
| Additive inverse property | If z = a+ib is a complex number, then -z = -a – ib is its additive inverse. | z+(-z) = 0 |
| Additive identity | When a value is added to a complex number and the outcome is the same complex number, it is called additive identity. | (a+ib) + (0 + i0) = a + ib |
Subtraction of Complex Numbers
The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part. For the complex numbers
z1 = a + ib, z2 = c + id ,
we have z1 - z2 = (a - c) + i(b - d)
Multiplication of Complex Numbers
The expansion of (a+b) (c+d) =ac+ad+bc+bd
Similarly, take the complex numbers z1 = a1+ib1 and z2 = a2+ib2
Then, the product of z1 and z2 is defined as:
z1 z2=(a1+ib1) (a2+ib2)
z1 z2 = a1 a2+a1 b2 i+b1 a2i+b1 b2 i2
Since, i2 = -1, therefore, z1 z2 = (a1 a2 – b1 b2) + i (a1 b2 + a2 b1)
Properties of Multiplication of Complex Numbers
| Name of the Property | Description | Expression |
|---|---|---|
| Closure property | Only a complex number can be formed by multiplying two complex numbers. | z1*z2 = z |
| Commutative property | Changes in the order of complex numbers have no effect on the outcome of their product. | z1*z2 = z2*z1 |
| Associative property | Regrouping complex numbers has no effect on the outcome of their product. | z1(z2*z3) = (z1*z2) z3 |
| Distributive property | When a complex number is multiplied by the sum of two complex numbers, the result is | z1(z2+z3) = z1*z2 + z1*z3 |
Division of Complex Numbers
Imagine the complex number y1 = a1 + ib1 and y2 = a2 + ib2, then the quotient of y1/y2 is written as, y1/y2=y1 × 1/y2
Therefore, to determine y1/y2, we have to multiply y1 with the multiplicative inverse of y2.
Check the example below to understand the concept of Division of complex numbers in detail:
Let y1 = a1+ib1 and y2 = a2+ib2, then y1/y2 can be written as:
y1/2 = (a1+ib1)/(a2+ib2)
So, (a1+ib1)/(a2+ib2) = [(a1+ib1) (a1-ib2)]/[(a2+ib2) (a2-ib2)]
(a1+ib1)/(a2+ib2) = [(a1a2) -(a1b2i) +(a2b1i) +b1b2)]/[(a22+b22)]
(a1+ib1)/(a2+ib2) = [(a1a2) +(b1b2) +i(a2b1-a1b2)]/(a22+b22)
Therefore, y1/y2=a1a2+b1b2/a22+b22 + i a2b1−a1b2/a22+b22
The video below explains this:
Algebraic Operations on Complex Numbers Detailed Video Explanation:
The conjugate of a complex number
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The conjugate of a complex number a + bi is the complex number a - bi.
For example, the conjugate of 3 + 7i is 3 - 7i.
If a complex number is multiplied by its conjugate, the result will be a positive real number which, of course, is still a complex number where the b in a + bi is 0.
Things to Remember
- All real numbers are complex numbers but all complex numbers need not be real numbers.
- The equality of complex numbers indicates that if a1+ib1 = a2+ib2, then a1=a2 and b1=b2.
- A complex number is a number that comprises of both real and imaginary variables
- The real part of the complex number is equal to the sum of the real parts of each complex number, and the imaginary part is equal to the sum of the imaginary parts of each complex number.
- The conjugate of a complex number a + bi is the complex number a - bi.
- Suppose Y1 = a1+i b1 and Y2 = a2+i b2 are two complex numbers:
- Addition- Y1 + Y2= a1+a2 +i (b1 +i b2)
- Subtraction- Y1 - Y2= a1-a2 +i (b1 -i b2)
- Multiplication- Y1Y2= (a1a2 - b1b2)+i (a1b2 +a2b1)
- Division- Y1 / Y2= (a1a2 + b1b2) /(a22 +b22)+i [(a2b1 -a1b2)/(a22 +b22)]
Also Read:
| Related Articles | ||
|---|---|---|
| Branches of Algebra | Sum of Squares | |
| Geometric Progression | Polynomials | |
| Geometric mean | Sequence and series | |
Important Questions
Ques: Add 2+4i and -1+3i. (1 marks)
Ans: Given: two complex numbers,
2+4i and -1+3i
Sum of the given numbers= (2+4i) +(-1+3i)
=(2-1) +(4i+3i)
=1 + 7i
Ques: Simplify: 7 + i + 4 + 4. (1 marks)
Ans: Given that:
x= 7 + i + 4 + 4
= (7+4+4) + i
= 15 + i
Ques: Multiply the complex numbers (5+3i) * (3+4i). (2 marks)
Ans: Given two complex numbers: (5+3i) and (3+4i)
(5+3i) * (3+4i) = 15+20i+9i-12
(5+3i) * (3+4i) = (15-12) + i (20+9)
(5+3i) * (3+4i) = 3+29i
The product of (5+3i) and (3+4i) is 3+29i.
Ques: Multiply (1 + 4i) and (3 + 5i). (2 marks)
Ans: (1 + 4i) ∗ (3 + 5i) = (3 + 12 i) + (5i + 20i2)
= 3 + 17i − 20
= −17 + 17i
Ques: Subtract (2+5i) from (7+15i). (2 marks)
Ans: We know that:
(a+bi) – (c+di) = (a-c) + i(b-d).
Hence,
(7+15i) – (2+5i) = (7-2) +i (15-5)
(7+15i) – (2+5i) = 5+10i
Ques: Compute: (2 + i)2 and express answer in a + bi form. (3 marks)
Ans: Let x be an imaginary number
Given that:
x= (2 + i)2
= (2 + i) • (2 + i)
= 2(2 + i) + i(2 + i)
= 4 + 2i + 2i + i2
= 4 + 4i + (-1) = 3 + 4i
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