Complex Numbers are the numbers of the form (a + i b) where a & b are the real numbers and i is an imaginary unit called iota that represents √-1. Complex numbers contain both a real part and the imaginary part. Complex numbers are a part of the number system. Number systems have sets of natural numbers, whole numbers, integers, real numbers, and complex numbers.
A number system is a combination of natural numbers, whole numbers, integers, real numbers, and complex numbers.
Natural numbers are also known as counting numbers. Counting numbers are the numbers that we count from 1 to some extent of numbers.
Whole numbers are the numbers in a set of 0 and natural numbers. These whole numbers have 0 added as additional with natural numbers.
Integers are the numbers that have positive and negative numbers. This integer set has positive numbers like natural numbers, zero, and negative numbers.
Real numbers are numbers having both rational and irrational numbers. Rational numbers are in the form of p/q and terminating numbers whereas irrational numbers are not in the form of p/q and non-terminating numbers.
Complex numbers are numbers that are a combination of a real part and an imaginary part. The imaginary part is like the multiplication of i=√(-1) and coefficient. The real part is like coefficient where coefficient may be real numbers, integers, whole numbers, and natural numbers. The real part and the imaginary part are combined with “+”. Representation of complex numbers in the form of a+ ib which is a general form of a complex number.
Complex numbers
Complex numbers are defined as the combination of real parts and imaginary parts. Complex numbers are in the form of a + ib. There are three types of complex numbers
Polar form: The polar form of a complex number is represented as r(cosθ + isinθ). Here rcosθ is a real part of complex numbers and rsinθ is the imaginary part of complex numbers.
Exponential form: The exponential form of complex numbers is represented as reiθ. Here reiθ is represented as rcosθ+i rsinθ where rcosθ is the real part and rsinθ is the imaginary part.
General form: The general form of complex numbers is represented as a+ib. Here a is the real part and b is an imaginary part coefficient.
How complex numbers are evaluated?
- There are three forms of complex numbers. In those forms, complex numbers can able to perform normal calculations like addition, subtraction and multiplication, and even division also.
- Complex numbers are evaluated by using convenient forms for every calculation we perform these days.
Method
- Write given complex numbers in the form of a+ib for additional purposes.
- If complex numbers are not in the form of a+ib then convert the given complex number into a+ib.
- If a complex number is in the form of exponential write it as cosθ+i sinθ form so that you can able to write in a+ib form.
- If complex number is in the form of polar form then convert r(cosθ + i sinθ) into a+ib form by the formulas where θ =tan-1(b/a) , r = √(a2 + b2)
- For additional calculation of a group of complex numbers write the real part left most side after equal to the symbol and write the imaginary parts next to the real parts.
- The above point 5 is easy to calculate addition and subtraction calculations.
How to calculate the Multiplication of complex numbers
- Write given complex numbers in the form of a+ib for multiplication purposes.
- Multiply complex numbers (a+ib) and (p+iq).
- After multiplying complex numbers (a+ib) and (p+iq) we will get (ap+iaq+pbi+i2bq)=(ap-bq)+i(pb+aq)
How to calculate the Division of complex numbers
- Write given complex number in the form of a+ib for division purposes.
- Division complex numbers (a+ib) and (p+iq).
- In the division of complex numbers (a+ib)/(p+iq) we have to multiply and divide with denominator conjugate which is p-iq.
- In adding p-iq as multiplication and division of (a+ib)/(p+iq) as (a+ib)×(p-iq)/(p+iq)*(p-iq)=(ap-bq)+i(pb+aq)/(p2+q2).
Complex conjugate
Definition: Complex conjugate is defined as the inverse of the sign of the imaginary part. It is represented as Z*. Let’s say the conjugate of a+ib is a-ib.
Examples:
- The complex conjugate of a complex number 2+i7 is 2-i7.
- The complex conjugate of a complex number 4+i9 is 4-i9.
How to find complex conjugates of a complex number.
Step 1: Write the complex number given.
Step 2: To find a complex conjugate we have to inverse the sign of the imaginary part of a complex number a+ib as a-ib
Step 3: Complex number of Z= a+ib is find as Z*= a-ib.
Examples: Find the complex conjugate of Z = 3+i9.
Answer:
Given complex number Z = 3+i9.
Complex conjugate Z* is found as the inverse of the imaginary part of complex number = 3-i9.
Therefore complex conjugate Z*= 3-i9
Practice Problems based on Complex numbers
Question 1: Find the addition of two complex numbers 7+i9 and 2+i8.
Answer:
Given complex numbers
Let p=7+i9, q= 2+i8
Addition of p+q = 7+i9+2+i8
= 7+2+i9+i8
= 9+i17
Therefore addition of 7+i9 and 2+i8 is 9+i17
Question 2: Find subtraction of two complex numbers 6+i1 and 2+i6
Answer:
Given complex numbers
Let p=6+i1
q= 2+i6
subtraction of p-q = 6+i1-(2+i8)
= 7-2+i1-i8
= 5-i7
Therefore subtraction of 6+i1 and 2+i6 is 5-i7
Question 3: Find subtraction of three complex numbers 6+i1, 2+i6 and 4+i0
Answer:
Given complex numbers
Let p=6+i1
q= 2+i6
r= 4+i0
First subtract the s = p-q and then subtract result s with r which is s-r
subtraction of s = p-q = 6+i1-(2+i8)
s = 7-2+i1-i8
s = 5-i7
subtraction of s-r = 5-i7-(4+i0)
=5-4-i7-i0
=1-i7
Therefore subtraction of 6+i1, 2+i6 and 4+i0 is 1-i7
Question 4: Find the addition of three complex numbers 7+i7, 2+i8, and 5+i5.
Answer:
Given complex numbers
Let p=7+i7
q= 2+i8
r= 5+i5
Addition of p+q+r = 7+i7+2+i8+5+i5
= 7+2+5+i7+i8+i5
= 14+i20
Therefore addition of 7+i7 , 2+i8 and 5+i5 is 14+i20
Question 5: Find the Multiplication of two complex numbers 6+i4 and 5+i2
Answer:
Given that complex numbers p=6+i4, q=2+i9
Multiplication of complex number p and q is p×q = (6+i4)×(2+i9)
= (6×2+i9×6+ i4×2+i2 ×36 )
= (12+ 54i+ 8i-36)
= ( 24+ 62i)
Multiplication of two complex numbers 6+i4 and 5+i2 is ( 24+ 62i)
Question 6: Find the Division of two complex numbers 1+3i and 9+i1.
Answer:
Given that complex numbers p= 1+3i, q= 9+i1.
Division of complex number p and q is p/q = 1+3i/( 9+i1)
= (1+3i/( 9+i1))×(9-i1)/(9-i1)
= ((1+3i)×(9-i1) /( 9+i1)×(9-i1))
= ( 9-i1+27i-i23)/(81-9i+9i-i21)
= (9+3+26i)/(81+1+0)
= (12+26i)/82
= (12/82) + (26i/82)
= 0.146+ 0.317i
Division of two complex numbers 1+3i and 9+i1 is 0.146+ 0.317i.