How to evaluate Complex Numbers?
Last Updated :
06 Sep, 2024
Answer : Methods to evaluate complex numbers:
- Write given complex numbers in the form of a+ib
- If a complex number is in the form of an exponential write it in cosθ+i sinθ form so that you can able to write in a+ib form.
- If a 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)
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 an 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 the 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.
Related Articles:
Similar Reads
How to distribute complex numbers?
Complex numbers are used to find the square root of negative numbers. It comprises the real and imaginary parts. The complex number is of the form a + ib where a is the real part and b is the imaginary part. The real part of the complex number is any number and is represented by Re(z) where 'z' is a
5 min read
Conjugate of Complex Numbers
In the world of mathematics, complex numbers are one of the most important discoveries by mathematicians as they help us solve many real-life problems in various fields such as the study of electromagnetic waves, engineering, and physics.The Conjugate of a Complex Number is also a complex number obt
6 min read
How to Use Complex Numbers in math.js?
Complex numbers are essential in various fields of science and engineering, particularly in signal processing, quantum mechanics, and control theory. A complex number consists of a real part and an imaginary part, typically written in the form a+bi, where i is the imaginary unit. Math.js, a powerful
3 min read
Complex Numbers
Complex numbers are an essential concept in mathematics, extending the idea of numbers to include solutions for equations that don't have real solutions. Complex numbers have applications in many scientific research areas, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and v
12 min read
Modulus and Conjugate of a Complex Number
Complex numbers find application in different areas of mathematics and engineering science. With complex numbers, there are two significant values: modulus and conjugate. The modulus gives the ‘size’ or ‘degree of magnitude’ of the complex number, the conjugate on the other hand involves the changin
5 min read
Complex Number Formula
The sum of a real number and an imaginary number is defined as a complex number, and the numbers that are not real numbers are called imaginary numbers. The number can be written in the form of b+ic, where b and c are real numbers ic is an imaginary number, and †i†is an imaginary part which is cal
13 min read
Complex Number Power Formula
Complex Numbers are numbers that can be written as a + ib, where a and b are real numbers and i (iota) is the imaginary component and its value is √(-1), and are often represented in rectangle or standard form. 10 + 5i, for example, is a complex number in which 10 represents the real component and 5
6 min read
What are Complex Numbers?
A complex number is a term that can be shown as the sum of real and imaginary numbers. These are the numbers that can be written in the form of a + ib, where a and b both are real numbers. It is denoted by z. Here the value 'a' is called the real part which is denoted by Re(z), and 'b' is called the
6 min read
Dividing Complex Numbers
Complex numbers are those with the formula a + ib, where a and b are real numbers and i^2 (iota) is the imaginary component and represents (-1), and are often represented in rectangular or standard form. 10 + 5i, for example, is a complex number in which 10 represents the real component and 5i repre
3 min read
How to convert a complex number to exponential form?
Complex numbers are a way of numerical representation in the form of a + ib in which a and b stands for real numbers and stands for imaginary number. For example, 2 + 5i is a complex number in which 2 and 5 are the real numbers in the place of a and b. And, i is the imaginary number. Different forms
3 min read