How to Add Subtract Multiply and Divide Complex Numbers with Formulas and Solved Examples
Algebraic Of Complex Numbers
Have you ever heard of complex numbers? Do you know what an iota is? What kind of number is \[\sqrt{-2}\]? Does the number even exist? To get all your answers, let’s first understand the entire number system.
Number System
The number system is broadly divided into two parts: Real Numbers and Complex Numbers.
Real numbers
Real numbers are those which can be shown on a number line. On the other hand, complex numbers are those which can not be expressed on a number line or be experienced in real life. Real Numbers are further divided into two categories called rational and irrational numbers.
Rational numbers are numbers which can be expressed as fractions and their denominators are not equal to 0. All the real numbers which are not rational are Irrational numbers. Rational numbers are made by dividing two integers. Integers include all negative and positive natural numbers along with zero. Integers are further divided into two sub-categories: whole numbers and natural numbers. Whole numbers are positive counting numbers along with 0. When you remove 0 from whole numbers, we obtain positive counting numbers which are known as natural numbers.
Complex Numbers
Complex numbers are also known as Imaginary numbers. Now that we know the definition of complex numbers and that complex numbers are the part of the Number System, let’s see some examples.
All the negative numbers under root are imaginary numbers.
\[\sqrt{-2}\] and \[\sqrt{-2}\] are two very different things. The first one is a real number. Since it’s a negative number under root the second one is a complex number. A complex number is represented in the following way: a+bi, where a is the real part and b is the imaginary part.
You can write the complex number \[\sqrt{-2}\] in a+bi form. 0+2i is equal to \[\sqrt{-2}\]. You must be wondering why are we using the symbol ’i’? What does it mean? Well, it is iota. Have you ever heard of iota? If not, then is all that you need to know about iota.
IOTA
Iota is a greek letter which is used to represent the imaginary part of a complex number. Iota(i) is considered to be the square root of -1. It may also be defined as a number whose square is -1.
i=\[\sqrt{-1}\]
i²=-1
i³=i
i⁴=1
Algebraic Operations On Complex Numbers:
Four types of algebraic operations can be done on complex numbers. These four algebra of complex numbers are:
Addition
Subtraction
Multiplication
Division
There are several properties that algebra on imaginary numbers follow:
Closure law
The sum or product of two imaginary numbers will always get you an imaginary number.
Commutative Law
If you change the order of imaginary number while adding or multiplying the result will not change that is the answer you get will always be the same.
Associative Law
If you add or multiply any three complex numbers in any order the result will always remain the same.
Existence of Additive Identity
This property tells us that if we add zero to any complex we will get the same complex number. This shows that there’s a number that can be added to get the same number back. It is also known as zero complex number and is denoted as 0 (or 0 + i0).
Existence of Additive Inverse
A complex number has the opposite sign for its both real and imaginary parts. This is known as the Existence of Additive inverse.
Multiplicative Identity
Multiplicative Identity is a property which talks about the existence of a complex number that when multiplied to another will get the same result. it is denoted as 1 (or 1 + i0)
Multiplicative Inverse
It is a property of any non- zero complex number to have a reciprocal. This is known as the multiplicative inverse.
Distributive Property
When you split the multiplication of a complex number by another term this property is known as the distributive property.
Note: Subtraction follows all the properties followed by addition.
Fun Facts:
Both real and imaginary parts are present in the square root of i.
The N-th root can have N number of unique solutions and any root of i has multiple unique solutions
The result may vary depending on whether i is present in the numerator or denominator in an imaginary fraction.
When you raise i to the i power, the number you get is a real number.
Numbers like \[\pi \], i and e are all related to one another.
FAQs on Algebraic Operations on Complex Numbers with Rules and Applications
1. What are algebraic operations on complex numbers?
Algebraic operations on complex numbers are the rules for addition, subtraction, multiplication, and division of numbers in the form a + bi, where i² = −1. A complex number has:
- a = real part
- b = imaginary part
2. How do you add complex numbers?
To add complex numbers, add the real parts together and the imaginary parts together separately. For (a + bi) + (c + di):
- Real part: a + c
- Imaginary part: b + d
Example: (3 + 2i) + (1 + 4i) = (3 + 1) + (2 + 4)i = 4 + 6i.
3. How do you subtract complex numbers?
To subtract complex numbers, subtract the real parts and subtract the imaginary parts separately. For (a + bi) − (c + di):
- Real part: a − c
- Imaginary part: b − d
Example: (5 + 6i) − (2 + 3i) = (5 − 2) + (6 − 3)i = 3 + 3i.
4. How do you multiply complex numbers?
To multiply complex numbers, use the distributive property (FOIL method) and simplify using i² = −1. For (a + bi)(c + di):
- Expand: ac + adi + bci + bdi²
- Since i² = −1, replace bdi² with −bd
Example: (2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i² = 2 + 11i − 12 = −10 + 11i.
5. How do you divide complex numbers?
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For (a + bi)/(c + di):
- Multiply top and bottom by (c − di)
- Denominator becomes c² + d²
Example: (1 + 2i)/(3 + 4i) = (1 + 2i)(3 − 4i)/(3² + 4²) = (11 + 2i)/25 = 11/25 + 2/25 i.
6. What is the conjugate of a complex number?
The conjugate of a complex number a + bi is a − bi. It is formed by changing the sign of the imaginary part.
- Conjugate of 3 + 5i is 3 − 5i
- Conjugate of 4 − 2i is 4 + 2i
7. What is the modulus of a complex number?
The modulus of a complex number z = a + bi is |z| = √(a² + b²). It represents the distance of the point from the origin on the complex plane.
- For z = 3 + 4i
- |z| = √(3² + 4²) = √(9 + 16) = 5
8. What are the properties of complex number operations?
Algebraic operations on complex numbers follow the same basic properties as real numbers. These include:
- Commutative property (z₁ + z₂ = z₂ + z₁)
- Associative property
- Distributive property
- Additive identity: 0 + 0i
- Multiplicative identity: 1 + 0i
9. What is the standard form of a complex number?
The standard form of a complex number is a + bi, where a and b are real numbers and i² = −1. In this form:
- a is the real part
- b is the imaginary coefficient
10. Can you give a worked example of all algebraic operations on complex numbers?
Yes, for z₁ = 2 + 3i and z₂ = 1 − i, the results of algebraic operations are as follows:
- Addition: (2 + 3i) + (1 − i) = 3 + 2i
- Subtraction: (2 + 3i) − (1 − i) = 1 + 4i
- Multiplication: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i
- Division: (2 + 3i)/(1 − i) = (2 + 3i)(1 + i)/(1 + 1) = (−1 + 5i)/2 = −1/2 + 5/2 i
