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
1. What is a complex number and what is its standard algebraic form?
A complex number is a number that can be expressed in the form a + ib, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i.e., i² = -1). In the standard form a + ib, 'a' is called the real part and 'b' is called the imaginary part of the complex number. This form is essential for performing algebraic operations.
2. What are the four basic algebraic operations on complex numbers?
The four fundamental algebraic operations that can be performed on complex numbers are:
- Addition: Combining the real parts together and the imaginary parts together.
- Subtraction: Subtracting the real parts and the imaginary parts separately.
- Multiplication: Expanding the product similar to binomials, remembering that i² = -1.
- Division: Multiplying the numerator and denominator by the conjugate of the denominator to make the denominator a real number.
3. How do you add or subtract two complex numbers? Provide an example.
To add or subtract two complex numbers, you simply combine their corresponding real and imaginary parts. For two complex numbers, z₁ = a + ib and z₂ = c + id:
- Addition: z₁ + z₂ = (a + c) + i(b + d)
- Subtraction: z₁ - z₂ = (a - c) + i(b - d)
Example: If z₁ = 3 + 2i and z₂ = 1 - 4i, then their sum is (3+1) + i(2-4) = 4 - 2i.
4. How is the multiplication of two complex numbers performed?
To multiply two complex numbers, z₁ = a + ib and z₂ = c + id, you can treat them like binomials and use the distributive property. The key is to substitute i² with -1.
z₁ * z₂ = (a + ib)(c + id) = a(c + id) + ib(c + id)
= ac + adi + bci + bdi²
= ac + (ad + bc)i + bd(-1)
= (ac - bd) + i(ad + bc)
The result is another complex number where (ac - bd) is the real part and (ad + bc) is the imaginary part.
5. Why is the conjugate necessary for the division of complex numbers?
The conjugate of a complex number (a + ib) is (a - ib). It is necessary for division because our goal is to eliminate the imaginary part 'i' from the denominator. Multiplying a complex number by its conjugate always results in a real number (a² + b²). This transforms the division problem into a simpler form where we just divide the real and imaginary parts of the new numerator by this real number, expressing the final answer in the standard a + ib form.
6. What are the powers of iota (i) and why do they follow a cycle?
The powers of iota (i) follow a repeating cycle of four distinct values. This cyclic pattern arises from the fundamental definition that i² = -1.
- i¹ = i
- i² = -1
- i³ = i² * i = -1 * i = -i
- i⁴ = (i²)² = (-1)² = 1
- i⁵ = i⁴ * i = 1 * i = i (the cycle restarts)
This four-step cycle (i, -1, -i, 1) repeats indefinitely, which is a crucial property used for simplifying higher powers of i.
7. How does the Argand plane differ from the Cartesian plane?
The Argand plane and the Cartesian plane are both two-dimensional coordinate systems, but they represent different types of numbers. The key difference is:
- The Cartesian plane plots ordered pairs of real numbers (x, y). Both the x-axis and y-axis represent real number lines.
- The Argand plane (or complex plane) plots complex numbers of the form a + ib. The horizontal axis is the real axis (representing 'a'), and the vertical axis is the imaginary axis (representing 'b').
Essentially, the Argand plane provides a geometric interpretation for complex numbers, which do not exist on a single real number line.
8. Do standard algebraic properties like the commutative and associative laws apply to complex numbers?
Yes, the fundamental algebraic properties that apply to real numbers also hold true for complex numbers. For addition and multiplication:
- Commutative Law: z₁ + z₂ = z₂ + z₁ and z₁ * z₂ = z₂ * z₁
- Associative Law: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃) and (z₁ * z₂) * z₃ = z₁ * (z₂ * z₃)
- Distributive Law: z₁ * (z₂ + z₃) = z₁ * z₂ + z₁ * z₃
These properties ensure that algebraic manipulations with complex numbers are consistent and predictable, similar to algebra with real numbers.
