Table of Contents
Introduction to Complex numbers
Introduction to complex numbers: Complex numbers extend the concept of real numbers by introducing the imaginary unit “i” (√-1). They are represented in the form a + bi, where “a” and “b” are real numbers. The real part “a” represents an ordinary real number, while the imaginary part “bi” involves the imaginary unit. Complex numbers find applications in mathematics, physics, engineering, and signal processing, offering solutions to a broader range of problems..
Definition of Complex numbers
Complex numbers are numbers that add an imaginary unit “i” (√-1) to the idea of real numbers. A complex number is written as a + bi, where “a” and “b” are real numbers and “i” is the imaginary unit. The real component “a” denotes a regular real number, but the imaginary part “bi” denotes an imaginary unit. Complex numbers are crucial in mathematics and have several applications in engineering, physics, and signal processing..
Complex numbers in math
Complex numbers in math extend the concept of real numbers by introducing an imaginary unit “i” (√-1). Represented as a + bi, they have a real part (a) and an imaginary part (bi). Complex numbers find applications in various mathematical fields, allowing solutions to problems involving both real and imaginary components.
Notation of complex numbers
- The notation of complex numbers is expressed as a + bi, where “a” represents the real part, “b” is the coefficient of the imaginary unit “i” (√-1). The term “bi” denotes the imaginary part. The real and imaginary parts together form a unique complex number.
- In the complex plane, the real part is plotted along the horizontal axis (x-axis), and the imaginary part along the vertical axis (y-axis).
- Additionally, complex numbers can be represented in polar form as re^(iθ), where “r” is the magnitude (modulus) and “θ” is the argument (angle) of the complex number. This notation facilitates solving complex arithmetic and analysing their geometric properties.
- Condition for purely real: A complex number a+bi is purely real if b = 0
- Condition for purely imaginary: A complex number a+bi is purely imaginary if a = 0.
What are real numbers?
Real numbers are the set of all rational and irrational numbers that can be plotted on the number line. They include integers, fractions, decimals, and square roots of positive numbers. Real numbers encompass both positive and negative values and form the basis of mathematical calculations and measurements..
What are Imaginary numbers?
Imaginary numbers are numbers that involve the imaginary unit “i” (√-1). They cannot be represented on the number line but are crucial in mathematics and engineering. Imaginary numbers are expressed as bi, where “b” is a real number, and they play a significant role in complex numbers and solving certain equations.
Absolute value of complex number
The absolute value (modulus) of a complex number a + bi is denoted by |a + bi| and is equal to the square root of the sum of the squares of its real and imaginary parts (√(a^2 + b^2)). It represents the distance of the complex number from the origin in the complex plane.
Algebraic operations on complex numbers
Algebraic operations on complex numbers include addition, subtraction, multiplication, and division.
Addition:
(a + bi) + (c + di) = (a + c) + (b + d)i
- Example: (2 + 3i) + (1 – 2i) = (2 + 1) + (3 – 2)i = 3 + i
Subtraction:
(a + bi) – (c + di) = (a – c) + (b – d)I
- Example: (5 – 3i) – (2 + 7i) = (5 – 2) + (-3 – 7)i = 3 – 10i
Multiplication:
(a + bi) * (c + di) = (ac – bd) + (ad + bc)i
- Example: (2 + 3i) * (1 – 2i) = 2 – 4i + 3i – 6i^2
= 2 – i – 6(-1) = 8 – i
Division:
(a + bi) / (c + di) = [(ac + bd) / (c^2 + d^2)] + [(bc – ad) / (c^2 + d^2)]i
Example:
(4 + 5i) / (3 – 2i) = [(4 * 3 + 5 * 2) / (3^2 + (-2)^2)]
+ [(5 * 3 – 4 * (-2)) / (3^2 + (-2)^2)]i
= (22/13) + (23/13)i
These algebraic operations allow us to perform arithmetic with complex numbers, enabling us to solve complex equations and analyze various mathematical and scientific problems.
Power of iota
The powers of the imaginary unit “i” (√-1) follow a cyclic pattern:
i^0 = 1 (Any number raised to the power of 0 is equal to 1).
- i^1 = i
- i^2 = -1 (i^2 is -1, as i * i = -1).
- i^3 = -i (i^3 is -i, as i^2 * i = -1 * i = -i).
- i^4 = 1 (i^4 is 1, as i^2 * i^2 = (-1) * (-1) = 1).
The powers of “i” repeat every four powers, forming a cycle: 1, i, -1, -i, and so on.
Argand plane
The Argand plane is a graphical representation of complex numbers. It is also known as the complex plane. It employs the Cartesian coordinate system, with the x-axis representing the real component and the y-axis representing the imaginary part of the complex number. Complex numbers are displayed as points on this plane, allowing their geometric features to be visualised. The complex number 0 corresponds to the origin (0, 0). The Argand plane simplifies complex number operations like as addition, subtraction, multiplication, and division, as well as the interpretation of magnitudes and angles using polar coordinates. This graphical tool is critical for comprehending complicated analytical and engineering applications.
Examples on complex numbers
Example 1:
Addition of Complex Numbers
Given two complex numbers: z1 = 3 + 2i and z2 = -1 + 4i
To find their sum (z1 + z2):
z1 + z2 = (3 + 2i) + (-1 + 4i) = (3 – 1) + (2 + 4)i = 2 + 6i
Example 2:
Subtraction of Complex Numbers
Given two complex numbers: z1 = 5 – 3i and z2 = 2 + 7i
To find their difference (z1 – z2):
z1 – z2 = (5 – 3i) – (2 + 7i) = (5 – 2) + (-3 – 7)i = 3 – 10i
Example 3:
Multiplication of Complex Numbers
Given two complex numbers: z1 = 2 + 3i and z2 = 1 – 2i
To find their product (z1 * z2):
z1 * z2 = (2 + 3i) * (1 – 2i) = 2 – 4i + 3i – 6i^2
Since i^2 = -1, the expression simplifies to:
z1 * z2 = 2 – 4i + 3i + 6 = 8 – i
Example 4:
Division of Complex Numbers
Given two complex numbers: z1 = 4 + 5i and z2 = 3 – 2i
To find their division (z1 / z2):
z1 / z2 = (4 + 5i) / (3 – 2i)
To eliminate the imaginary denominator, we multiply the numerator and denominator by the conjugate of z2:
z1 / z2 = (4 + 5i) * (3 + 2i) / (3 – 2i) * (3 + 2i)
Simplifying the expression:
z1 / z2 = (12 + 8i + 15i + 10i^2) / (9 + 4i^2)
Since i^2 = -1, the expression further simplifies to:
z1 / z2 = (12 + 23i – 10) / (9 + 4(-1))
z1 / z2 = (2 + 23i) / 13
These examples demonstrate the basic operations involving complex numbers. Remember that complex numbers have both real and imaginary parts, and these properties make them useful in various mathematical and engineering applications.
Also check: Co-Prime Numbers
Frequently asked questions about Complex numbers
Why 4 is a complex number?
The complex number 4 may be written as 4 + 0i, where 4 is the real component and 0 is the imaginary part. Because the imaginary element is zero, the result is a real number. To be called fully complex, numbers must have non-zero imaginary portions.
What are complex numbers and its types?
Solution: Because complex numbers are represented in the form a + bi, where a and b are real numbers and i represents the imaginary unit (-1) the number 4 is not commonly regarded a complex number. A complex number has a real (a) and an imaginary (bi) portion. The complex number 4 may be written as 4 + 0i, where 4 is the real component and 0 is the imaginary part. Because the imaginary element is zero, the result is a real number. Or 4 is a complex number with zero imaginary portion.
Is 3 √5 complex number?
Yes, the complex number 3 √5 may be written as 3 √5 + 0i, where 3 √5 is the real component and 0 is the imaginary part. Because the imaginary element is zero, the result is a real number. 3 √5 is a complex number with zero imaginary portions
What are all complex numbers?
All complex numbers have the formula a + bi, where a and b are real numbers and i represents the imaginary unit (-1). The symbol C represents the set of complex numbers. In this form, a represents the complex number's real portion, while b is its imaginary part. The complex number becomes a real number if the imaginary element (b) is zero. When the real component (a) is zero and the imaginary part (b) is non-zero, the result is a pure imaginary number.
Why are complex numbers called so?
Complex numbers are called complex because they are an extension of the real numbers, introducing the concept of the imaginary unit i (√-1). When mathematicians were trying to solve polynomial equations, they encountered situations where the solutions were not expressible using real numbers. To overcome this limitation, they introduced the imaginary unit i and created the concept of complex numbers. The name complex comes from the fact that these numbers have both a real part and an imaginary part, making them more intricate than ordinary real numbers. The term complex does not imply that they are complicated or difficult to work with, but rather highlights their richer mathematical structure, allowing for solutions to a broader range of mathematical problems, including those that involve non-real solutions.
Is zero is complex number
Yes, zero (0) is considered a complex number. A complex number is of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Zero can be expressed as 0 + 0i, where the real part a is 0, and the imaginary part bi is also 0.
What is complex numbers and its properties?
Complex numbers are numbers that extend the concept of real numbers by introducing an imaginary unit i (√-1). A complex number is represented in the form a + bi, where a and b are real numbers, and i is the imaginary unit. Properties of complex numbers: Addition, Subtraction, Multiplication, Division, Complex Conjugate, Modulus (Absolute Value), Complex Plane, Euler's Formula
Is square root of 2 is complex number?
Yes, the square root of 2 (√2) is considered a complex number. A complex number is of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). In the case of √2, it can be expressed as √2 + 0i, where the real part a is √2 and the imaginary part bi is 0. Since √2 can be represented in the form of a complex number, it is included in the set of complex numbers (ℂ). However, it is also a real number since the imaginary part is zero (b = 0). All real numbers can be considered as complex numbers with a zero imaginary part.
What is the formula for complex number?
The formula for a complex number is: a + bi where a is the real part of the complex number, b is the imaginary part of the complex number, and i is the imaginary unit (√-1). In this representation, a and b are real numbers, and i is used to denote the imaginary unit. Complex numbers are often written in this form to distinguish between the real and imaginary components. The real part represents the ordinary real number, and the imaginary part represents the imaginary number multiplied by i.
