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 a symbol known as the imaginary unit that solves the equation i² = −1. René Descartes referred to ‘I’ as an imaginary number because no real number satisfies this equation.
This article will teach us about the definition, terminology, visualization, properties, and operations of complex numbers.
What is a Complex Number?
The product of a real and an imaginary number is known as a complex number. A complex number has the formula a + ib and is usually represented by the symbol z. Both a and b are real numbers in this case. The value ‘a’ is basically known as the real part, which is denoted by Re(z), and the value ‘b’ is known as the imaginary part Im (z). ib is also known as an imaginary number.
Power of I Used in Complex Numbers
The letter ‘I’ is known as the iota, and it is used to represent the imaginary part of a complex number. Furthermore, the iota(i) function is very useful for calculating the square root of negative numbers. We have the value of i2 which equals -1, and this is used to find the value of √-4 equals √i24 = +2i The value of i2 is equal to 1 which is the fundamental aspect of a complex number. Further, let’s try to understand more about the increasing powers of i.
- i equals √-1
- i2 equals -1
- i3 equals i.i2 equals i(-1) = -i
- i4 equals (i2)2 equals (-1)2 = 1
- i4n equals 1
- i4n + 1 equals i
- i4n + 2 equals -1
- i4n + 3 equals -i
Modulus of the Complex Number
The modulus of the complex number is the distance of the complex number represented as a point in the argand plane (a, ib).
Argument of a Complex Number
The angle formed by the line connecting the geometric representation of the complex number and the origin with the positive x-axis in the anticlockwise direction is known as the complex number’s argument.
Graphing of Complex Numbers
The complex number is made up of a real and an imaginary part, which can be represented as coordinate points in the euclidean plane as an ordered pair (Re(z), Im(z)). The complex plane, named after Jean-Robert Argand, is the euclidean plane with reference to complex numbers.
The complex number z = a + ib is represented by the real part – ‘a’ on the x-axis and the imaginary part – ‘ib’ on the y-axis.
Use of Complex Numbers
In mechanical engineering, the use of complex numbers is limpse.
- Potential theory and aerofoil design are examples of fluid mechanics.
- Damping, loss factor, and tan delta are all terms used to describe viscoelasticity.
- Application in vibration theory – Random and harmonic vibration
- It is also used in data processing with tools such as FFT and Laplace transform.
- Any type of harmonic excitation
Polar Form and Argand Plane
The Argand plane, also known as a complex plane, like the XY plane, is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.
We calculate the real and complex components in terms of r and θ, where r is the vector’s length and θ is the angle formed with the real axis.
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