How to Plot Complex Numbers in Argand Plane and Convert to Polar Form
In the subject of mathematics, the z-plane or complex plane is the type of plane related to the complex coordinate system that is established by the original axis and the perpendicular imaginary axis. It may be assumed as a modified Cartesian plane, including the real portion of a complex number which is represented by a displacement along the x-axis, and the imaginary portion which is represented by a displacement along the y-axis. The concept of the complex plane permits the geometric interpretation of complex numbers.
Argand Plane in Addition
In the case of an addition, they add like vectors and the multiplication of two complex numbers can be understood most easily in polar coordinates.
The modulus or magnitude of the product is the final product of the two absolute values or the two moduli, and the angle or the argument of the product is the sum of the two arguments or angles. This complex plane as discussed above is known as the Gauss plane or the Argand plane.
What is Meant By Complex Number?
Complex numbers are a kind of number that comprises two parts; which are a real number and an imaginary number. Complex numbers are considered to be the building blocks of further complicated calculations in the subject of mathematics, such as algebra. Complex Number itself has various ways through which they can be expressed.
Among these, the four most commonly used methods are listed below:
The Cartesian, Algebraic, or the Rectangular Form: The most straightforward way to calculate the Argand Diagram of Complex numbers.
The Exponential Form: The alternative method involving the usage of exponents to calculate the result.
Trigonometric or the Polar Form: With the help of trigonometry, a Complex Calculation can be completed.
Vector Form: Calculation using a vector that consists of both direction and magnitude, to calculate argand planes of complex numbers.
Importance of Complex Numbers
Complex Numbers find use in both physics and mathematics.
Complex Numbers are used to minimize the use of Calculus.
Complex Numbers are a much sought-after option in solving geometric equations in Mathematics, and make the process of calculation easier.
Complex Numbers have gained a legitimate place in the subject of mathematics.
Complex Numbers are relatively used in multiplying angles and vectors.
Complex Numbers are also used to describe real-life quantities through the representation of real numbers.
Complex Numbers introduce geometric interpretations of arithmetic operations.
Complex Numbers can represent real numbers on a two-dimensional plane.
Complex Numbers satisfy all three laws: commutative, associative and distributive laws.
What is Meant by Polar Representation?
Apart from the rectangular form, the polar form of a complex number is another approach to represent a complex number. Typically, complex numbers are represented in the manner z = x+iy, where 'i' is an imaginary number. Complex numbers, on the other hand, are expressed in polar form as a combination of modulus and argument.
A complex number's modulus is also known as its absolute value. This polar form is represented in the coordinate system by polar coordinates of real and imaginary integers.
What is meant by Modulus?
The term Modulus of the complex number or the magnitude is referred to the distance of the particular point on the argand plane that represents the complex number z from the point of origin, in the Argand diagram. In simpler words, a Modulus is the measurement of the distance of the complex number from the point of its origin.
If there is a complex number z= x+iy, then the modulus is given with the formula:
$|z|=\sqrt{a^{2}+b^{2}}$
What you Should Remember:
x and y both will always be real numbers
x is considered the real segment of z (complex number)
y when used with i is known as the imaginary segment of z (complex number)
Hence, complex numbers = real segment + imaginary segment
Solved Example:
Suppose you have to find the complex number z of the equation where, x = -3 &
yi = 4i then, z= -3+4i where, z is the complex number, (-3) is the real segment and 4i is the imaginary segment. The formula is explained below on how to find the distance between the origin and the complex number.
$|z|=\sqrt{a^{2}+b^{2}}$
$r= |z|=\sqrt{-3^{2}+4^{2}}$
$r= |z|=\sqrt{{9}+16}$
$r= |z|=5$
where,
r = the length of the vector
Importance of Argand Diagrams
Argand Diagrams are often used to plot the positions of poles of a function and the zeros in the complex plane.
Example Determining the Coordinates of Complex Numbers
If z = 7+i is represented in the form of an Argand Diagram, the Cartesian Coordinates are:
x and y are the axes of the diagram
Im = the imaginary plane
R = the real plane
The value z = 7+i has been plotted.
Practice Question MCQs
1. In the argand plane, which axis is known as the real axis?
z-axis
y-axis
x-axis
None of these
Answer: C)
2. In the argand plane, which axis is known as the real axis?
Z-axis
Y-axis
X-axis
None of these
Answer: C)
FAQs on Argand Plane and Polar Form of Complex Numbers Explained
1. What is the Argand plane in complex numbers?
The Argand plane is a graphical representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part. In this plane:
- The x-axis is the real axis.
- The y-axis is the imaginary axis.
- A complex number z = a + ib is plotted as the point (a, b).
2. How do you represent a complex number on the Argand plane?
A complex number z = a + ib is represented by plotting the point (a, b) on the Argand plane. Follow these steps:
- Identify the real part a (x-coordinate).
- Identify the imaginary part b (y-coordinate).
- Plot the point (a, b).
3. What is the modulus of a complex number in the Argand plane?
The modulus of a complex number is its distance from the origin in the Argand plane and is given by |z| = √(a² + b²). For z = a + ib:
- Square the real part: a²
- Square the imaginary part: b²
- Add and take the square root
4. What is the argument of a complex number?
The argument of a complex number is the angle θ that the line joining the point to the origin makes with the positive real axis. It is calculated using θ = tan⁻¹(b/a) (with quadrant consideration). For z = a + ib:
- Find tan⁻¹(b/a).
- Adjust θ based on the quadrant.
5. What is the polar form of a complex number?
The polar form of a complex number is written as z = r(cosθ + i sinθ), where r is the modulus and θ is the argument. Here:
- r = |z|
- θ = arg(z)
6. How do you convert a complex number from rectangular form to polar form?
To convert z = a + ib to polar form, calculate its modulus and argument, then write it as r(cosθ + i sinθ). Steps:
- Find r = √(a² + b²).
- Find θ = tan⁻¹(b/a) (adjust quadrant).
- Substitute into r(cosθ + i sinθ).
7. How do you convert a complex number from polar form to rectangular form?
To convert from polar form z = r(cosθ + i sinθ) to rectangular form, expand using cosine and sine values. Steps:
- Multiply r by cosθ to get the real part.
- Multiply r by sinθ to get the imaginary part.
8. What is Euler’s form of a complex number?
Euler’s form expresses a complex number as z = re^{iθ}, where r is the modulus and θ is the argument. It comes from Euler’s identity:
- e^{iθ} = cosθ + i sinθ
9. Why is polar form useful for multiplying and dividing complex numbers?
Polar form simplifies multiplication and division because you multiply moduli and add arguments. If z₁ = r₁e^{iθ₁} and z₂ = r₂e^{iθ₂}:
- Multiplication: z₁z₂ = r₁r₂ e^{i(θ₁ + θ₂)}
- Division: z₁/z₂ = (r₁/r₂) e^{i(θ₁ − θ₂)}
10. What is the geometric meaning of a complex number in polar representation?
In polar representation, a complex number represents a vector with magnitude r and direction θ in the Argand plane. Specifically:
- r is the length (distance from origin).
- θ is the angle from the positive real axis.
