What is Meant By an Argand Plane and Polar Representation?
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 Representation of Complex Number
1. What is an Argand plane?
An Argand plane, also known as the complex plane, is a two-dimensional coordinate system used to represent complex numbers geometrically. In this plane, the horizontal axis is called the real axis (representing the real part of the complex number) and the vertical axis is called the imaginary axis (representing the imaginary part).
2. How do you plot a complex number like z = a + ib on an Argand plane?
To plot a complex number z = a + ib on an Argand plane, you treat the real part 'a' as the x-coordinate and the imaginary part 'b' as the y-coordinate. You then plot the point (a, b) on the plane. This point uniquely represents the complex number. For example, the complex number 3 + 4i would be plotted at the point (3, 4).
3. What is the polar representation of a complex number?
The polar representation is an alternative way to describe a complex number. Instead of using its real and imaginary coordinates (a, b), it uses the number's distance from the origin and the angle it makes with the positive real axis. This form is written as z = r(cos θ + i sin θ), where:
- r is the modulus (distance from the origin).
- θ (theta) is the argument (angle with the positive real axis).
4. What is the key difference between the Cartesian and polar representation of a complex number?
The key difference lies in what each form emphasizes. The Cartesian form (z = a + ib) describes a point by its horizontal and vertical distances from the origin. The polar form (z = r(cos θ + i sin θ)) describes the same point by its direct distance and direction (angle) from the origin. While addition and subtraction are easier in Cartesian form, multiplication, division, and finding powers are significantly simpler in polar form.
5. What do the modulus and argument of a complex number represent on an Argand plane?
On an Argand plane, for a complex number z = a + ib:
- The modulus (|z| or r) represents the straight-line distance from the origin (0,0) to the point (a, b). It is calculated using the formula r = √(a² + b²) and is always a non-negative value.
- The argument (arg(z) or θ) represents the angle that the line segment from the origin to the point (a, b) makes with the positive real axis. It is calculated using trigonometry, often as θ = tan⁻¹(b/a), carefully considering the correct quadrant.
6. Why is polar representation particularly useful for multiplying complex numbers?
Polar representation simplifies multiplication because of a simple rule: to multiply two complex numbers, you multiply their moduli and add their arguments. For example, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), their product is z₁z₂ = r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)). This is much more efficient than the distributive multiplication required in the Cartesian (a + ib) form.
7. How do you convert a complex number from its rectangular form (a + ib) to its polar form?
To convert a complex number z = a + ib to its polar form z = r(cos θ + i sin θ), follow these two steps:
- Calculate the modulus (r): Use the Pythagorean theorem to find the distance from the origin: r = √(a² + b²).
- Calculate the argument (θ): Find the angle using the formula θ = tan⁻¹(|b/a|) and then adjust it based on the quadrant where the point (a, b) lies to get the principal argument.
Once you have 'r' and 'θ', you can write the number in its polar form.
8. Can two different complex numbers have the same modulus? What about the same argument?
Yes, two different complex numbers can have the same modulus. The modulus represents the distance from the origin, so all points on a circle centered at the origin have the same modulus. For example, 3 + 4i and 4 + 3i are different numbers but both have a modulus of 5. However, two different non-zero complex numbers will always have different combinations of modulus and argument. They can share the same argument only if they lie on the same line extending from the origin, but they would have different moduli (e.g., 2+2i and 5+5i).
9. Why is the complex plane specifically called the Argand plane?
The plane is named after Jean-Robert Argand, an amateur mathematician who published a paper in 1806 describing this geometric interpretation of complex numbers. While other mathematicians like Caspar Wessel and Carl Friedrich Gauss had developed similar concepts earlier, Argand's work was particularly influential in popularizing the idea of visualizing complex numbers as points on a plane, leading to his name being associated with it.
