The Argand plane is a mathematical concept used to graphically represent complex numbers. It is similar to the Cartesian coordinate system used for real numbers but is specially specified to handle the two-dimensional nature of complex numbers. Argand Plane is also known as the complex plane.
It is a graph with real numbers on the horizontal axis and imaginary numbers on the vertical. It helps understand complex numbers visually. We have explained in detail about Argand Plane with the properties and examples below.
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Argand Plane
Argand plane is used to represent a complex number in a two-dimensional plane.
Argand plane id is named for Jean-Robert Argand, a mathematician. It is a graphical method to represent complex numbers. It's similar to a two-dimensional plane, having the real numbers on the horizontal axis and the imaginary numbers on the vertical axis.
These numbers are then positioned as the new points on the plane, while each point has the coordinates (a, b) which corresponds to the real and imaginary parts of the number. It is a very good tool for the visualization of abstract arithmetic and geometry.
Diagram of Argand Plane
Polar Representation In Argand Plane
In the Argand plane, which is a graphical representation of complex numbers, polar representation is a way to express complex numbers using their magnitude (distance from the origin) and angle (or argument) with respect to the positive real axis.
Imagine a two-dimensional plane where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. In this plane, the origin (0,0) represents the complex number 0.
Now, let's consider a complex number ? = ? + ??, where a is the real part and b is the imaginary part. The magnitude (or modulus) of this complex number, denoted by ∣z∣, is the distance from the origin to the point representing the complex number in the plane.
It's calculated using the Pythagorean theorem: ∣?∣ = √a2 + b2
Properties Of Argand Plane
- Argand plane shows visually the relationship between complex numbers and their real and imaginary parts.
- It brings understanding of number properties, e.g., conjugates, more clearly through geometric visualization.
- The points' distance from the origin equals the modulus of a complex number, the absolute value of its imaginary part.
- The angle whose vector points along the positive real axis is the argument of a complex number.
- The representation of addition and subtraction of complex numbers by vectors resembles vectorial operations.
- Scaling and the rotation of multiplication are implied, while division is its inverse in the Argand plane.
- Complex conjugates occur as a mirror image on the real axis.
- The notion of roots and powers is illustrated using the division of modulus circle and the scale and the rotation.
- Complex numbers can be dealt with geometrically in argand's plane regarding functions that are complex.
- Geometric representation of the solutions is used to help understand equations in their solutions.
Solved Examples on Argand Plane
Example 1: Addition of Complex Numbers on the Argand Plane
Let's add two complex numbers: z1 = 3 + 4i and z2 = 2 - i.
Solution:
To add them, we simply add their real and imaginary parts separately:
= (3 + 4i) + (2 - i)
= (3 + 2) + (4i - i)
= 5 + 3i
So, the result of adding z1 and z2 is the complex number 5 + 3i.
On the Argand plane, this corresponds to moving 5 units to the right along the real axis and 3 units up along the imaginary axis from the origin.
Example 2: Multiplication of Complex Numbers on the Argand Plane
Let's multiply two complex numbers: z1 = 2 + 3i and z2 = 1 - 2i.
Solution:
To multiply them, we use the distributive property:
= (2 + 3i) × (1 - 2i)
= 2 × 1 + 2 × (-2i) + 3i × 1 + 3i × (-2i)
= 2 - 4i + 3i - 6i2
Remembering that i2 = -1,
we simplify: = 2 - 4i + 3i + 6 = 8 - i
So, the result of multiplying z1 and z2 is the complex number 8 - i.
On the Argand plane, this corresponds to moving 8 units to the right along the real axis and 1 unit down along the imaginary axis from the origin.
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