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Complex Plane

Introduction

If a and b are natural numbers such that a > b, then the equation x + a = b is not solvable in N, the set of natural numbers.  This means that there is no natural number satisfying the equation x + a = b. so, the set of natural numbers is extended to form the set I of integers in which every equation of the form x + a = b, for a and b being natural numbers is solvable. However, the equations of the form x a = b, where a ≠ 0 and a and b being integers are not solvable for I also. Hence, the set I of integers is extended to obtain the set Q of all rational numbers in which every equation of the form x a  = b, where a ≠ 0 and a and b being integers is solvable. But, the equations of the form x 2 = 2, x 2 = 3 etc. are not solvable in Q because there is no rational number whose square is 2. Such numbers are known as irrational numbers. The set Q of all rational numbers is extended to obtain the set of R of real numbers that include both rational and irrational numbers. However, the equations of the form x 2 + 1 = 0, x 2 + 4 = 0, etc. are not solvable in R which means that there is no real number whose square is a negative real number. Euler was the first mathematician to introduce the symbol i ( iota ) for the square of – 1, i.e. a solution of x 2 + 1 = 0 with the property i 2 = – 1. He also called this symbol as the imaginary unit. This gave the concept of complex numbers and the complex plane. Let us learn more about them. 

Definition

If a, b are two real numbers, then a number of the form a + i b is called a complex number. For example, 7 + 2 I, – 1+ I, 3 – 2 I, 0 + 2 I are complex numbers. Real and imaginary parts of a complex number : If z = a + i b is called a complex number, then “ a “ is called the real part of z and “ b “ is known as the imaginary part of z. the real part of z is denoted by Re ( z ) and the imaginary part of z is denoted by Im ( z ). The complex plane is named after a Paris-based amateur mathematician Jean-Robert Argand (1768 – 1822).

Modulus of a Complex Number

The modulus of a complex number z = a + i b is denoted by | z | and is defined as 

| z |  = $\sqrt{a^2+ b^2} = \sqrt{{ Re (z) }^2+ { Im (z)}^2}$

From above we can see that | z | ≥ 0 for all z ∈ C.

What is a complex plane?

The complex plane (also known as the Gauss plane or Argand plane) is a geometric method of depicting complex numbers in a complex projective plane. Let us learn more about the complex plane

Geometrical representation of a complex number ( Argand Plane )

A complex number z = x + i y can be represented by a point ( x, y ) on the plane which is known as the Argand plane. To represent z = x + i y in geometric form, we take mutually perpendicular straight lines. Now we will plot a point whose x and y coordinates are represented by the real and the imaginary parts of z. This point P ( x, y ) represents the complex number  z = x + i y. Below is the geometric representation of the point P ( x, y ) on the complex plane.

Some important points to remember here are – 

  1. If the complex number is purely real, then its imaginary part will be 0. This means that a purely real number will be represented by a point on the x – axis. This is why x – axis is known as the real axis.
  2. If the complex number is purely imaginary, then its real part will be 0. This means that a purely imaginary number will be represented by a point on the y – axis. This is why y – axis is known as the imaginary axis.
  3. If P ( x, y ) is a point on the complex plane, then the point P ( x, y )  represents a complex number z = x + i y. the complex number z = x + i y is known as the affix of the point P.
  4. The plane in which we represent a complex number in geometrical form is known as the complex plane or Argand plane or the Gaussian plane. The point P plotted on the Argand plane is called the Argand diagram.

Argument or Amplitude of a Complex number for different signs of real and imaginary parts

Let us now discuss the Argument or Amplitude of a Complex number for different signs of real and imaginary parts. There can be four different conditions depending upon the signs of real and imaginary parts. These four conditions are – 

  1. Argument of z = x + i y when x > 0 and y > 0
  2. Argument of z = x + i y when x < 0 and y > 0
  3. Argument of z = x + i y when x < 0 and y < 0
  4. Argument of z = x + i y when x > 0 and y < 0

Let us discuss these one by one

Argument of z = x + i y when x > 0 and y > 0

Since x and y are both positive, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the first quadrant. Let  be the argument of z and let be the acute angle satisfying tan = | y / x | . Below is the geometrical representation of this case – 

Argument of z = x + i y when x < 0 and y > 0

Since x is less than zero, which means it is negative and y is positive, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the second quadrant. Let  be the argument of z and let be the acute angle satisfying tan = | y / x | . Below is the geometrical representation of this case – 

Argument of z = x + i y when x < 0 and y < 0

Since x and y are both negative, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the third quadrant. Let  be the argument of z and let be the acute angle satisfying tan = | y / x | . Below is the geometrical representation of this case – 

Argument of z = x + i y when x > 0 and y < 0

Since y is less than zero, which means it is negative and x is positive, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the fourth quadrant. Let  be the argument of z and let be the acute angle satisfying tan = | y / x | . Below is the geometrical representation of this case – 

From the above, we can define an algorithm for finding the argument of a complex number using the complex plane.

Finding the argument of a complex number using the complex plane

Below we have the algorithm that can be used to find the argument of a complex number z = x + i y using the complex plane – 

  1. The first step is to find the acute angle which is given by tan = | y / x |.
  2. The next step is to determine the quadrant in which the point P ( x, y ) lies. 
  3. If P ( x, y ) belongs to the first quadrant, then arg ( z ) = α
  4. If P ( x, y ) belongs to the second quadrant, then arg ( z ) = π – α
  5. If P ( x, y ) belongs to the third quadrant, then arg ( z ) = – ( π – α) or π + α
  6. If P ( x, y ) belongs to the fourth quadrant, then arg ( z ) = -α or 2 π – α

Let us understand it through an example

Example

Suppose we wish to modulus and argument of the complex number 1 + i √3 

Solution

Let z = 1 + i √3 and let be the acute angle given by tan α = | $\frac{Im ( z )}{Re ( z )}$ |. Then,

tan α = √3 ⇒ α = $\frac{π}{3}$

Also, | z | = $\sqrt{1^2+ (√3)^2}$ = 2

Solved Examples

Example 1 Find the modulus and argument of z = 4 + 3 i.

Solution We have been given the complex number z = 4 + 3 i. Let us first plot this complex number in a complex plane. Below is the geometrical representation of z = 4 + 3 i – 

The point Q represents the coordinates ( 4, 3 ). Now, we know that the length of the line OQ will represent the modulus of the complex number z. Applying Pythagoras theorem in the above graph we will have,

OQ 2 = 4 2 + 3 2 = 16 + 9 = 25 

⇒ OQ = 5

Therefore, we can say that the modulus of the complex number z = 4 + 3 i is 5 . . . . . . . . . . . . . . . . . . . . ( 1 )

The next step is to find the argument of the complex number z = 4 + 3 i . In order to find the argument we must calculate the angle between the x axis and the line segment OQ.

Considering the right angled triangle OQN, we will have,

tan θ = $\frac{3}{4}$

Now, we know that tan θ = $\frac{3}{4}$ = 36.97o

Therefore, the argument of the complex number z = 4 + 3 i  is  36.97o . . . . . . . . . . ( 2 )

From ( 1 ) and ( 2 ), we have, the modulus of the complex number z = 4 + 3 i  is   5 and its argument is 36.97o

Example 2 Find the modulus and argument of z = 3 – 2 i.

Solution We have been given the complex number z = 3 – 2 i.. Let us first plot this complex number in a complex plane. Below is the geometrical representation of z = 3 – 2 i. – 

The point P represents the coordinates ( 3, – 2 ). Now, we know that the length of the line OP will represent the modulus of the complex number z. Applying Pythagoras theorem in the above graph we will have,

OP 2 = 3 2 + 2 2 = 9 + 4 = 13 

⇒ OP =  √13

Therefore, we can say that the modulus of the complex number z = 3 – 2 i is  √13 . . . . . . . . . . . . . . . . . . . . ( 1 )

The next step is to find the argument of the complex number z = 3 – 2 i. In order to find the argument, we must calculate the angle between the x axis and the line segment OP.

Considering the right angled triangle OPN, we will have,

tan θ = $\frac{3}{4}$

Now, we know that tan θ = $\frac{3}{4}$ = 36.97o

Since the angle is negative, therefore, the argument of the complex number z = 3 – 2 i is  – 33.67o . . . . . . . . . . ( 2 )

From ( 1 ) and ( 2 ), we have, the modulus of the complex number z = 3 – 2 i is  √13 and its argument is – 33.67o

Key Facts and Summary

  1. If a, b are two real numbers, then a number of the form a + i b is called a complex number.
  2. If z = a + i b is called a complex number, then “ a “ is called the real part of z and “ b “ is known as the imaginary part of z. the real part of z is denoted by Re ( z ) and the imaginary part of z is denoted by Im ( z ).
  3. The modulus of a complex number z = a + i b is denoted by | z | and is defined as | z |  = $\sqrt{a^2+ b^2} = \sqrt{{ Re (z) }^2+ { Im (z)}^2}$
  4. The complex plane (also known as the Gauss plane or Argand plane) is a geometric method of depicting complex numbers in a complex projective plane.
  5. A complex number z = x + i y can be represented by a point ( x, y ) on the plane which is known as the Argand plane.
  6. If the complex number is purely real, then its imaginary part will be 0. This means that a purely real number will be represented by a point on the x – axis. This is why x – axis is known as the real axis.
  7. If the complex number is purely imaginary, then its real part will be 0. This means that a purely imaginary number will be represented by a point on the y – axis. This is why y – axis is known as the imaginary axis.
  8. If P ( x, y ) is a point on the complex plane, then the point P ( x, y )  represents a complex number z = x + i y. the complex number z = x + i y is known as the affix of the point P.
  9. The plane in which we represent a complex number in geometrical form is known as the complex plane or Argand plane or the Gaussian plane. The point P plotted on the Argand plane is called the Argand diagram.
  10. Argument of z = x + i y when x > 0 and y > 0 – Since x and y are both positive, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the first quadrant.
  11. The argument of z = x + i y when x < 0 and y > 0 – Since x is less than zero, which means it is negative and y is positive, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the second quadrant.
  12. Argument of z = x + i y when x < 0 and y < 0 – Since x and y are both negative, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the third quadrant.
  13. The argument of z = x + i y when x > 0 and y < 0 – Since y is less than zero, which means it is negative and x is positive, therefore, the point P ( x, y ) representing z = x + i y in the Argand plane lies in the fourth quadrant.

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