**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 are 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 –

- 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 – the axis is known as the real axis.
- 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.
- 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.
- 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.

**Multiplication of Complex Numbers**

Let us now understand the multiplication of complex numbers. For this purpose let us consider two complex numbers, say, z_{1} = ( a + i b ) and z_{2} = ( c + i d ). Let us how we can multiply these two complex numbers.

Since we need to multiply z_{1} and z_{2}, we will have,

z_{1} x z_{2} = ( a + i b ) ( c + i d ) . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

⇒ z_{1} x z_{2} = a c + i ( a d ) + i ( b c ) + i ^{2} ( b d )

We know that i ^{2} = – 1 , therefore, substituting the value of i ^{2} in the above equation, we will have,

z_{1} x z_{2} = a c + i ( a d ) + i ( b c ) + ( – 1 ) ( b d )

Now, we will combine the real numbers with the imaginary numbers, we will get,

z_{1} x z_{2} = ( a c – b d ) + i ( a d + b c ) . . . . . . . . . . . . . . . . ( 2 )

**So, we can say that for any two complex numbers, z**_{1}** = ( a + i b ) and z**_{2}** = ( c + i d ), z**_{1}** x z**_{2}** = ( a c – b d ) + i ( a d + b c )**

Let us now discuss the associative property of multiplication of complex numbers.

**What is associative property?**

Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter.** **In the case of multiplication, this means that if we want to multiply 3 numbers, two of them can be chosen first, one as a multiplier and the second as a multiplicand. The result of the multiplication would serve as a multiplier and the third number as a multiplicand to get the final answer. Is multiplication **of complex numbers **associative? Let us find out.

**Is multiplication of complex numbers associative?**

For verifying the associative property of multiplication of complex numbers, let us consider three complex numbers.

So, let z _{1} = a + i b, z _{2} = c + i d and z _{3} = e + i f be any three complex numbers. Then in order for the multiplication of these complex numbers to be associative, the following statement should be true –

( z _{1} z _{2} ) z _{3} = z _{1} ( z _{2} z _{3} )

Let us verify the above statement using the three complex numbers, z _{1} = a + i b, z _{2} = c + i d and z _{3} = e + i f

So, we have,

( z _{1} z _{2} ) z _{3} = { ( a + i b ) ( c + I d ) } ( e + i f )

= { ( a c – b d ) + i ( a d + c b ) } ( e + i f )

= { ( a c – b d ) e – ( a d + c b ) f ) + i { ( a c – b d ) f + ( a d + c b ) e )

= { a ( c e – d f ) – b ( c f + e d ) } + i { b ( c e – d f ) + a( e d + c f )

= ( a + i b ) { ( c f – d f ) + i (c f + e d ) }

= z _{1} ( z _{2} z _{3} )

Hence, for any three complex numbers, z _{1} = a + i b, z _{2} = c + i d and z _{3} = e + i f, we have,

( z _{1} z _{2} ) z _{3} = z _{1} ( z _{2} z _{3} )

Hence, we can say that the multiplication of complex numbers satisfies the associative property.

**Verifying the associative property of multiplication of complex numbers as ordered pair**

By the definition of complex numbers, the complex numbers, x, y, z ∈ C are identified by ordered pairs x = ( a, b ), y = ( c, d ) z = ( e, f ) for some real numbers a, b, c, d, e, f ∈ R

So, for the ordered pairs to be associative, we need to show that

( x . y ) . z = x . ( y . z )

We have,

( x . y ) . z = [ ( a , b ) ⋅ ( c , d ) ] ⋅ ( e , f ) ( by definition of complex numbers )

⇒ ( x . y ) . z = ( a c – b d , a d + b c ) ⋅ ( e , f ) ( by definition of multiplication of complex numbers )

⇒ ( x . y ) . z = ( ( a c – b d ) e − ( a d + b c ) f , ( a c – b d ) f + ( a d +b c ) e ) (by definition of multiplication of complex numbers )

⇒ ( x . y ) . z = ( ( a c e – b d e ) − ( a d f + b c f ) , ( a c f – b d f ) + ( a d e + b c e ) ) ( by distributive law of real numbers )

⇒ ( x . y ) . z = ( a c e – b d e – a d f – b c f , a c f – b d f + a d e + b c e ) ( by distributive law of real numbers and associativity of addition )

⇒ ( x . y ) . z = ( a c e – a d f – b c f – b d e , a c f + a d e + b c e – b d f ) ( by commutativity of addition of real numbers )

⇒ ( x . y ) . z = ( ( a c e – a d f ) − ( b c f + b d e ) , ( a c f + a d e ) + ( b c e – b d f ) ) (by distributive law of real numbers )

⇒ ( x . y ) . z = ( a ( c e – d f ) – b ( c f + d e ) , a ( c f + d e ) + b ( c e – d f ) ) ( by distributive law of real numbers )

⇒ ( x . y ) . z = ( a , b ) ⋅ ( c e – d f , c f + d e ) ( by definition of multiplication of complex numbers )

⇒ ( x . y ) . z = ( a , b ) ⋅ [ ( c , d ) ⋅ ( e , f ) ] ( by definition of multiplication of complex numbers )

⇒ ( x . y ) . z = x ⋅ ( y ⋅ z ) ( by definition of complex numbers )

Hence, we can say that the multiplication of ordered pairs of complex numbers satisfies the associative property.

**Key Facts and Summary**

- If a, b are two real numbers, then a number of the form a + i b is called 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 modulus of a complex number z = a + i b is denoted by | z | and is defined as

| z | = $\sqrt{a^2+ b^2} = \sqrt{{ Re ( z )}^2 + { Im ( z )}^2}$ - 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- So, we can say that for any two complex numbers, z
_{1}= ( a + i b ) and z_{2}= ( c + i d ), z_{1}x z_{2}= ( a c – b d ) + i ( a d + b c ) - Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter.
- For any three complex numbers, z
_{1}= a + i b, z_{2}= c + i d and z_{3}= e + i f, we have, ( z_{1}z_{2}) z_{3}= z_{1}( z_{2}z_{3})

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