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Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

Basic Algebraic Properties of Complex Numbers

The properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the basic algebraic properties and verify some of them.

Basic Algebraic Properties of Complex Numbers

The properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the basic algebraic properties and verify some of them.

 

Properties of complex numbers


Let us now prove some of the properties.

 

Property

The commutative property under addition

For any two complex numbers z1 and z2 , we have z1 + z2 = z2 + z1.

Proof

Let z1 = x1 + iy1 , z2 = x2 + iy2 , and x1 , x2 , y1 , and y2 R ,

z1 + z2  = ( x1  + iy1 ) + ( x2 + iy2 )

= ( x1 + x2 ) + i ( y1 + y2 )

= ( x2 + x1 ) + i ( y2 + y1 )                 (since x1 , x2 , y1 , and y2 R )

= ( x2 + iy2 ) + ( x1 + iy1 )

= z2 + z1.

 

Property

Inverse Property under multiplication

Prove that the multiplicative inverse of a nonzero complex number z = x + iy is


Proof

The multiplicative inverse is less obvious than the additive one. 

Let z−1 = u + iv be the inverse of z = x + iy

We have z z-1 = 1

That is ( x + iy )(u + iv) = 1

( xu - yv) + i(xv + uy) = 1+ i0

Equating real and imaginary parts we get

xu - yv = 1and xv + uy = 0 .

Solving the above system of simultaneous equations in u and v


Note that the above example shows the existence of z-1 of the complex number z . To compute the inverse of a given complex number, we conveniently use z-1 = 1/z. If z1 and z2 are two complex numbers where z2 ≠ 0, then the product of z1 and 1/ z1 is denoted by z1/z2. Other properties can be verified in a similar manner. In the next section, we define the conjugate of a complex number. It would help us to find the inverse of a complex number easily.

Complex numbers obey the laws of indices


 


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12th Mathematics : UNIT 2 : Complex Numbers


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