Basic Algebraic Properties of 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.

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