Define multiplication numbers.

1.	Define multiplication numbers.
Answer» Let z₁= a + ib and z₂ = c + id be any two complex numbers. Then, the product z₁z₂ is defined as follows: z₁z₂ = (ac – bd) + i(ad + bc) Properties of product of complex numbers: Closure property: The product of two complex numbers is always a complex number. Commutative law: For any two complex numbers z₁ and z₂ we have z₁ z₂ = z₂z₁ Associative law: For any three complex numbers z₁ z₂ and z₃ we have (z₁z₂)z₃= z₁(z₂z₃). Existence of multiplicative identity: For every complex number z, we have z . 1 = 1. z = z. Thus, 1 is the multiplicative identity. Existence of multiplicative inverse: Let z = a + ib. Then z^-1 = 1/z = 1/(a + ib) = 1/(a + ib) . (a - ib)/(a - ib) = (a - ib)/(a² + b²) Clearly z. 1/z = 1/z . z = 1 Thus, every z = a + ib has its multiplicative inverse, given z^-1 = 1/z = a/(a² + b²) - a/(a² + b²)i (z ≠ 0)

Answer»

Let z₁= a + ib and z₂ = c + id be any two complex numbers. Then, the product z₁z₂ is defined as follows:

z₁z₂ = (ac – bd) + i(ad + bc)

Properties of product of complex numbers:

Closure property: The product of two complex numbers is always a complex number.
Commutative law: For any two complex numbers z₁ and z₂ we have z₁ z₂ = z₂z₁
Associative law: For any three complex numbers z₁ z₂ and z₃ we have (z₁z₂)z₃= z₁(z₂z₃).
Existence of multiplicative identity: For every complex number z, we have z . 1 = 1. z = z. Thus, 1 is the multiplicative identity.
Existence of multiplicative inverse: Let z = a + ib. Then

z^-1 = 1/z = 1/(a + ib) = 1/(a + ib) . (a - ib)/(a - ib) = (a - ib)/(a² + b²)

Clearly

z. 1/z = 1/z . z = 1

Thus, every z = a + ib has its multiplicative inverse, given

z^-1 = 1/z = a/(a² + b²) - a/(a² + b²)i (z ≠ 0)

Discussion