**Theorem1: (Closure property): **If z1 and z2 ϵ C, then the product of two complex numbers is a complex number, i.e.

z1**. **z2 ϵ C

**Proof: **Let z1 = (a + ib)

And,

z2 = c + id

where

a, b, c, d ϵ R

therefore,

z1 **.** z2 = (a + ib) (c + id)

= (ac – bd) + i (ad + bc) ϵ C

As stated by the closure property of addition, subtraction and multiplication of real numbers (ac – bd) and (ad + bc) both are real numbers.

Hence,

(ac – bd) + i (ad + bc) ϵ C

Therefore,

z1 ϵ C ; z2 ϵ C z1**.** z2 ϵ C for all z1, z2 ϵ C

**Theorem 2 :**(** Commutative Law):** If z1, z2 ϵ C then,

z1z2 = z2z1

**Proof: **Let z1 = (a + ib)

And,

z2 = c + id

z1 z2 = (a + ib) (c + id)

= (ac – bd) + i (ad + bc)

= (ca – db) + i (da + cb)

= (c + id) (a + ib)

= z2 z1

Thus,

z1z2 = z2z1

**Theorem3: (Associative law): **If z1, z2, z3 ϵ C, then

(z1 z2) z3 = z1 (z2 z3)

**Proof: **Let z1 = a + ib

And,

z2 = c + id

z3 = e + if

(z1 z2) z3 = [(a + ib) (c + id)] (e + if)

= [(ac – bd) + i (ad + bc)] + (e +if)

= {(ac – bd) e – (ad + bc) f}+ i {(ac – bd)f + (ad +bc)e}

= (ace – bde – adf – bcf) + i (acf – bdf + ade + bce)

= {a (ce – df) – b (cf + de)} + i {a(cf + de) + b(ce – df)}

= (a + ib) {(ce – df) +i (cf +de)}

= z1 (z2 z3)

Thus,

(z1 z2) z3 = z1 (z2 z3)

**Theorem4: (Multiplicative Inverse): **For each z ϵ C, there exist a complex number denoted by z^{-1} such that

z ^{z-1} = z^{-1} z = 0

**Proof: **z = a + ib ϵ C

And,

z ≠ 0

let z^{-1} = (*x* + iy) ϵ C

Then,

z z^{-1} = 1

(a + ib) (*x* + iy) = 1

(a*x* – by) + i(b*x* + ay) = 1 + i0

a*x* – by = 1

and

b*x* + ay = 0

**To Schedule a Multiplication properties tutoring session
To submit Multiplication properties assignment click here.**

Assignment Writing Help

- Science Assignment Help
- Math Assignment Help
- Chemistry Assignment Help
- Physics Assignment Help
- Biology Assignment Help
- Psychology Assignment Help
- History Assignment Help
- Geography Assignment Help
- English Assignment Help
- Humanities Assignment Help
- Nursing Assignment Help
- Social Science Assignment Help
- Arts and Architecture Help
- Statistics Assignment Help
- Law Assignment Help
- Computational Mathematics Assignment Help

Engineering Assignment Services

- Programming Assignment Help
- Database Help
- Data Structure Assignment Help
- Operating Systems Assignment Help
- Computer Network Assignment Help
- UML Diagram Assignment Help
- IT Assignment Help
- Game Programming Help
- Computer Science Assignment Help
- Information Systems Assignment Help
- Chemical Engineering Assignment Help
- Civil Engineering Assignment Help
- Electrical, Electronics Help
- Mechanical Engineering Assignment Help
- Petroleum Engineering Assignment Help
- Biochemical and Biotechnology Help

Do My Assignment Help

- Accounting Assignment Help
- Finance Assignment Help
- Economics Assignment Help
- Marketing Assignment Help
- Human Resources Assignment Help
- Operations Management Assignment Help
- Strategy and Planning Help
- Project Management Help
- Case Studies Writing Help
- Political science
- Referencing Help
- Assignment Help Websites
- Online Assignment Help
- Do My Assignment
- Do My Homework

Write My Essay Services

- Essay Writing Help
- Business Essay Writing Help
- Assignment Writing Services
- Plagiarism Free Essay Writing
- Essay Editing Service
- Dissertation Writing Services
- Thesis Writing Help
- Custom Writing Help
- Write My Essay
- Write My Paper
- Paper Writing Service
- Academic Writing Help
- College Essay Writing
- Cheap Essay Writing