Complex numbers are added, subtracted, and multiplied in a similar way. To simplify complex-valued expressions, combine "like" terms and apply the various other methods that we have learned for working with polynomials.

1.6.1 Addition Properties:

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

z1 + z2 ϵ C

Proof: let z1, z2 ϵ C

Where, z1 = a +ib

And,

z2 = c + id

where

a, b, c, d ϵ R

therefore,

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

= (a + c) + i (b + d) ϵ C

Hence,

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,

z1 + z2 = z2 + z1

**Proof:** let z1 = a + ib,

z2 = c + id

then,

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

= (a + c) + i (b + d)

= (c + a) + i (d + b)

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

= z2 + z1

**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)

= [(a + c) + i (b + d)] + (e +if)

= [(a + c + e) + i (b + d + f)]

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

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

= z1 + (z2 + z3)

**Theorem4: (Additive inverse): ** For each z ϵ C, there exist –z ϵ C such that

z + (- z) = (- z) + z = 0

**Proof: **z = a + ib

Then,

-z = (-a) + i (-b) ϵ C

Therefore,

z + (-z) = (a + ib) + [(-a) + i (-b)]

= [a + (-a)] + i [b + (-b)]

= 0 + i 0

= 0

Thus,

z + (- z) = 0

Similarly,

(-z) + z = 0

Therefore,

z + (-z) = (-z) + z = 0

**To Schedule a Complex Number tutoring session
To submit Complex Number 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
- Health Informatics

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