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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsComplex NumbersProperties of Complex Number Arithmetic

### Multiplication : Closure Law

Complex multiplication is closed : Product or quotient of two complex number is another complex number

click on the content to continue..

What does 'closure' mean?

• closed
• not open
• both the above
• both the above

Given complex numbers z_1=a_1+b_1i and z_2=a_2+b_2i, a_1,a_2,b_1,b_2 in R, will z_1 xx z_2 a complex number?

• No. When multiplied, the result has i^2
• Yes. When multiplied, the result consists of real numbers, as real and imaginary parts
• Yes. When multiplied, the result consists of real numbers, as real and imaginary parts

The answer is 'Yes. When multiplied, the result consists of real numbers as real and imaginary parts'

•  Product or quotient of two complex number is another complex number - closed

Closure property of Multiplication and Division: For any given complex numbers z_1, z_2 in CC,  z_1 xx / -: z_2 in CC.

Solved Exercise Problem:

Given complex numbers z_1=a_1+b_1i and z_2=a_2+b_2i, a_1,a_2,b_1,b_2 in R, will z_1 -: z_2 a complex number?

• No. Complex division is not defined
• Yes. complex number division is equivalently multiplication by the conjugate
• Yes. complex number division is equivalently multiplication by the conjugate

The answer is 'Yes. When multiplied, the result consists of real numbers as real and imaginary parts'

slide-show version coming soon