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Thought-Process to Discover Knowledge

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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.

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mathsComplex NumbersAlgebra of Complex Numbers

### Exponent of a complex Number

Exponent of a complex number is explained.

click on the content to continue..

Given z_1 = a_1+ib_1 and z_2 = a_2+ib_2, Can the exponent z_1^z_2 be written in a+ib form?

• No. Exponent to the power of complex number is not possible
• Yes. By converting to polar form z_1 = r e^(i theta).
• Yes. By converting to polar form z_1 = r e^(i theta).

The answer is 'Yes. By converting to polar form z_1 = r e^(i theta)'

Given z_1 = a_1+ib_1 and z_2 = a_2+ib_2, exponent
z_1^z_2
quad quad = (r_1e^(i theta_1) )^(a_2+ib_2)
quad quad = r_1^a_2 
quad quad quad quad xx r_1^(ib_2)
quad quad quad quad xx e^(i theta_1 a_2)
quad quad quad quad xx e^(i theta_1 i b_2)

quad quad = r_1^a_2 
quad quad quad quad xx e^(ib_2 ln r_1) (
quad quad quad quad xx e^(i theta_1 a_2)
quad quad quad quad xx e^(- theta_1 b_2)

quad quad = r_1^a_2 e^(- theta_1 b_2) 
quad quad quad quad xx e^(i(b_2 ln r_1 + theta_1 a_2))

The result is in the polar form and can be converted to coordinate form.

Given z_1 = a_1+ib_1 and z_2 = a_2+ib_2, Can the root root(z_2)(z_1) be computed in a+ib form?

• No. root to a complex number is not possible
• Yes. By considering root as power of (1/z_2)
• Yes. By considering root as power of (1/z_2)

The answer is 'Yes. By considering root as power of (1/z_2)'

Given z_1 = a_1+ib_1 and z_2 = a_2+ib_2, root
root(z_2)(z_1)
quad quad = z_1^(1/z_2)
quad quad = z_1^(bar(z_2)/(|z_2|^2))
By following the rules of exponent of a complex number, the root can be solved.

To find exponent and root of complex numbers, the rules of numerical expression is used to arrive at the coordinate form a+ib.

Exponent and Roots of Complex number
•  For z_1^(z_2), convert z_1 to polar form re^(i theta)
•  For a^(ib), convert a to e^(ln a) form
•  For z_1^(1/z_2), convert 1/z_2 to a complex number in numerator bar(z_2)/(|z_2|^2)

Solved Exercise Problem:

What is (1+i)^(3/2)

• 3/2+3/2i
• 2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))
• 2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))
• 2^(3/2)(cos (3 pi/4)+i sin (3 pi/4))
• 2^(3/4)(cos (3 pi/2)+i sin (3 pi/2))

The answer is '2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))`'

slide-show version coming soon