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Thought-Process to Discover Knowledge

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mathsComplex NumbersIntroduction to Complex Numbers

### Complex Numbers: Equivalent representations

Complex number in the form a+ib can be equivalently represented in the form r(cos theta + i sin theta). This representation, has an advantage to understanding the generic form of complex numbers.

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Which of the following equals 2?

• 4-2
• 1+1
• 6/3
• all the above
• all the above

The answer is 'All the above'.

A number can be equivalently represented with numerical expressions of various forms.

Given hypotenuse is c. Which of the following represent the length of side opposite to the angle theta in a right angle triangle?

• c sin theta
• c tan theta cos theta
• c sqrt(1 - cos^2 theta)
• all the above
• all the above

The answer is 'all the above'.

A constant can be represented as expressions.

It was established that solution to quadratic equation is in the form a+b i. What is an equivalent expression of this?

• sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)
• sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)
• The option given above is not correct.

The answer is 'sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)'

A number a+bi is equivalently given as
color(darkorange)(sqrt(a^2+b^2)) (color(blue)(a/(sqrt(a^2+b^2)))
quad quad quad + color(green)(b/(sqrt(a^2+b^2))) i)

This form looks similar to color(blue)(cos) and color(green)(sin) of a right angled triangle.

It is equivalently represented as color(darkorange)(r)(color(blue)(cos theta) + i color(green)(sin theta)) where color(darkorange)(r = sqrt(a^2+b^2)) and theta = tan^(-1)(b/a).

sin theta equals ...

• sin (theta + 2n pi) where n=0,1,2,...
• sin (theta + 2n pi) where n=0,1,2,...
• it is not the above

The answer is 'sin (theta + 2n pi)'

A complex number in the form a+ib  is equivalently given in the form r (cos theta + i sin theta)

A number in the form a+bi is equivalently given as
quad quad = r (cos (theta+ 2n pi)
quad quad quad quad + i sin (theta+ 2n pi))
where r = sqrt(a^2+b^2), theta = tan^(-1)(b/a)

slide-show version coming soon