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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 NumbersIntroduction to Complex Numbers

### Euler's Formula

To understand the generic form of complex numbers as a+i b, one should know Euler's Formula. This helps to convert numerical expressions of various forms to the form a+i b.

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First, The Euler's Formula is proven in abstract form -- where manipulation of expressions and equations help to prove.

Then, the significance and the intuitive understanding of Euler's formula is discussed.

Proof for Euler's formula :
Consider y=cos x + i sin x
Differentiate this
dy = (-sin x + i cos x)dx
dy = (i^2 sin x + i cos x)dx
dy = (i sin x + cos x)idx
dy = y i dx
(dy)/y = i dx
integrate this
ln y = ix
y = e^(ix)

we started with y=cos x + i sin x and ended with y = e^(ix). So,
e^(ix) = cos x + i sin x

Euler's Formula : e^(ix) = cos x + i sin x

Euler's Formula :
e^(ix) = cos x + i sin x

To develop intuitive understanding of Euler's formula, let us examine the properties of the form of complex number r (cos theta + i sin theta).

For an intuitive understanding of Euler's formula: Multiplication of two complex number leads to addition of theta -- similar to exponents where multiplication of two numbers leads to addition of powers

Prove that cos theta + i sin theta has the properties of a number in exponent form (Students are required to work this out)

(cos theta_1 + i sin theta_1)xx (cos theta_2 + i sin theta_2)
quad quad = (cos(theta_1+theta_2) + i sin(theta_1+theta_2))

(cos theta + i sin theta)^n
quad quad = (cos n theta + i sin n theta)

Similarly
e^(i theta_1) xx e^(i theta_2)
quad quad = e^(i(theta_1+theta_2))

[e^(i theta)]^n
quad quad = e^(i n theta)

For an intuitive understanding of Euler's formula. Rate of change of the function with respect to theta equals i times the function itself -- similar to base of natural logarithm where (d e^(ax))/dx = a e^(ax).

Prove that rate of change of z = cos theta + i sin theta is iz.
(dz)/(d theta)
quad quad = -sin theta + i cos theta
quad quad = i^2 sin theta + i cos theta
quad quad = i(i sin theta + cos theta)
quad quad = i z

Similarly
d/(d theta) e^(i theta)
quad quad = i e^(i theta)

Intuitive understanding of Euler's formula :

On examining the properties of the form of complex number r (cos theta + i sin theta).

•  The representation is defined by ordered pair of numbers (r,theta)

•  Multiplication of two complex number leads to addition of theta -- similar to exponents where multiplication of two numbers leads to addition of powers

•  rate of change with respect to theta equals i times the function itself -- similar to base of natural logarithm where (d e^(ax))/dx = a e^(ax).

These properties along with the abstract derivation given in a page earlier, it is understood that a complex number can equivalently be represented as an exponent with base e.

Significance of Euler's Formula:

Remember that irrational numbers have so many different variants like root(3)(4), root(4)(4), root(5)(4), etc. Note that, 20+root(2)(3)-4root(4)(3) is an irrational number.

Irrational numbers does not have an uniform representation and so they are represented with numerical expressions.

In case of complex numbers, all complex numbers can be represented in the form a+ib. Each of i^i, 3^(3+4i), etc. can be represented in the form a+ib. This is made possible by Euler's Formula that connects the exponent form to the coordinate form.

The coordinate form of complex number is equivalently represented in an exponential form by Euler's Formula.

Polar form: A complex 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))
quad quad = r e^(i (theta+2n pi))
where r = sqrt(a^2+b^2), theta = tan^(-1)(b/a)

Solved Exercise Problem:

In which of the following forms a complex number can be given?

• a+ib
• r(cos theta + i sin theta)
• r e^(i theta)
• All the above
• All the above

The answer is 'All the above'

Can real numbers be represented in the form of a+b i?

• Yes, where b=0
• Yes, where b=0
• No, Real numbers cannot be in the form a+b i

The answer is 'Yes, Where b=0'.
All real numbers can be equivalently represented in the form a+bi where b=0.

Real numbers are also part of complex numbers with imaginary part 0.

Real number in Complex Number System : A real number a is equivalently given as
a+0i
quad quad = a(cos 2n pi + i sin 2n pi)
quad quad = a e^(i 2n pi), where n=0,1,2,3...

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