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Introduction to Complex Numbers

Introduction to Complex Numbers

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Euler's formula


 »  exponent form of complex number

 »  `re^(i theta)``= r(cos theta + i sin theta)`
    →  `e` is the base of natural logarithm
    →  abstracted based on the properties of polar form `r(cos theta + i sin theta)`

The coordinate form, polar form, and exponent form help to establish a generic representation `a+ib` for all complex numbers.

Euler's Formula

plain and simple summary

nub

plain and simple summary

nub

dummy

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

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

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

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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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Starting on learning "Eulers formula". ;; To understand the generic form of complex numbers as a+i b, one should know oy-lers Formula. This helps to convert numerical expressions of various forms to the form a+i b

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`

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.

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

  • 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`.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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


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)`

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...`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

The answer is 'All the above'

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First, The oylers Formula is proven in abstract form -- where manipulation of expressions and equations help to prove. ;; Then, the significance and the intuitive understanding of oylers formula is discussed.
Proof for oylers formula : ;; Consider y=cos x + i sine x ;; Differentiate this ;; d y = minus sine x + i cos x d x;; Manipulate this to express in terms of y and use minus 1 equals i squared. ;; d y = i squared sine x + i cos x d x ;; d y = i sine x + cos x , i d x ;; d y = y i d x ;; d y by y = i d x ;; integrate this ;; log base e, y = i x ;; y = e power i x ;; we started with y=cos x + i sine x and ended with y = e power i x. ;; So, e power i x = cos x + i sine x
oylers formula is e power i x = cos x + i sine x.
oylers formula is e power i x = cos x + i sine x.
To develop intuitive understanding of oylers formula, let us examine the properties of the form of complex number r cos theta + i sine theta
For an intuitive understanding of oylers 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 sine theta has the properties of a number in exponent form. ;; Students are required to work this out. ;; cos theta 1 + i sine theta 1 multiplied by cos theta 2 + i sine theta 2 ;; equals ;; cos theta 1+theta 2 + i sine theta 1+theta 2 ;; cos theta + i sine theta power n ;; equals ;; cos n theta + i sine n theta ;; Similarly ;; e power i theta 1 multiplied by e power i theta 2 ;; equals ;; e power i, theta 1 + theta 2 ;; e power i theta power n ;; equals ;; e power i n theta
For an intuitive understanding of oylers 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 power a x, by dx = a, e power a x.;; Prove that rate of change of z = cos theta + i sine theta is i z. ;; dz by d theta ;; equals ;; -sine theta + i cos theta ;; equals ;; i power 2 sine theta + i cos theta ;; equals ;; i multiplied by i sine theta + cos theta ;; equals ;; i z ;; Similarly ;; d by d theta e power i theta ;; equals ;; i e power i theta
Intuitive understanding of oylers formula : On examining the properties of the form of complex number r cos theta + i sine 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 rate of change is proportional. ;; 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 oylers Formula: ;; Remember that irrational numbers have so many different variants like third root of four, fourth root of four, fifth root of 4, et cetera. Note that, 20 + square root of 3 minus 4 times 4th root of 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+i b. Each of i power i, 3 power 3+4i, et cetera can be represented in the form a+i b. This is made possible by oylers Formula that connects the exponent form to the coordinate form.
The coordinate form of complex number is equivalently represented in an exponential form by oylers Formula.
Polar form: A complex number in the form a+b i is equivalently given as r cos theta+ 2n pi + i sine theta+ 2n pi ;; equals ;; r e power i theta+2 n pi ;; where r = square root of a squared + b squared, theta = tan inverse b by a.
In which of the following forms a complex number can be given?
a;b;bee
a plus i b
cos;cause;sine;theta
r cos theta plus i sine theta
power;e
r e power i theta
all;above
All the above
The answer is 'All the above'
Can real numbers be represented in the form of a+b i ?
yes;s;where;0;zero
Yes, where b=0
no;real;cannot;can;form
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+b i 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 ;; equals, cos 2n pi + i sine 2n pi ;; equals, a e power i 2n pi, where n=0,1,2,3 et cetera

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