Euler s Explanation

7/30/2019 Euler s Explanation

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An Explanation of Eulers Identity

Explaining Eulers Identity 2 = 1 requires examining several concepts...1. The idea of imaginary number i.First, we introduce the imaginary number, . If we solve the equation 2+ 1 = 0, we get

that 2 = 1 . However, we find no solution among the real numbers, so mathematiciansdeveloped the concept of an imaginary number, where 2= 1. The resulting number systemis known as the Complex Numbers, where a complex number = + , where and are realnumbers.

Second, let us define a point in the complex plane. We start by illustrating what it means to

multiply by i, geometrically. First, look at the real number line, a geometry with which we are

more familiar.

We have a horizontal line containing all the real numbers, with the positive reals to the right of

the origin and the negative reals the left of the origin. Now let us look at number, a positivereal number. If we multiply by -1 we get number , the additive inverse of. We know that thedistance from to the origin is the same as the distance from to the origin, so the points aresymmetric about the origin on the number line, thus if we rotate number180we will get .

So we see that multiplying by -1 is the same as rotating 180.

But now let us look at what happens when we want to rotate to 1 by using 2 rotations, sorotate 90then rotate another90to end at 180or multiplying by 1. Let us call a rotation of90the same as multiplying by , just as a rotation of180is the same as multiplying by 1.Then if we rotate and then rotate again we get


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= 1 or 2= 1.Now to solve for it seems logical to take the square root of both sides, sor=1.We know that i =

1 so a rotation of90

must be equal to multiplying by i.

Now back to complex numbers, let us consider = + . When we go to plot this on theCartesian plane ( plane) we have a problem because both the axis and the axis are realnumber lines. There is no place to put a complex number. What happens is that we get a whole

new graphing system, known as the complex plane, where the horizontal axis is the real number

line and the vertical axis is the imaginary number line. Since we know that multiplying by isthe same as rotating 90 if we take the real horizontal axis and multiply the entire line by , weget the imaginary vertical axis.


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This means that we can represent any point = + as a point in the complex plane, wherethe value determines the distance moved in the horizontal direction and the value of determines the distance moved in the vertical (imaginary) direction. It is important to note that in

the complex plane each point represents an individual value, whereas a point in the Cartesian

plane represents a coordinate, or two values (an value and a value).2. The number and the derivative of function() = .Let us look at the number first. The number can be expressed in several different ways.

For now we define e such that e = limn 1 + 1

n

n.

What is 1 + 1nn

?

Using binomial theorem we expand1 + 1nn.

1 + 1

= 0 1(1)

0 + 1 11(1)

1 + 2 12(1)

2 + 3 13(1)

3 + + 10(1)

= 1 +1! (

1)

1 +(1)

2! (1)

2 +(1)(2)

3! (1)

3 + + (1)(2)..321! (1)

=1+1+

1

2! +(1

)(1

)

3! + +1

1

(1()

)

!

Now, when , 1 0. Then

(1 +1)

= 1 + 1 +12! +

13! + =


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lim(1 +1)

=

If we let = 1then we get lim0(1 + )= lim(1 +

1)

. That means

= lim0(1 + )

Now we define the derivative of() = . According to the definition of derivative,

derivation is a measure of how a functions values change as its input changes. Let = ().

Then

= ( + ) () = + = ( 1)

=

(1)

= lim0 = lim0(1)

= lim0

1 ----------------(1)

Let 1 = . Then = + 1

When = , then do on both sides.

ln = ln( + 1)

= ln( + 1)

Therefore, 0 when 0.

So lim01

= lim0

ln(+1) =lim01

ln(+1)/ =lim01

ln(+1)

= lim01

ln(+1)

=

1

lim ln(+1)

=1

lnlim(+1)

From the above ( = lim0(1 + ) ), we know lim0( + 1)

= . So

lim01

=1

ln =11 = 1 -----------(2)

Now we turn back to (1). = lim0 1 , but we change the base b to e. Then we have


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= lim0 1 .

Plug in (2) to (1), then we have

=

1 =

Thus the derivative of is . This is probably the most important property of() = , it isthe only exponential function whose derivative is the same as the function itself.

But what happens when we generalize this real valued function to also take in complex

numbers? We want to preserve the property that the function has its own derivative. Now when

dealing with functions of real numbers, we can only vary the value of the input variable, the

independent variable, by changing the value of that input . However when we instead look atfunction whose input are complex numbers, we can change look at how the function varies as

,

the real part, changes, or as , the imaginary part changes. So if we have a complex number = + we can have either

+ the exponential property (+ = ) also holds

for complex and imaginary numbers so+ = and thus + =

by

properties of derivative. Now since does not depend on , as changes does not changeat all, it is constant and thus

= 1. Now

is saying what is the derivative of where is a real number. This is what we showed above, and we said it is the function itself. So

= .

Now plugging this into

= 1 = .

Now if we look at

+ = () =

. Since does not depend on ,

as varies will remain the same and thus = 1. Now since we know that in the real

numbers

= and since = in the complex plane,

= . So thus

= 1

=

.

We know from above that the multiplying by is the same as rotating90. So if we look atthe derivative, it means that the value of the derivative is the value of the function at a 90rotation. Since we are always moving at an 90 angle, we are never getting further away or closerto where we started, instead we just move in a constant circle.


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3. The unit circle, radians, and .The Unite circle in the Cartesian plane and in the Complex Plane

The unite circle on the Cartesian plane is a circle with radius 1.

We know that the distance formula is = (2 1)2 + (2 1)2. Because the radius is 1, = 1. Therefore, any line that starts at the origin and ends at a point on the circle must be lengthone.

On the other hand, we know that for any angle in a right triangle (where is not the rightangle), the cosine function returns the ratio of the length of adjacent side to the length of the

hypotenuse. That is cos() = adjacenthypotenuse. By the same measure, the sine function returns the

ratio of the length of opposite side to the length of hypotenuse. That is sin(x) =opposite

hypotenuse. As

the pictures showing below, when a point is on the unit circle, the hypotenuse of the right

triangle which is radius, is equal to 1. Therefore, the x-coordinate of a point on the unit circle isequal to the cosine of the angle and likewise the -coordinate is equal to the since of the angle,for any angle.

The same property holds in the complex plane. We can rewrite all points + as (, ) andthus ( cos , sin ) cos + sin since we know that every coordinate in the complexplane represents one number. That is known as polar form.

Let us look back at the complex number+.


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We said before that the complex number+ = and thus this must be equivalent to thevalue in polar form. So

= cos + sin = (cos + sin ) .Now is a real number since is a real number, thus = , the only real part in polar form.So = cos + isin. Now in the complex number we are interested in 2, = 0 and thusthe radius, = 0 = 1 for all possible values of.

This means that for every value of we are always 1 unit away from the origin of the circle.

That means that the values of map onto a circle in the complex plane with radius 1.

Another way to think about this is to look again at when = 0 so at 0 = 1. Since the rateof change is , which is equal to , there is not a component that brings it closer to the origin,just a rotation. Thus it stays a constant distance, 1, from the radius. This implies that we are

looking at a circle with radius 1 and that the magnitude of and is 1 for all values of.


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Since is always changing, we can assume that every point along the circle represents some value and as changes we move around the circle. But how do we know what relation the value has to the location on the circle? Since the derivative of is and the length of is1 for all values of

, the length of

must also be 1 just at a 90

angle from

. This implies

that the rate of change has the same magnitude, or length, of, just moving at a 90 angle. Sothis means that the rate at which the function is changing, or the magnitude of the derivative,is the same as the amount the independent variable has changed. So we know that we need to go

all the way around a circle with radius 1, or 2 units around the circle. Since we know that the

rate or the speed we are moving at is the same as the amount has changed, we know must bethe number of radians we have rotated in the counterclockwise direction from the positive x-axis.

Another way to think about this is to take the real number line of all possible values of. Nowalso make a circle with radius 1 centered about the origin of the complex plane. For every 1 unit

you move on the real line, also move 1 unit around the circle. Since the circle has radius 1, the

circumference is equal to 2r, or 2 units sowe have to move 2 units around the circle, and thus

2 units on the real number line. Since is our only input value, it must correspond directly tothe angle measure. Since there are 2 radians in a circle and we must move 2 units. So the

change of distance in the range must correspond to the change of distance in the domain.

Radians and 2


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Let us look at what a radian is. Taking any circle with center at the vertex of the angle, a

radian measure of the angle is equal to the ratio of the length of the subtended arc to the radius of

the circle.

What happen when we look at the entire circle? How can we calculate the arc length of the

circle? Well, the arc length is equal to the circumference and the circumference of a circle is

equal to 2. If the circle is a unit circle, which means the radius is 1, then the circumference isequal to 2. According to the definition of radian, the number of radians is equal to the arc length,

which is 2.

Eulers Identity =

Let us back to we are interested complex number2

. From the pictures above we see thatafter rotating from the positive horizontal axis 2 radians, we get the value of2.

In above we got = cos + isin . Now we know that is equal to out input . So thismeans that = cos + isin. If we plug in our desired value 2, then we get

2 = cos(2) + sin(2) = 1 + (0) = 1.

PS: if you are interested in another way to prove Eulers identity using calculus, you can go

here:http://en.wikipedia.org/wiki/Euler%27s_formula.
http://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Euler%27s_formula

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Euler s Explanation