You can only proof this identity after you clearly defined what e^(ix) means.
In this sense arguments using calculus and using differential equations are incomplete, without at least a clear definition any argument becomes an heuristic argument.
I know of three ways of defining e^(ix).
(1) From the complex series of e^(z) defined as sum of z^n/n!
(2) as the limit of n to infinity of (1 + ix/n)^n
(3) or directly as cos(x) + i sin(x)
Either way only after defining e^(ix) is that you should show that it has the properties like e^(ia)*e^(ib)=e^(i(a+b)), or (e^ix)'=ie^(ix) that you would expect it to have, and some use them fearlessly.
Number (1) is the most common one, I think number (2) would be a nice thing to use since it is analogous to the real case and can also be interpreted geometrically (as Richard Feynman does for a reference). But using (3) is totally misleading because it doesn't show why it should be true.
That is why I think heuristic arguments are needed to provide "a reason" for us to believe that such a thing should be true. Using circular motion as in my next post seems to me much more simple and much less "out of the blue". While the proof by itself is heuristic it can be made rigorous.
Showing posts with label Complex numbers. Show all posts
Showing posts with label Complex numbers. Show all posts
Proof of Euler's formula
This is a proof of Euler's formula that uses the circular motion
Suppose
P(t)= e^(it)
is a trajectory on the complex plane, using some definition of the exponential (the Taylor series definition for example) it is possible to show that
P'(t)=(e^(it))'= i e^(it) =i P(t)
and
P(0)= e^(i0)= e^(0)=1
In fact any reasonable definition of exponential should have these properties if you want to take these for granted all this becomes an heuristic proof.
then P(t) describes a trajectory in the complex plane that at t = 0 is at 1 and which velocity P(t) is always perpendicular, (90 degrees counter-clockwise) to P(t) and the velocity P(t) also has the same modulus as P(t). Such trajectory can only be the circular movement with radius one, unit velocity, starting at the point 1: cos(t) + i sin(t)!! If you want rigor again use the uniqueness of solution of differential equations the Picard–Lindelöf theorem.
e^(it)= cos(t) + i sin(t).
Suppose
P(t)= e^(it)
is a trajectory on the complex plane, using some definition of the exponential (the Taylor series definition for example) it is possible to show that
P'(t)=(e^(it))'= i e^(it) =i P(t)
and
P(0)= e^(i0)= e^(0)=1
In fact any reasonable definition of exponential should have these properties if you want to take these for granted all this becomes an heuristic proof.
then P(t) describes a trajectory in the complex plane that at t = 0 is at 1 and which velocity P(t) is always perpendicular, (90 degrees counter-clockwise) to P(t) and the velocity P(t) also has the same modulus as P(t). Such trajectory can only be the circular movement with radius one, unit velocity, starting at the point 1: cos(t) + i sin(t)!! If you want rigor again use the uniqueness of solution of differential equations the Picard–Lindelöf theorem.
e^(it)= cos(t) + i sin(t).
Complex Numbers 3
So we saw that with the rotation-stretching interpretation multiplying and taking powers of complex numbers makes more sense.
Obs. While this interpretation makes multiplication easier it does not help when making addition. Happily addition is easy in the algebraic form "a + bi".
Now we are going to see three bonus of looking at complex numbers in this way.
First bonus: Our theory is somewhat arbitrary why we defined i to rotate 90 o counterclockwise ? It could be clockwise couldn't it ? The answer is yes! Lets say you defined a different clockwise imaginary number j, because you didn't like mine, then you would have a complex number theory just like mine with j's in the place of i's! Your drawings would be the reflection of my drawings. But the arithmetic would be the same.
But your j does the same job as my -i! So in a way the -i does the same job as i, but clockwise. Changing i's to -i's changes the orientation of my rotations. So if I take a number like 3+4i, then 3+4(-i) = 3 - 4i has opposite rotation but the same stretching factor. The operation changing i's to -i's is called conjugation and this reasoning explain most of it's properties directly.
Second bonus: Let's say our stretching factor is 1 (no stretching at all) in this case our complex number only rotate vectors by angles a and b. So they are:
cos(a) + i sin(b)
cos(b) + i sin(b)
An their product by our interpretation:
cos(a+b) + i sin(a+b)
But we can also multiply
cos(a)+i sin(b))*(cos(b)+i sin(b)) by distribution and that is:
(cos(a)cos(b) - sin(a)sin(b)) + i(cos(a)sin(b) + sin(b)cos(a))
And we have the formulas for cos(a+b) and sin(a+b)!,
Although the geometrical demonstration for those formulas is really nice too (there is at least one nice geom. dem. that I know of).
As an exercise you can find what happens when b = -a.
Third bonus: if you remember the First bonus a+bi and a-bi have same stretching but opposite rotation angles u and -u say. Remembering a+bi forms a right triangle with legs a and b lets call c its hypotenuse, that is also its stretching factor.
Then the product of a+bi and a-bi has u + (-u) = 0 rotation and stretching c*c = c2 , so it is really c2 , but algebraically
(a+bi)*(a-bi) = a2 - abi + abi - (bi)2 = a2 -(-b2 )=a2 +b2 .
So a2 + b2 = c2
That is Pythagoras theorem.
Obs. While this interpretation makes multiplication easier it does not help when making addition. Happily addition is easy in the algebraic form "a + bi".
Now we are going to see three bonus of looking at complex numbers in this way.
First bonus: Our theory is somewhat arbitrary why we defined i to rotate 90 o counterclockwise ? It could be clockwise couldn't it ? The answer is yes! Lets say you defined a different clockwise imaginary number j, because you didn't like mine, then you would have a complex number theory just like mine with j's in the place of i's! Your drawings would be the reflection of my drawings. But the arithmetic would be the same.
But your j does the same job as my -i! So in a way the -i does the same job as i, but clockwise. Changing i's to -i's changes the orientation of my rotations. So if I take a number like 3+4i, then 3+4(-i) = 3 - 4i has opposite rotation but the same stretching factor. The operation changing i's to -i's is called conjugation and this reasoning explain most of it's properties directly.
Second bonus: Let's say our stretching factor is 1 (no stretching at all) in this case our complex number only rotate vectors by angles a and b. So they are:
cos(a) + i sin(b)
cos(b) + i sin(b)
An their product by our interpretation:
cos(a+b) + i sin(a+b)
But we can also multiply
cos(a)+i sin(b))*(cos(b)+i sin(b)) by distribution and that is:
(cos(a)cos(b) - sin(a)sin(b)) + i(cos(a)sin(b) + sin(b)cos(a))
And we have the formulas for cos(a+b) and sin(a+b)!,
Although the geometrical demonstration for those formulas is really nice too (there is at least one nice geom. dem. that I know of).
As an exercise you can find what happens when b = -a.
Third bonus: if you remember the First bonus a+bi and a-bi have same stretching but opposite rotation angles u and -u say. Remembering a+bi forms a right triangle with legs a and b lets call c its hypotenuse, that is also its stretching factor.
Then the product of a+bi and a-bi has u + (-u) = 0 rotation and stretching c*c = c2 , so it is really c2 , but algebraically
(a+bi)*(a-bi) = a2 - abi + abi - (bi)2 = a2 -(-b2 )=a2 +b2 .
So a2 + b2 = c2
That is Pythagoras theorem.
Complex Numbers 2
Last post Complex Numbers 1, we saw that:
Positive numbers like 3 stretches vectors.
Pure positive i numbers like 2i rotate by 90o and stretches.
Negative numbers like -3 rotate by 180o and stretches.
Pure negative i numbers like -2i rotate a vector by 270o and stretches.
What you guess composite numbers like 1+i or 2+3i do?
Returning with our vector v we can phrase the question as what is (1+i)*v ? It is (1+i)*v = v + iv, by distribution.
The vector iv forms 90o with v, so v + iv forms a square! and the sum is the diagonal of that square. That is v + iv has 45o, with v and is a little longer by a stretching factor of sqrt(2)=1,41 approximately.
And (1+i)2 should rotate by 45o + 45o = 90o and stretch by sqrt(2) * sqrt(2) = 2.
So it should be 2i!! Checking algebraically
(1+i)2 = (1+i)*(1+i) = 1 + 2i + i2 = 1 + 2i -1 = 2i.
Now what is (3+4i)*v ? It is the diagonal of the rectangle with sides 3v and 4iv, so it is a rotation by the angle of the diagonal with v and a stretching by 5 of v. So what is (3+4i)2 ? Before doing the multiplication we now that the angle will be the double of 3+4i and the stretching will be of 25!
Take a complex number z1 that rotates by 30o and stretches by 7 and another z2 that rotates by 120o and stretches by 4 then z1 * z2 rotates by 30o +120o = 150o and stretches by 4*7= 28.
In the last paragraph I didn't say what those numbers were, with a little trigonometry you find that they are:
7(cos(30o) + i sin(30o))
4(cos(120o) + i sin(120o))
28(cos(150o) + i sin(150o))
And that is the rule of sum the angles and multiply the stretching factors.
This way to write complex numbers is called the polar form it is more useful when we want to multiply or divide complex numbers, and it also makes powering much more easier.
Like if we want to calculate
(2(cos(30o) + i sin(30o)))15
We get 215 of stretching and an angle of 15*30o = 450o = 90o since 360o is a complete turn.
More on the next post.
Positive numbers like 3 stretches vectors.
Pure positive i numbers like 2i rotate by 90o and stretches.
Negative numbers like -3 rotate by 180o and stretches.
Pure negative i numbers like -2i rotate a vector by 270o and stretches.
What you guess composite numbers like 1+i or 2+3i do?
Returning with our vector v we can phrase the question as what is (1+i)*v ? It is (1+i)*v = v + iv, by distribution.
The vector iv forms 90o with v, so v + iv forms a square! and the sum is the diagonal of that square. That is v + iv has 45o, with v and is a little longer by a stretching factor of sqrt(2)=1,41 approximately.
And (1+i)2 should rotate by 45o + 45o = 90o and stretch by sqrt(2) * sqrt(2) = 2.
So it should be 2i!! Checking algebraically
(1+i)2 = (1+i)*(1+i) = 1 + 2i + i2 = 1 + 2i -1 = 2i.
Now what is (3+4i)*v ? It is the diagonal of the rectangle with sides 3v and 4iv, so it is a rotation by the angle of the diagonal with v and a stretching by 5 of v. So what is (3+4i)2 ? Before doing the multiplication we now that the angle will be the double of 3+4i and the stretching will be of 25!
Take a complex number z1 that rotates by 30o and stretches by 7 and another z2 that rotates by 120o and stretches by 4 then z1 * z2 rotates by 30o +120o = 150o and stretches by 4*7= 28.
In the last paragraph I didn't say what those numbers were, with a little trigonometry you find that they are:
7(cos(30o) + i sin(30o))
4(cos(120o) + i sin(120o))
28(cos(150o) + i sin(150o))
And that is the rule of sum the angles and multiply the stretching factors.
This way to write complex numbers is called the polar form it is more useful when we want to multiply or divide complex numbers, and it also makes powering much more easier.
Like if we want to calculate
(2(cos(30o) + i sin(30o)))15
We get 215 of stretching and an angle of 15*30o = 450o = 90o since 360o is a complete turn.
More on the next post.
Complex Numbers 1
In this post we will try to find some light in those strange complex numbers. There is a nice way to present them as rotations and stretchings of vectors, where complex numbers concepts and properties are all very natural, look for that later in this post.
Let's begin by telling my first experiences with complex numbers.
I think my first impressions were based on those mysterious names, after all we would learn about "complex numbers", about the number " i " that was the "square root of -1", that the teacher said before it didn't existed, and also about the "imaginary" numbers in opposition to the more "real" numbers learned before.
The names "complex numbers" and "imaginary numbers" were not a good choice of words from the pedagogical point of view. Well, "i" is the number that squared is -1, what is the logic in that? Lets make some experiments.
For example: 32 = 9 and 52 = 25 so positive numbers squared are positive. (-3)2 = (-3)x(-3) = 9, because negative times negative is positive, so negative numbers squared are also positive!
Concluding that all numbers squared are positive! So no number squared is negative! And in particular no number squared is -1! at worst it is 0.
So they actually invented a number " i " not a "normal number" that squared would be -1, that is what my teacher said to me.
After some algebra facts they let you know that you can plot a complex number like " 3+ 4i " in an x-y axis were the point (3,4) is, unfortunately that was not so helpful. Later on that class my teacher showed me lots of even more bizarre properties of complex numbers.
All that together was a dazzling sequence of "out of the blue" concepts and relations. Actually, complex numbers have a pretty decent history, real flesh and bone people invented it you know, and that history is not "out of the blue" I may tell something of that story latter.
Now I will present complex numbers in a different and I hope better way, first lets explore vectors.
A vector can represent something as velocity or force on a body and lots of other concepts. Lets think of it as a force on an object (which I am not drawing) to fix ideas.
When we apply two forces on an object what happens?
We have a resulting force! That depends on the strength of the original ones and on their direction. There is a way to add them, say u = AB v = AC move the vector u (maintaining direction) to the endpoint of the other C making a vector CD the sum u + v is AD that is the parallelogram law.
If we add v + v we have a vector twice the size in the same direction or v + v = 2v, also, v + v + v = 3v is the vector in the same direction three times longer. 1.5v is the vector with 1.5 the size, -v = -1v is the vector with opposite direction and same size (v + -v = 0), and -5v has opposite direction and is 5 times the size of v.
So when we multiply by a positive number we "stretch" the vector, or maybe shrink it if we multiply by a number smaller than 1, and when we multiply by a negative number the vector inverts its direction and is stretched appropriately.
If we multiply by 3 and after that multiply by 2 we get a vector six times larger, or just 2(3v) = 6v.
Imagine what we get with -3(4v), 5(-1v) and -3(-2v)? Think geometrically!!
-12v, -5v and 6v respectively
Obs. A negative number times a negative number should be positive since we invert the vector two times! - a good justification for that rule/theorem in my opinion.
If we put another vector w somewhere else multiplying by real numbers would also work exactly in the same way.
We can also think that multiplying by -1 rotates vectors by 180o degrees, in that spirit we could rotate vectors by other angles like 90o or 120o . Let's say the number "j" rotates vectors by 90o degrees counterclockwise.
If we multiply our vector by j two times we rotate it by:
90o + 90o = 180o that is -v, or
j*jv = -v = -1v
So j*j = j2 should be -1, and we have a number that squared is -1! j is really our imaginary number i, time for more experiments.
2i rotates 90o and multiplies or stretches by 2
2i*2i rotates by 90o + 90 o =180 o and multiplies by 2x2=4 or 2ix2i = -4.
-i rotates by 90o and inverts the direction so it rotates by 270o .
i3 = i*i*i rotates by 90o + 90o + 90o = 270o also, so i3 = -i.
i4 = i*i*i*i rotates 4*90o = 360o so i4 = 1 = i2 * i2 .
You can guess i5 , i6 , and i37 , note that the algebra coincides with our geometrical ideas.
All properties of complex numbers can be understood in this geometric "rotation-stretching" way, more on that on next post.
Let's begin by telling my first experiences with complex numbers.
I think my first impressions were based on those mysterious names, after all we would learn about "complex numbers", about the number " i " that was the "square root of -1", that the teacher said before it didn't existed, and also about the "imaginary" numbers in opposition to the more "real" numbers learned before.
The names "complex numbers" and "imaginary numbers" were not a good choice of words from the pedagogical point of view. Well, "i" is the number that squared is -1, what is the logic in that? Lets make some experiments.
For example: 32 = 9 and 52 = 25 so positive numbers squared are positive. (-3)2 = (-3)x(-3) = 9, because negative times negative is positive, so negative numbers squared are also positive!
Concluding that all numbers squared are positive! So no number squared is negative! And in particular no number squared is -1! at worst it is 0.
So they actually invented a number " i " not a "normal number" that squared would be -1, that is what my teacher said to me.
After some algebra facts they let you know that you can plot a complex number like " 3+ 4i " in an x-y axis were the point (3,4) is, unfortunately that was not so helpful. Later on that class my teacher showed me lots of even more bizarre properties of complex numbers.
All that together was a dazzling sequence of "out of the blue" concepts and relations. Actually, complex numbers have a pretty decent history, real flesh and bone people invented it you know, and that history is not "out of the blue" I may tell something of that story latter.
Now I will present complex numbers in a different and I hope better way, first lets explore vectors.
A vector can represent something as velocity or force on a body and lots of other concepts. Lets think of it as a force on an object (which I am not drawing) to fix ideas.
When we apply two forces on an object what happens?
We have a resulting force! That depends on the strength of the original ones and on their direction. There is a way to add them, say u = AB v = AC move the vector u (maintaining direction) to the endpoint of the other C making a vector CD the sum u + v is AD that is the parallelogram law.
If we add v + v we have a vector twice the size in the same direction or v + v = 2v, also, v + v + v = 3v is the vector in the same direction three times longer. 1.5v is the vector with 1.5 the size, -v = -1v is the vector with opposite direction and same size (v + -v = 0), and -5v has opposite direction and is 5 times the size of v.
So when we multiply by a positive number we "stretch" the vector, or maybe shrink it if we multiply by a number smaller than 1, and when we multiply by a negative number the vector inverts its direction and is stretched appropriately.
If we multiply by 3 and after that multiply by 2 we get a vector six times larger, or just 2(3v) = 6v.
Imagine what we get with -3(4v), 5(-1v) and -3(-2v)? Think geometrically!!
-12v, -5v and 6v respectively
Obs. A negative number times a negative number should be positive since we invert the vector two times! - a good justification for that rule/theorem in my opinion.
If we put another vector w somewhere else multiplying by real numbers would also work exactly in the same way.
We can also think that multiplying by -1 rotates vectors by 180o degrees, in that spirit we could rotate vectors by other angles like 90o or 120o . Let's say the number "j" rotates vectors by 90o degrees counterclockwise.
If we multiply our vector by j two times we rotate it by:
90o + 90o = 180o that is -v, or
j*jv = -v = -1v
So j*j = j2 should be -1, and we have a number that squared is -1! j is really our imaginary number i, time for more experiments.
2i rotates 90o and multiplies or stretches by 2
2i*2i rotates by 90o + 90 o =180 o and multiplies by 2x2=4 or 2ix2i = -4.
-i rotates by 90o and inverts the direction so it rotates by 270o .
i3 = i*i*i rotates by 90o + 90o + 90o = 270o also, so i3 = -i.
i4 = i*i*i*i rotates 4*90o = 360o so i4 = 1 = i2 * i2 .
You can guess i5 , i6 , and i37 , note that the algebra coincides with our geometrical ideas.
All properties of complex numbers can be understood in this geometric "rotation-stretching" way, more on that on next post.
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