Studying Mathematics Thread

[polymath257]

A very, very nice post. I only have a slight quibble about the history.

Plato was a couple of centuries *before* Euclid. So the geometry that Plato wanted for entrance to his Academy was not that of Euclid, but more likely that of Theatetus.

Second, Euclid's Elements *did* include a considerable amount of number theory. For example, his proof that there are infinitely many primes is simple and direct and used today. He also gave a condition when an even number is perfect (later showed to be the only situation where an even number is perfect by Euler).

Given your discussion of the 5th Postulate, I was also surprised that you didn't mention the rise of non-Euclidean geometry and the effect it had on the formalization movement leading up to Russell.

All said, though, well done!

Looks like the forums ate my last attempt at a reply, hopefully this one fairs better.

I didn't mention the number theory in Elements because, while the number theory was also rigorously approached and there was a significant amount of it, it wasn't axiomatized the same way that geometry was.  And I didn't mention non-Euclidean geometry because I'd already decided to reserve the 5th Postulate's epic for a potential later post, and also because I didn't feel like spending half the post trying to explain spherical and hyperbolic geometry and how that differs from the Euclidean postulates.  And I attributed the Academy's obsession with geometry to Euclid because... .... well, that one was a major screwup on my part.  I've believed that one for years but apparently it's false and I never, well, never did the math on that.  Good catch.

Well, Euclid made a distinction between axioms (which were general statements) and postulates (which were specific to the area of study). He actually had 5 axioms and 5 postulates for his geometry. My impression is that he regarded the axioms as the foundation for number theory and that geometry needed the additional postulates to make it work.

I definitely understand why you left out non-Euclidean geometry. Another aspect is that Euclid had  a number of 'hidden assumptions' that weren't really brought out until the 19th century, having to do with 'betweenness' properties. He implicitly used that a line meeting one side of a triangle will also meet at least one of the other sides (maybe at the corner).

In any case, I look forward to your later post!

[GrandizerII]

Ooh, I see a competition in math nerdom going on between polymath and Reltzik.

On a different note, good to be back here. I've been thinking of getting a degree in mathematics (with the eventual desire to maybe pursue a higher degree in some advanced math field of interest and become a qualified online tutor or something), but can't seem to find an Australian institution that provides an accredited math program online, and just don't have the time to study on-campus because of my job.

Yeah, they're talking a bit above my pay grade.

I had a mini-meltdown last night when I couldn't figure out why De Moivre's theorem worked. Like, why can you just take those exponents and drop them all willy nilly into the equation like that? The formula looked familiar enough, but I couldn't for the life of me remember what it iwas. Surely it's not so simple as Euler's formula. I would have recognized that right away. So I spent all night typing variations of "why the fuck does de moivre's theorem work?" into search engines only to realize that there was a common theme in the answers.

---

It's Euler's formula.

---

So yeah, their conversation is a bit over my head. But that's okay! I'm learning.

I've been slacking off on maths lately, and I only faintly remember something about De Moivre's theorem. Having read your post, I now want to really review that bit and make as much sense out of it as possible.

[polymath257]

Ooh, I see a competition in math nerdom going on between polymath and Reltzik.

On a different note, good to be back here. I've been thinking of getting a degree in mathematics (with the eventual desire to maybe pursue a higher degree in some advanced math field of interest and become a qualified online tutor or something), but can't seem to find an Australian institution that provides an accredited math program online, and just don't have the time to study on-campus because of my job.

Yeah, they're talking a bit above my pay grade.

I had a mini-meltdown last night when I couldn't figure out why De Moivre's theorem worked. Like, why can you just take those exponents and drop them all willy nilly into the equation like that? The formula looked familiar enough, but I couldn't for the life of me remember what it was. Surely it's not so simple as Euler's formula. I would have recognized that right away. So I spent all night typing variations of "why the fuck does de moivre's theorem work?" into search engines only to realize that there was a common theme in the answers.

---

It's Euler's formula.

---

So yeah, their conversation is a bit over my head. But that's okay! I'm learning.

Alternatively, it is the cosine and sine sum formulas applied repeatedly by induction.

[Aliza]

Yeah, they're talking a bit above my pay grade.

I had a mini-meltdown last night when I couldn't figure out why De Moivre's theorem worked. Like, why can you just take those exponents and drop them all willy nilly into the equation like that? The formula looked familiar enough, but I couldn't for the life of me remember what it was. Surely it's not so simple as Euler's formula. I would have recognized that right away. So I spent all night typing variations of "why the fuck does de moivre's theorem work?" into search engines only to realize that there was a common theme in the answers.

---

It's Euler's formula.

---

So yeah, their conversation is a bit over my head. But that's okay! I'm learning.

Alternatively, it is the cosine and sine sum formulas applied repeatedly by induction.

But wait... seriously... why do you just get to put the exponent in front of the cos and isin? How does that work? See, I once had this professor who insisted on proving to us why a formula worked, and I'd sit there in class thinking, "I don't give a shit! This is boring and confusing. Just give me the formula and I'll plug in my little values and get an A in your class. Cause that's what I do!"

But now I'm in this place where I'm seeing things and I can't just take DeMovire's word for it. I'll grant that I'm more inclined to take Euler's word for it, but I'd still like to know why this formula works.

[GrandizerII]

Alternatively, it is the cosine and sine sum formulas applied repeatedly by induction.

But wait... seriously... why do you just get to put the exponent in front of the cos and isin? How does that work? See, I once had this professor who insisted on proving to us why a formula worked, and I'd sit there in class thinking, "I don't give a shit! This is boring and confusing. Just give me the formula and I'll plug in my little values and get an A in your class. Cause that's what I do!"

But now I'm in this place where I'm seeing things and I can't just take DeMovire's word for it. I'll grant that I'm more inclined to take Euler's word for it, but I'd still like to know why this formula works.

Have you checked this link?

https://proofwiki.org/wiki/De_Moivre%27s_Formula

If that doesn't help, what do you mean by "exponent in front of the cos and isin"? Because, from what I see, only r has a variable exponent.

[polymath257]

Alternatively, it is the cosine and sine sum formulas applied repeatedly by induction.

But wait... seriously... why do you just get to put the exponent in front of the cos and isin? How does that work? See, I once had this professor who insisted on proving to us why a formula worked, and I'd sit there in class thinking, "I don't give a shit! This is boring and confusing. Just give me the formula and I'll plug in my little values and get an A in your class. Cause that's what I do!"

But now I'm in this place where I'm seeing things and I can't just take DeMovire's word for it. I'll grant that I'm more inclined to take Euler's word for it, but I'd still like to know why this formula works.

Well, first, do you know the sine and cosine sum formulas?

sin(A+B)=sin(A)cos(B)+cos(A)sin(B)
cos(A+B)=cos(A)cos(B)-sin(A)sin(B)

If not, then we can go through those separately. If so, then see what happens when you multiply out

[cos(A)+i*sin(A)] * [cos(B)+i*sin(B)]

You should get

cos(A)cos(B) + i* sin(A)cos(B) +i*cos(A)sin(B) +i^2 *sin(A)sin(B)

=[cos(A)cos(B)-sin(A)sin(B)]+i*[sin(A)cos(B)+cos(A)sin(B)] =cos(A+B) +i *sin(A+B)

(remember that i^2 =-1)

In other words, if you multiply two expressions of the form cos(A)+i*sin(A), the result can be obtained by *adding* the angles involved.

Now, what happens if you multiply the *same* expression over and over again? The angle adds up again and again, however many times you did the multiplication.

That is why

[cos(A)+i*sin(A)]^n = cos(n A)+i*sin(n A)

Each multiplication corresponds to an addition of the angles.

[Aliza]

In other words, if you multiply two expressions of the form cos(A)+i*sin(A), the result can be obtained by *adding* the angles involved.

Now, what happens if you multiply the *same* expression over and over again? The angle adds up again and again, however many times you did the multiplication.

That is why

[cos(A)+i*sin(A)]^n = cos(n A)+i*sin(n A)

Each multiplication corresponds to an addition of the angles.

So I took out paper and pencil and worked it out along with your example. I can clearly see how cosine and sine sum/difference formulas fit into DeMoivre's theorem. Sometimes I need it spelled out, but once I started following along with you, it was clear where the steps would take me. I was still scratching my head over how the exponent plays into this, but then it hit me like a ton of bricks. I see the little cycle there of adding and multiplying.

In short, you've effectively explained it to me, so thank you.

[polymath257]

In other words, if you multiply two expressions of the form cos(A)+i*sin(A), the result can be obtained by *adding* the angles involved.

Now, what happens if you multiply the *same* expression over and over again? The angle adds up again and again, however many times you did the multiplication.

That is why

[cos(A)+i*sin(A)]^n = cos(n A)+i*sin(n A)

Each multiplication corresponds to an addition of the angles.

So I took out paper and pencil and worked it out along with your example. I can clearly see how cosine and sine sum/difference formulas fit into DeMoivre's theorem. Sometimes I need it spelled out, but once I started following along with you, it was clear where the steps would take me. I was still scratching my head over how the exponent plays into this, but then it hit me like a ton of bricks. I see the little cycle there of adding and multiplying.

In short, you've effectively explained it to me, so thank you.

I'm really glad I could help!

---

But wait... seriously... why do you just get to put the exponent in front of the cos and isin? How does that work? See, I once had this professor who insisted on proving to us why a formula worked, and I'd sit there in class thinking, "I don't give a shit! This is boring and confusing. Just give me the formula and I'll plug in my little values and get an A in your class. Cause that's what I do!"

But now I'm in this place where I'm seeing things and I can't just take DeMovire's word for it. I'll grant that I'm more inclined to take Euler's word for it, but I'd still like to know why this formula works.

Have you checked this link?

https://proofwiki.org/wiki/De_Moivre%27s_Formula

If that doesn't help, what do you mean by "exponent in front of the cos and isin"? Because, from what I see, only r has a variable exponent.

The biggest problem here is that the identification

exp( i*A)=cos(A)+i*sin(A)

is not at all easy to see. There are two main ways I know of: one uses the power series expansions of exp, sin, and cos. The other does it by solving a differential equation.

But DeMoivre's theorem is something that can be appreciated well before those topics are covered.

[Fireball]

In other words, if you multiply two expressions of the form cos(A)+i*sin(A), the result can be obtained by *adding* the angles involved.

Now, what happens if you multiply the *same* expression over and over again? The angle adds up again and again, however many times you did the multiplication.

That is why

[cos(A)+i*sin(A)]^n = cos(n A)+i*sin(n A)

Each multiplication corresponds to an addition of the angles.

So I took out paper and pencil and worked it out along with your example. I can clearly see how cosine and sine sum/difference formulas fit into DeMoivre's theorem. Sometimes I need it spelled out, but once I started following along with you, it was clear where the steps would take me. I was still scratching my head over how the exponent plays into this, but then it hit me like a ton of bricks. I see the little cycle there of adding and multiplying.

In short, you've effectively explained it to me, so thank you.

Watch out, or he'll have you studying analytical algebraic topology of infinitely differentiable Riemannian Manifolds! Boyja Moi!

Finally got a chance to post that. Special thanks go to Tom Lehrer for making a song about it!

[polymath257]

So I took out paper and pencil and worked it out along with your example. I can clearly see how cosine and sine sum/difference formulas fit into DeMoivre's theorem. Sometimes I need it spelled out, but once I started following along with you, it was clear where the steps would take me. I was still scratching my head over how the exponent plays into this, but then it hit me like a ton of bricks. I see the little cycle there of adding and multiplying.

In short, you've effectively explained it to me, so thank you.

Watch out, or he'll have you studying analytical algebraic topology of infinitely differentiable Riemannian Manifolds! Boyja Moi!

Finally got a chance to post that. Special thanks go to Tom Lehrer for making a song about it!

As I recall, it was

Analytic and algebraic topology *of locally Euclidean metrizations* of infinitely differentiable Riemannian manifolds....