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The paper that keeps showing up

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Re: The paper that keeps showing up

#31

Post by cestith »

cyberbanjo wrote: post #27
lordnacho wrote: post #23

Earlier quoted context omitted.

Someone on HN proposed "Lateral Number" which I thought was a good one. It's a number, just to the side of the numbers you know.

"If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." --Gauss

One of the most helpful things for me when handling imaginary and complex numbers came not from a math class, but a physics class. My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. This was literally an aside to something else he was talking about at the time.

Calling the imaginary numbers "lateral" and saying they were "to the side of" other numbers would've been helpful, too. I'm not sure it would have been quite as helpful to me, personally, but either is better than "imaginary" just to juxtapose with the "real" numbers.

Re: The paper that keeps showing up

#32

Post by zozbot234 »

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

You just need to familiarize yourself with the prereqs. If you have no idea what a "group homomorphism" is, or how to spot one in the wild, you aren't really expected to grok this stuff.

Re: The paper that keeps showing up

#33

Post by lordnacho »

cestith wrote: post #31
cyberbanjo wrote: post #27

Earlier quoted context omitted.

"If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." --Gauss

One of the most helpful things for me when handling imaginary and complex numbers came not from a math class, but a physics class. My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. This was literally an aside to something else he was talking a…

> My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers.

IIRC you often encounter a system where the energy is constant, but only if you consider the imaginary part of some equation holding the potential energy, which then exchanges with "actual" kinetic energy.

Re: The paper that keeps showing up

#34

Post by contravariant »

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

Do you know any group theory? Otherwise it's no wonder you see most of this as meaningless symbol manipulation, as you wouldn't be able to relate any of the symbols to meaningful concepts.

Though if you do know group theory you may have been put off by their choice to use addition and multiplication instead of multiplication and integer powers, as is more common.

I'm also somewhat annoyed they choose to phrase things in terms of a "Group of prime order p with associated field of scalars Fq". Since there is exactly 1 group of order p they may want to justify why it's worth bothering to phrase it like that instead of just doing addition mod q. And in this case Fq is also just integers mod q, and you don't use any of its field properties, so why not just use integers?

Re: The paper that keeps showing up

#35

Post by cronokirby »

contravariant wrote: post #34
club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

Do you know any group theory? Otherwise it's no wonder you see most of this as meaningless symbol manipulation, as you wouldn't be able to relate any of the symbols to meaningful concepts. Though if you do know group theory you may have been put off by their choice to use addition and multiplication instead of multiplication and integer powers, as is more common. I'm also somewhat annoyed they choose to phrase things…

A subtle point which I don't really explain in the post, since it was initially targeted to people who had done some cryptography with groups before, is that you need to carefully distinguish a group as a mathematical object, and a group as a computational object.

In cryptography, you not only need a mathematical group, but you also need a group as a concrete object, with ways of representing elements as bits, and efficient algorithms for manipulating elements of the group.

In fact, you also assume that computing other representations of the group is difficult. If you take the kind of group I mention in the post, it is the same group, mathematically, as just taking the integers mod q, along with addition. However, it should be very difficult to figure out how to convert the representation of the group you have into the the "simpler" representation as integers.

As for the field properties, you do need to be able to invert for some things later on in the post.

Re: The paper that keeps showing up

#36

Post by lynguist »

Wow, Ueli Maurer. That was a deeply nostalgic name to show up in this article.

I took the mandatory first-year Discrete Math course from him, and I would say that it was the tastiest math class I’ve ever taken in my life.

His script for the class as well, with its famous chapters 1-8. Even though it was riddled with Helvetisms, each sentence in the script was crystal clear and Ueli Maurer’s passion could be felt through and through.

Re: The paper that keeps showing up

#37

Post by contravariant »

cronokirby wrote: post #35
contravariant wrote: post #34

Earlier quoted context omitted.

Do you know any group theory? Otherwise it's no wonder you see most of this as meaningless symbol manipulation, as you wouldn't be able to relate any of the symbols to meaningful concepts. Though if you do know group theory you may have been put off by their choice to use addition and multiplication instead of multiplication and integer powers, as is more common. I'm also somewhat annoyed they choose to phrase things…

A subtle point which I don't really explain in the post, since it was initially targeted to people who had done some cryptography with groups before, is that you need to carefully distinguish a group as a mathematical object, and a group as a computational object. In cryptography, you not only need a mathematical group, but you also need a group as a concrete object, with ways of representing elements as bits, and ef…

Ah you need division for the extractability, fair enough. Though if it can be extended to arbitrary groups the field structure can't be crucial.

Re: The paper that keeps showing up

#38

Post by dylan604 »

lordnacho wrote: post #12
club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

I guess whoever taught you complex numbers didn't do it with a focus on diagrams? I agree math should be taught as visually as possible, with algebra-only explanations being minimised. Even things like the expansion of (a+b)^2 = a^2 + 2ab + b^2 or a^2 - b^2 = (a + b)(a - b) should be motivated by diagrams. Anyway it turns that multiplication of complex numbers is adding the polar angles. This explains why multiplying…

While it has been multiple decades now since I've been in school, this by far, was the best explanation I've read on how/when/where/why of imaginary numbers. I don't recall ever hearing polar anything related to discussions of "i". Thanks. You've now filled my "learn something new every day" quota.

Re: The paper that keeps showing up

#39

Post by est31 »

hamilyon2 wrote: post #9
marosgrego wrote: post #7

Earlier quoted context omitted.

You can think of `i` as a rotation by 90 degrees counterclockwise. You do it twice and the real number line gets reflected (as if multiplied by -1). It is what multiplying by i actually does to the complex plane.

I like how your illustration also implies that -i is also square root of -1, which is true

Yes! In fact, you can rotate i (or 1) by any angle around the origin and obtain a so-called "root of unity". If the angle was rational, you can now apply this rotation multiple times to itself and eventually you'll obtain 1 or -1.

Re: The paper that keeps showing up

#40

Post by klyrs »

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

> My question is: did I just kinda hit the limit of my IQ / abstract thinking skills? Do other people smarter than me think all this makes sense on an intuitive level?

I honestly don't think so. I was like that at one point, too, and "quit" math. Got a job and spent a few years programming full-time. I then went back to college, and lo and behold, I was required to do math for the courses I was interested in. I then had a couple of teachers who really made us grind, like, all the even-number problems in each chapter.

One teacher gave what seemed to be impossibly long exams; he took pride in people dropping his course. After the first exam, I got an okay score, but he put a note on the exam and I chatted with him after class. He was like, "your handwriting sucks on the last half of the exam, you should practice the material more so you don't feel so rushed." So, on the following exam, I literally did speed drills on the practice exam until I could knock it dead in an hour. Took 4 tries, if I recall correctly.

Eventually, I was able to "see" through the symbols. It took hundreds, perhaps thousands, of hours of dedicated practice. It wasn't easy. I was enraged, distraught, I was brought to tears, but I kept at it. I used to tell myself that I was too stupid to do math. In retrospect, I wasn't motivated enough to do math. I found that it's hard, perhaps incredibly hard, but not impossible. Though it wasn't my goal to begin with, I eventually got a PhD in math.

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