algebra life 4

I am reading a lot of maths lately, trying to get as much as possible before my birthday.

Algebra and analysis.

Algebra and geometry.

Algebra and number theory.

What is algebra? It comes from arabic, meaning the reunion of broken parts. “In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.” Wikipedia.

So far in reading algebra, I have been reading about groups, different types of groups, subgroups. Basically a lot of definitions and theorems and proofs regarding groups which follow from the group axioms.

Fairly abstract.

Groups, along with rings and fields (which I have not read yet) are descriptions of the simplest algebraic structures which allow equations like polynomial equations to be solved.

My life is broken, can algebra help to reunion the broken parts of my life?

algebra life 3

Whenever I read some sort of text, like mathematics, the beginning is usually enticing and interesting enough to keep my attention.

And then it gets tough, and some things like abstract algebra gets very abstract quickly, the learning curve is fairly steep for someone with no guidance.

I then try to skim through it and see if I can find any beacon of light where I can latch onto or motivate me to persevere through the text.

For modern/abstract algebra, this will have to be Lagrange’s theorem which states that for any finite group G, the order of every subgroup H of G should divide the order of G.

I understand what it means, but I don’t think I truly appreciate it yet. Which means I will have to backtrack and see if I can get a better intuition through means of maybe drawing some pictures or construct a more complete picture (story) of group theory.

algebra life 2

Associativity

A lot of algebraic structures, like the addition and multiplication of real numbers or the rational numbers have the associative property, which is often taken for granted.

It basically says that given a binary operation ∗ , is associative if for all three elements x, y, and z in a given set S, (x ∗ y)  ∗ z = x ∗ (y ∗ z). Then the products can be unambigiously written as x  ∗ y  ∗ z.

From this, the general associative law follows in that for any number of elements, their products can be unambiguiously written without any parentheses.

This can be proven by mathematical induction, but it is not the easiest one to formulate.

As a non-math major, it took me some time to absorb this fact.

Non-associativity

Without the associative property, as the number of elements increase, the number of possible products increase quickly (look at the Catalan numbers for a formula).

Operations that don’t have associativity include subtraction, division and exponention.

For example, we can see that (2/3)/5 does not equal 2/(3/5). The first equates to 2/15 and the second equates to 10/3.

algebra life 1

My life now revolves around reading algebra. This series of posts and probably the remaining posts for this blog will be titled “algebra life n”, where n is an element of the natural numbers {1, 2, 3, …}

I have nothing else good to do.

I am reading chapter 3 from the Judson textbook. I start from Chapter 3 because the MIT course said so, but I realise I have to backtrack to the previous chapters anyway to understand some concepts in chapter 3, like reviewing key concepts like mathematical induction, the well-ordering principle, and the division algortihm.

I’ve seen those mathematical concepts many times already, but it is not drilled into my head yet.

Looking at problems recursively and inductively instead of explicitly often results in better understanding of complex issues.

modern algebra course

I am trying to self-study this course.

The pre-requisites for this course is multi-variable calculus, for which I am not well-versed, but I will cover necessary material when necessary.

If you want to study this course with me, just do these readings from the free Judson textbook.

I am also doing the assignments from the same course which require the Herstein textbook (not available for free). You can do them if you want. I am doing them to consolidate my knowledge on the algebraic structures.

Why are you doing this course? I don’t know. There are some interesting theorems that act as beacons of light and I hope that they will be good enough for me to persevere through this sea of difficult abstraction. Join me!

euler’s formula

Preface

Whenever I do a post like this, I hope I don’t make a silly mistake; but then again, posts of this nature contain content copied and pasted from other sources.

I first encountered the following equation in high school. The class was further mathematics, and only had 3 people. I believe it was for the A-levels module FM1 (sadly, this was ages ago, FM stands for further mathematics). Our teacher was teaching the Taylor series and he proved the equation that way. Later, in this post, I will just prove the equation via differentiation.

I don’t remember encountering this equation ever again, not even at university (though I didn’t study mathematics, sorry for being so cryptic, maybe later I will tell you what I studied at uni). Actually, maybe we touched it at university, but it didn’t stick in my mind that much.

Recently, I was researching mathematical proofs (because this is what I do in my life now), and came across this neat equation again.

 

Main point: Euler’s formula

e^{i\theta } = cos\theta + isin\theta discovered by Swiss mathematician Euler, born in 1707.

 

Beauty

It is beautiful for many reasons. When you set \theta = \pi, you get

e^{i\pi}+1=0.

This equation contains:

  • the basic arithmetic operations (each only occuring once):
    • addition,
    • multiplication, and
    • exponentiation
  • the five fundamental mathematical constants:
    • 0 (additive identity),
    • 1 (multiplicative identity),
    • π (3.141592653),
    • e (2.718281828),
    • i (square root of negative 1, basic unit of imaginary part of complex numbers)

 

Proof

By differentiation.

  • define function f(\theta)=e^{-i\theta}(cos\theta+isin\theta)
  • with product rule, it is easy to get the function’s derivative f'(\theta)=0
  • this means the function is constant in \theta
  • evaluate f(0)=1
  • this function is 1 everywhere
  • rearrange this to get Euler’s identity

QED.

 

Picture

euler's identity

Euler’s equation uses polar coordinates (distance, angle). e^{i\theta } = cos\theta + isin\theta separates the complex growth into its components: real growth (distance from origin) and imaginary growth (anti-clockwise rotation).

The equation in that picture is explained by starting at (1, 0). Operating (1, 0) with e^{i\pi}. This explains why e^{i\pi} is -1.

But there is more! Let’s not stop there… since this is periodic (goes around in a circle), this also applies to anything that is +2\pi.

It also shows why e^{\pi/2} = i.

Then we can show that i^i = e^{-\pi/2} = 0.20787... a real number!

Other usefulness of this equation is it replaces a lot of dreadful trig formulas you had to memorise in high school like double angle formulas – yuk! Just substitute in 2x in the theta and do some expansions, comparing imaginary parts with imaginary parts etc… and you will get what you want!

 

Salt

You might want to check other sources to gain a better intuitive understanding of the equation – it covers a lot of mathematical ground so it is worthwhile doing that. I warn you now that I do not have a complete understanding myself so take everything above with a grain of salt.

lies

I feel pained whenever I catch myself blatantly telling a lie to someone else. And I did yesterday when my neighbour asked me how the winter was for me.

I blurted some detail as a reply that was an outright lie. My neighbour might have sensed that it was a lie. I definitely knew it was a lie. It was not some innocent reply like “it was okay”. Anyways I added so much detail to that answer as if to compensate for something.

Doesn’t matter what I said. I lied and I added way too much detail that makes no sense. I could have just said the winter was ok. But I blabbled to fill the silence and those blabbles were lies.

What is wrong with me…

Human nature.

Maybe that is why I don’t talk much with other people. It’s because all we do is tell lies unless we talk about some abstract thing like mathematics. I have a feeling that we all tell lies because we inherently do not accept imperfection yet we, undoubtly, are.

I way around this is to either accept your imperfections (though my hypothesis is that we inherently cannot) or just accept that we tell lies. Is there another option?