How to Maths at Oxford

So the number sets kinda look like this,

you have 

a [all the real numbers]

a + bi [the complex numbers which include i the imaginary number which is really just another way of looking at how algebra can function (we can now take the square root of negative numbers), which is a different take on algebra where we simply had the set of real numbers]

and then you get 

a + bi + cj + dk [quaternions, which is the next step up from the complex numbers in algebra, and are a new set of numbers that have a new way of looking at algebra, and are also infinite]

So about the complex numbers, thats numbers that include i, the ~imaginary number~ which is a terrible name for it, it’s not imaginary, just a completely different bit of algebra that doesnt work with what was previously thought to be algebraic laws and rules. Because before we had i you couldn’t take the square root of a negative number, because thats just not how the real numbers worked. It changed how algebra functioned with this new set of numbers. 

You can just change the set of rules for math for it to work, by creating something new, as long as it all logically follows.

Anyway, once you get past the complex numbers you hit this cool thing called quaternions, which is another set of numbers, except in this set of numbers, which are also infinite, but a different type of infinity, they don’t follow all of the rules of algebra we were previously taught to believe, in this set of numbers we don’t have the commutativity property,

[commutativity property is where a*b = c, and b*a = c]

without this it means if you multiply numbers together together in different orders, you get different answers. Which isn’t how any of the other previous sets of numbers work in algebra.

in regular algebra which works with all those numbers up to the complex numbers: 2*3 = 6, and 3*2 = 6 In quaternions: j*k = i, but k*j = -i

So with quaternions 

i^2 = -1

j^2 = -1

k^2 = -1

But you know whats wild about this, in this set of numbers:

i*j*k = -1

And the order does matter in this set of numbers.

because in this set of algebra

i*j = k  BUT heres the wild part j*i = -k

we can see it again:

j*k = i, BUT k*j = -i

When you multiply j*k you get i, when you multiply k*j you get -i, which isn’t how any of the other number sets behind this one in algebra work, isn’t that wild? 

There’s a part of math where basic definitions of algebra don’t exist for these types of numbers to exist, and the further you keep going into new numbers, the more fundamental rules of algebra you start losing for the number sets to exist.

and this guy just came up with it walking across a bridge

Okay, let me tell you a story about Hillary Clinton.  She first made national headlines when she was something like 22 and the student graduation speaker at Wellesley College.  Now Wellesley is the most famous women’s college in the United States and at the time when women were not admitted to most of the Ivy League it was considered one of the top schools in the country for a woman to attend.  She used her platform to eviscerate the sitting Senator of Massachusetts, a Republican, for his support of the Vietnam War.  To his face because he was on the stage with her.

She was identified as a future leader of the Democratic Party from when she was in her early twenties.  After law school at one of the top law schools in the country she was hired as a staff worker on the Watergate Commission.  Now everyone in American politics in 1973 knew that the young people who worked on the investigation of Richard Nixon’s criminal activity were the future leaders of the Democratic Party.  

And from that moment, in her twenties, the American right decided to try and destroy her.  As an aside the right has spread unfounded rumors that she was fired from the Watergate Commission, the man who claims to have done it never had the authority to fire her and she worked on the commission until it closed down only leaving when all the staff at her level left, but facts have never gotten in the way of a good demonization of this woman.

She was one of the leading advocates for abused children and legal aid for poor people in the country in the late 1970s and 1980s when she was living in Arkansas and the narrative about her then was that she was the demon feminist who was masterminding letting children “divorce” their parents.  Because apparently there is never a circumstance in the minds of those on the right that a child might need to be emancipated from their parents or that children’s rights might be independent of those with the most power to control and abuse them.

When her husband was running for president she was attacked for her looks (and for the looks of her at the time twelve year old daughter).  Her career as a lawyer and as a law professor was used against her and the reason we have a “first ladies cookie recipe contest” every presidential campaign is because she had the audacity to say she was busy having a career and not baking cookies.  By the way this humiliating exercise is one she won both times when her husband was running and the magazine didn’t think that perhaps when she was running for president this little 24 year old personal attack exercise against her should be retired when she was making history.  Bill submitted the same recipe she won with in the 1990s.

And then the investigations started immediately.  

She lost several hundred thousand dollars in an investment scheme and she was accused of insider trading.

A personal friend committed suicide and she was accused both of having an affair with him and of killing him.

She was routinely accused of sleeping with … basically any female friend of hers.  Because making someone a scary lesbian wasn’t yet out of fashion in the mid 1990s.

She was attacked for trying to keep her family together as she was humiliated daily on national television.

And she was attacked for being mad at her husband as she was humiliated daily on national television.

They even attacked her cat because proper presidential families have dogs and it was clearly her witchcraft familiar.

The Republicans in congress have been launching fishing expeditions against this woman for the last 22 years and that’s not even counting the smear attacks that started in the late 1960s.  Today the Republicans in the House have spent more time and money investigating the attack on the embassy in Benghazi when she was secretary of state then was spent investigating the attacks on 9/11.  Ignoring that attacks on American embassies have occurred under every presidential administration for the last six decades and there were far more of them under George W Bush and for more people died in them under Ronald Reagan.

Oh.  And she was mocked in the late 1990s for daring to say there was a vast right wing conspiracy against her and her husband.  Because a woman should never acknowledge … you know… a provable fact.

So of course the average American voter has trouble distinguishing the truly horrific facts about Trump from the noise about Clinton.  We’ve been fed a steady diet of it for 45 fucking years with the express purpose of making sure this woman was never president of the United States.

drst:

Worth nothing that the email thing the Republicans are trying to hang her with? Her Republican predecessors (Colin Powell) set up their email the same way. The previous Republican president’s staff used a private server. This is literally “It’s OK If You’re A Republican” (IOKIYAR) in action. And the emails that are under investigation were never hacked. The Wikileaks crap is from the DNC hack, not from the State department.

The paid Russian trolls are still pushing the evil lesbian thing, btw, but the memes have shifted to paint her as a pedophile lesbian, because just “lesbian” isn’t enough to scare people anymore. And Trump keeps bringing up her husband’s infidelities because everything that happens in her life is somehow her fault (she stayed, so she’s a bad mother. If she’d left she’d be a disgrace who didn’t stand by her man).

No criminality in Clinton emails - FBI 

http://www.bbc.co.uk/news/election-us-2016-37892138

But in Sunday’s follow-up, Mr Comey wrote: “Since my letter, the FBI investigative team has been working around the clock to process and review a large volume of emails from a device obtained in connection with an unrelated criminal investigation.

“During that process, we reviewed all of the communications that were to or from Hillary Clinton while she was Secretary of State.

“Based on our review, we have not changed our conclusions that we expressed in July with respect to Secretary Clinton.”

Suppose you replace each 20 dollar bill with 20 1-dollar bills. You can make a 1:1 mapping between the dollar bills in your original collection of 1-dollar bills, and the ones in the collection that started as a collection of 20-dollar bills without ever running out of either: hence they have the same cardinality, which is to say, same amount of one-dollar bills.

(Cardinality is a word mathematicians use for ‘how big’ a set is. So ‘how many dollars do I have’ is asking for the cardinality of the set of dollars you have. If you have five dollars, your set has cardinality five.)

In other words, multiply aleph zero by any nonzero finite number and you still get aleph zero.

You have to do something much more dramatic to get to the next biggest infinity \(\aleph_1\), which if the continuum hypothesis is accepted is the cardinality of the real numbers. For example, assuming CH, you can get a set of cardinality \(\aleph_1\) by taking the power set of the set of natural numbers.

This means you can say stuff like, there are the same number of odd numbers, or in this case multiples of 20, as the number of all natural numbers (counting numbers 0 1 2 3…). In fact there is the same number of rational numbers (fractions) as natural numbers.

Another classical (i.e. not my invention) mindfuck with infinities goes like this: Two kingdoms, Infiniftyville and Foreverland are running low on reserves and so they decide to start saving money by stacking two coins on top of each other, every day, in their infinitely tall bank (which, honestly, is probably the reason why they’re having these financial problems to begin with, c’mon guys get your shit together.)

However, at the same time, a pair of thieves in the two towns realize what that infinitely tall bank is for and so they start going into their respective banks and stealing one coin from the stacks every day. But their methods are different: the Infinityville thief takes from the bottom of the stack, and the Foreverland thief takes from the top. 

They argue incessantly about which method is better, but both of them fair off equally well, of course: after doing this for an infinite number of days, they are both infinitely wealthy. However, the kingdoms do not share the same fate: one goes on to build yet another infinitely tall bank (an unpopular move, shockingly), but the other declares bankruptcy!

Which thief succeeded in destroying the monarchy? Answer/explanation below the cut.