The Mathematics Thread

[medhead]

Re: The totally off-topic thread

Don't think there is a misunderstanding just different attempts to demonstrate why 0 / 0 = 1 vs not undefined.

About the slope or gradient. As the slope of this function never changes then by calculating the slope/gradient by taking an infinitesimally small distance above and below any point of the function will generate the same result for all readings of the function other than X = 0, but only through accepting that 0 / 0 does not equal 1.

That is why you take the limits approaching a point from above and below. Taking the limits at a almost non-existent distance above and below the point achieves the same result for any point along the function other than when X=0.

So where do you stand? Should 0 / 0 = 1?

Even an infinitesimally small distance will result in a non-zero change in the vertical and horizontal axes. I'm a physicist, gradient is something that I use to determine physical quantities like g, etc.

Yes totally agree, so in calculating the gradient at the point of X = 0 and Y = 0 as given by the function Y = X, taking the limits approaching infinity gets you to 0 / 0.

Again, you do not calculate the gradient at a point. Y=X (or more correctly Y=m.X) is not a function for calculating the gradient. It the relationship between Y and X. This is all as per you post below. Gradient is deltaY/deltaX, the change in Y and X not the value of X and Y at a point. You're using y-y[SUB]1[/SUB] and x-x[SUB]1[/SUB]. No matter how small the limits, x will never equal x[SUB]1[/SUB] and y will never equal y[SUB]1[/SUB]. Those differences will always be non-zero.

In this case we'd end up with a really, really, small non-zero number divided by the same really, really small non-zero number. Any non-zero number divided by itself is 1, not 0/0.

Or it can be given by:

The gradient of the line is

Going from A to B the rise is y - y[SUB]1[/SUB], and the run is x - x[SUB]1[/SUB],

So,
gradient =

gradient = m, so
m =

m(x - x[SUB]1[/SUB]) = y - y[SUB]1[/SUB]
Let us look at this formula more closely.
It is telling us the equation of the line with gradient m which goes through the point (x[SUB]1[/SUB], y[SUB]1[/SUB]) is

I'm not even sure why you'd create that thing in the yellow box. It certainly does not tell us that is the equation for the line.

 

[Craigie]

I enjoy problem solving but I have been all like.... http://youtu.be/CMNry4PE93Y

 

[alieninperth]

While I can't say that I've understood much (if any) if the discussion over the last few pages, it has been fun. on a somewhat related note, I found these videos on youtube and found them to be quite interesting/informative. https://www.youtube.com/user/numberphile They may be of interest to others here.

 

[whatmeworry]

Re: The totally off-topic thread

So where do you stand? Should 0 / 0 = 1?

This is what Siri thinks.

[video=youtube;uEVeiiHZP68]https://www.youtube.com/watch?v=uEVeiiHZP68[/video]

 

[moa999]

Another fairly simple probability one is the number of people who play Blackjack but don't understand where the house edge is.

Tip: Its not the bonus you get when you get Blackjack - that is actually in the players favour

 

[JessicaTam]

Re: The totally off-topic thread

This is what Siri thinks.

[video=youtube;uEVeiiHZP68]https://www.youtube.com/watch?v=uEVeiiHZP68[/video]

Great answer!

 

[JessicaTam]

I have just learned that my high school maths teacher has died. Peter was passionate about mathematics, and while teaching us was also at uni furthering his own education. His passion certainly rubbed off on me and many other students - one of whom is literally now a rocket scientist in the USA.

The syllabus was something to be covered as the bare minimum. Instead of learning Pythagoras' theorem, we had to be able to prove it. I remember one matric* class where Peter showed us e ^ (i.pi) = -1. I think it took a double lesson to go through the proof. Amazing stuff.

When we struggled with a concept he would shake his head and say, "I don't understand why you don't understand". A saying I still use to this day.

Peter, thank you for sharing your love of all things numeric.

*Year 12 for those too young to remember the old SA system.

 

[JohnK]

Another fairly simple probability one is the number of people who play Blackjack but don't understand where the house edge is.

I don't play Blackjack that often but interested in the answer to the above. What is the house edge?

 

[moa999]

I don't play Blackjack that often but interested in the answer to the above. What is the house edge?

Given no-one else has touched this thread..

You bust, dealer takes your money, even if they subsequently bust themselves

Blackjack payout of 3:2 evens the odds a little bit, as do the dealer hit rules and the ability to split/double

 

[RAM]

Given no-one else has touched this thread..

You bust, dealer takes your money, even if they subsequently bust themselves

Blackjack payout of 3:2 evens the odds a little bit, as do the dealer hit rules and the ability to split/double

In Perth at the Casino in early 90s. Colleague of mine's idea of maths understanding was to reach for a calculator to add two 2 digit numbers.

He was freakishly lucky at times and that night was one of them AND he had no clue as to the probabilities of playing Black-jack

It was driving an Asian lady in her late 50s/early 60s wild.

He would keep asking for another card when he had 17 or 18 - regardless of whether the 5 packs had just been shuffled or was approaching the end of the run.

Must have been better than 2 out of 3 he was getting to 20 or 21. It drove the lady wild.

She would keep muttering, initially quite softly (I was playing right next to her) "What a fool."

He was 1st for the dealer, I was 5th and she was 6th. I am not sure if she was trying to count the cards and was just annoyed that he was using up the cards and running the decks down too quickly or whether she was just concentrating on the probabilities.

My colleague (also my boss) was having a whale of a time, liked the idea of getting lots of cards.

After what must have been just over two hours into the next day she got up, walked round to him, shouting quite loudly (turned heads on the floor); "Yull velly bad man, velly bad man, you never plays cards again." And stormed off.

My colleague was shocked, he'd been having so much fun (up around $1,000 from $20 stake) that he'd apparently never noticed what was happening at the table.

 

[anat0l]

..........

 

[RooFlyer]

I have just learned that my high school maths teacher has died. Peter was passionate about mathematics, and while teaching us was also at uni furthering his own education. His passion certainly rubbed off on me and many other students - one of whom is literally now a rocket scientist in the USA.

The syllabus was something to be covered as the bare minimum. Instead of learning Pythagoras' theorem, we had to be able to prove it. I remember one matric* class where Peter showed us e ^ (i.pi) = -1. I think it took a double lesson to go through the proof. Amazing stuff.

When we struggled with a concept he would shake his head and say, "I don't understand why you don't understand". A saying I still use to this day.

Peter, thank you for sharing your love of all things numeric.

*Year 12 for those too young to remember the old SA system.

Nice. Reminded me of one of my maths teachers at school. The teachers had been fairly uniformly 'normal' up till my matric ( ) years. Then the senior maths teacher went on long service leave and this apparition appeared in front of the first maths class of the year. About 22, over 6' tall, shoulder length dead straight black hair, modest handlebar moustache, granny glasses, black suit - he looked as though he'd just walked off the set of a Cheech and Chong film. In reality, he was a uni PhD student and no-one had seen anything like it in the class room before! Surprisingly (and a little disappointing for the class, at first) his teaching style was quite formal and methodical. We learned what 'Q.E.D.' meant .

Really got everyone's attention and I credit him for first getting me though Level III maths which in turn allowed me to do maths and physics at Uni etc etc.

 

[trixydoza]

A magnificent read:

http://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html?_r=0

The steady state of mathematical research is to be completely stuck.

 

[whatmeworry]

A VCE maths question.

 

[anat0l]

A VCE maths question.

Worked it out in 3 minutes, but unsure if I got the answer right, mind.

I actually got an answer using a fairly naive approach first, but after realising that my approach had too many assumptions, I had to do it another way.

What grade and (branch of maths) subject is this question for?

 

[whatmeworry]

Worked it out in 3 minutes, but unsure if I got the answer right, mind.

I actually got an answer using a fairly naive approach first, but after realising that my approach had too many assumptions, I had to do it another way.

What grade and (branch of maths) subject is this question for?

It was for year 12, further Maths. It is quite logical if you assume the sides are equal. Caused some students grief.


VCE maths problem: Can you solve the 50 cent question?

 

[TheRealTMA]

Worked it out in 3 minutes, but unsure if I got the answer right, mind.

I actually got an answer using a fairly naive approach first, but after realising that my approach had too many assumptions, I had to do it another way.

What grade and (branch of maths) subject is this question for?

THREE minutes! Jeez, what were you doing for the remaining 2:55? Trivial maths.

 

[david1]

It says in the question the sides are equal length! There is an assumption that all the angles are equal though.

It's simple geometry, disappointing that year 12 students would struggle with it!

 

[coles525]

Could the answer be B?

 

[anat0l]

It was for year 12, further Maths. It is quite logical if you assume the sides are equal. Caused some students grief.


VCE maths problem: Can you solve the 50 cent question?

I don't see why they are complaining. I've had more difficult questions than that in my senior exams. Perhaps this is a story that's a bit out of control or context. I mean, kids will find questions that they can't answer, then share them on Facebook. When I was their age, we usually shared them by word of mouth. Sometimes we'd have really tough questions that many people couldn't solve; the handful that could, well, naturally they scooped up the A or A+ grades.

But that is the purpose of exams - not only do they assess skill, but they should also be discriminating, clearly able to distinguish who should get an A and who should get a C. An exam might be marked way too tough if the percentage of those who get an A is very, very low, but I don't think it has happened too often yet. Sometimes the grade boundaries get relaxed anyway, sometimes on the insistence of a few problem parents (or those with backdoor connections and influence to members of the board of education).




Answer Spoiler!












At first, I assumed that if you joined the "base" of the triangle (the side opposite angle x) it would be an equilateral triangle, hence 60[sup]o[/sup]. However, I couldn't convince myself that that side would be the same length as the sides of the coin, even if the triangle is at least isosceles.

The longer method I did was similar to the second solution in the article. The internal angle of a dodecagon (12 sided regular polygon) is 150[sup]o[/sup]. (If you don't know how to find this, start with a corner of the polygon, draw a line joining that corner with another corner, then draw another line joining the first corner with another corner, and so on, until all corners are joined by a single line to a single corner. This forms a number of triangles - in the case of a 12-sided polygon, this makes 10 triangles. That means the sum of the angles in the polygon is 1800[sup]o[/sup], and divided by 12 gives 150[sup]o[/sup].)

Examining the point around x, we have two of the internal angles of the dodecagon, plus x, and all three of these must add up to 360[sup]o[/sup]. So the answer is 60[sup]o[/sup].