The "miraculous" in math

[wipeout]

Kinda like the word "infinitely". You hear it used carelessly all the time, as in "Eating a piece of apple pie is infinitely better than having your eye gouged out with a rusty spoon." If that's the case, then nothing is better than apple pie, not even apple pie with ice cream on top, or apple pie with ice cream on top and winning the lottery and having your worst enemy drop dead of envy.

Betcha ten bucks that the next time you hear someone say "infinitely", it will be misused.

It is a lot like that, with the term "infinitesimal" even being misused in science and mathematics books I've seen, but it works out okay as an intuitive idea.

Richard Feynman in his famous Feynman Lectures on Physics also uses the idea incorrectly but in that intuitively sense when introducing calculus. I'm not sure if that is intentional or not, but I tend to think it's the way he thought of it too as he usually mentions if something he says is not strictly correct if it's to get an idea across more easily.

 

[homer]

There are some nice ideas in topology .
ie, a cup is a torus but a glass (cup without an handle ) is a sphere .
Surfaces such as the Klein bottle can't exist in the real world but the math is OK .
In non-euclidean geometry you can have a triangle whose angles add up to zero .
An object with a finite area but an infinite length can be seen if you use ' Fractint ' to generate a Koch1 curve . The real thing could never be drawn in the real world since it would require an infinite number of terms , but this shows you what it would look like .
Actually the mandelbrot set itself is finite in area but the perimeter is infinitely long . Some very clever person has proved this I believe .

 

[homer]

That thing you mentioned about calculus and things getting infinitely small caused no end of trouble in mathematics . Eventually they came up with the idea of taking it to the limit since as you say there can never be a smallest .
Something similar occurs if you take the inverse of a number . As the numbers get bigger and bigger so the inverse approaches zero .

 

[Art Vandelay]

I think that "counterintuitive" would be a better term than "miraculous". It's not that these are things that can't happen, but that one might think can't happen. People, especially ones who haven't taken upper division math classes (said classes have a tendency to make people realize their intuition isn't as infallible as they thought it was), often think that because something "makes sense' or "is obvious", it must be true. One of the cardinal rules of mathematics is "Don't say it's true until it's proven". It's weird how people can be absolutely insistent that something is true, yet when pressed are unable to give a proof. On what are they basing their belief, if not on a proof?

For instance, suppose you look at the test scores at a school, and the average score for boys is higher than that for girls. You look at the rest of the schools in the district, and this holds for them too. Is it true that districtwide, boys average higher than girls? Most people would say "yes", with no doubt in their mind, but no proof either. Can you present a proof?

BPSCG said:

If that's the case, then nothing is better than apple pie, not even apple pie with ice cream on top, or apple pie with ice cream on top and winning the lottery and having your worst enemy drop dead of envy.

I don't see how that follows.

wipeout:

The "dx" of calculus often gets called an infinitesimal but my understanding is that it just means "if you take this smaller and smaller" and it always refers to finite quantities.

Not quite. It tells you not only what's getting smaller and smaller, but also by what the integrand is being multiplied. Take, for instance, the integral of sin x d(x^3). Taking x^3 smaller and smaller makes x get smaller and smaller, but it's not the same as the integreal of sin x dx.

 

[wipeout]

Not quite. It tells you not only what's getting smaller and smaller, but also by what the integrand is being multiplied. Take, for instance, the integral of sin x d(x^3). Taking x^3 smaller and smaller makes x get smaller and smaller, but it's not the same as the integreal of sin x dx.

I just wanted to make the general point that dx as most people will know it is referring to real finite numbers and is not an infinitesimal quantity in a true technical sense, despite what a lot of people get taught in science and engineering.

Of course, it's possible for people to use calculus fairly well all their lives without really knowing what "dx" or other notation means, what limits really mean, and so on.

That was the case for me in high school but I always knew I didn't really understand, and that just isn't good enough for me.

 

[EHocking]

No no. If you take ac from the side without an ac in it, you get -ac, have another look.
Drooper got the right answer. a-b-c=0, so you have ax0=bx0, then you remove the zeros to get a=b, which obviously is where the dodgy working comes in.

I'm going to try to blame my inexperience with this forum's posting screen, but am going to have to admit to a brain fart instead . (can believe that I posted what I did!)

 

[Pantastic]

I'm going to try to blame my inexperience with this forum's posting screen, but am going to have to admit to a brain fart instead . (can believe that I posted what I did!)

All is forgiven

 

[drkitten]

For instance, suppose you look at the test scores at a school, and the average score for boys is higher than that for girls. You look at the rest of the schools in the district, and this holds for them too. Is it true that districtwide, boys average higher than girls? Most people would say "yes", with no doubt in their mind, but no proof either. Can you present a proof?

I can present a counterexample, is that good enough?

 

[BPSCG]

BPSCG said:


quote:
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If that's the case, then nothing is better than apple pie, not even apple pie with ice cream on top, or apple pie with ice cream on top and winning the lottery and having your worst enemy drop dead of envy.
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I don't see how that follows

I was pointing up the misuse of the word "infinitely". My dictionary says it means "Without bounds or limits; beyond or below assignable limits..." So when someone says something is "infinitely better" than something else, he's saying the gap between them could not be greater. And that's almost never true. A Mercedes-Benz may be a thousand times better than a Yugo, or a jillion skillion times better. But if you say it's "infinitely" better, what do you say when next year's improved M-B comes out?

Everyone knows what you mean when you talk like that, but it's still sloppy usage.

 

[Tez]

I love the things that change dramatically as you change dimension. For example, you should only leave your home when drunk if you're going to stagger around in 2-d (in 3-d your chance of getting back home is <1).

There are tons of such things, especially in the theory of phase transitions. A pure math example: In every dimension except 4, there is a unique smooth structure on Euclidean space - in 4-d there is an uncountable infinity of such structures.

 

[drkitten]

I was pointing up the misuse of the word "infinitely". My dictionary says it means "Without bounds or limits; beyond or below assignable limits..." So when someone says something is "infinitely better" than something else, he's saying the gap between them could not be greater. And that's almost never true. A Mercedes-Benz may be a thousand times better than a Yugo, or a jillion skillion times better. But if you say it's "infinitely" better, what do you say when next year's improved M-B comes out?

Your dictionary is mathematically incorrect. As a simple example, the ratio 1/0 evaluates to "infinity" (technically, the value is unbounded as the denominator tends to zero in the limit), which makes 1 "infinitely greater" than 0, but 2 is still greater than 1 (and also "infinitely greater" than 0).

Your mistake is in assuming that infinity is a unique number.

One of the fundamental findings of early 20th century set theory (alluded to earlier) is that infinity itself comes in several different, comparable sizes --- the number of integers is infinite, while the number of real numbers is also infinite, but also provably larger than the number of integers. The number of functions from real numbers to real numbers is also infinite (and larger yet).

However, when we're talking about comparisons between this year's and last year's model, the question is not one of mathematics, but one of assessment. Your definition reads, "beyond or below assignable limits." Depending upon how you choose to measure distances, you can easily have two points, between which the distance measures as "infinite," and then a third point some distance from the second. In a space that obeys the triangle inequality, this third point will also be "infinitely" distance from the first, but still distinct (and possibly even "further" in an intuitive sense) from/than the second.

Mathematically, this isn't a problem because under any sensible definitions, adding any number to an infinite quantitiy still results in an infinite quantity.

 

[DaveW]

As a simple example, the ratio 1/0 evaluates to "infinity" (technically, the value is unbounded as the denominator tends to zero in the limit), which makes 1 "infinitely greater" than 0, but 2 is still greater than 1 (and also "infinitely greater" than 0).

This is wrong, at least as I was tought the definition of x/0... it was always described to me as "undefined," which is different than infinity. More technically correct would be as the denominator approaches 0, the ratio approaches infinity.

I'm not a math major, though, so feel free to correct me.

 

[drkitten]

This is wrong, at least as I was tought the definition of x/0... it was always described to me as "undefined," which is different than infinity. More technically correct would be as the denominator approaches 0, the ratio approaches infinity.

I'm not a math major, though, so feel free to correct me.

Consider yourself corrected. The "more technically correct" statement you present above is, in fact, exactly what I wrote ("the value is unbounded as the denominator tends to zero in the limit").

 

[DaveW]

No, I was arguing against the statement that 1/0 = infinity, which is different than saying "as the denominator approaches zero."

 

[TwoShanks]

There was an excellent series of short (15 minute) shows on BBC Radio 4 called "Five Numbers" a while ago. The series gives a quick look at some interesting numbers and mathematics of them. The shows are archived online here:
Radio 4 Archives

There was also a sequel series called "Another Five Numbers".

Good fun, I'm glad we have public broadcasts of this kind of thing.

 

[Crossbow]

Some of you may have seen this before, but I like it:

a = b + c

multiply both sides by (a-b)

a(a-b) = (a-b)(b+c)
a^2 - ab = ab + ac - b^2 - bc

take ac from both sides

a^2 - ab - ac = ab - b^2 - bc

factor out a-b-c from both sides

a(a-b-c) = b(a-b-c)

take a-b-c from both sides

a = b


Eh? Substitute any numbers you like for a, b and c (making sure a=b+c of course) and it works. Prove that 3=2. The reason is dodgy working at some point, but can you spot where?

IF
a = b + c

THEN
a-b-c = 0

THEREFORE
I have problem with at the part where
a(a-b-c) = b(a-b-c)

OR
a(0) = b(0)
Which would mean that
0 = 0

And while it is true that 0 = 0, however that fact does not lead to a = b being true as well.

I hope this helps!

 

[homer]

It's just the old division by zero thing . You can't do it because if you allow that 1/0 =infinity (OK I know ) then infinity + 1 is the same as infinity + 2 etc . You can add as many as you like , it will still be infinity .
Eg :It is easy to show that the odd numbers can be matched 1 to 1 with the integers :1-1, 3-2, 5-3 , 7-4, 9-5 ,(2n-1)-n....Hence the set of odd numbers is the same 'size' as the set of Integers. Infinity again!
Now the real numbers can't be matched this way so the set of Real numbers is an infinity that is larger than the infinity of integers .
The guy who discovered all this went insane , so you've been warned .

 

[sorgoth]

For instance, suppose you look at the test scores at a school, and the average score for boys is higher than that for girls. You look at the rest of the schools in the district, and this holds for them too. Is it true that districtwide, boys average higher than girls? Most people would say "yes", with no doubt in their mind, but no proof either. Can you present a proof?

Uh...can you explain this please?

 

[Felice]

Originally posted by sorgoth

Originally posted by Art Vandelay
For instance, suppose you look at the test scores at a school, and the average score for boys is higher than that for girls. You look at the rest of the schools in the district, and this holds for them too. Is it true that districtwide, boys average higher than girls? Most people would say "yes", with no doubt in their mind, but no proof either. Can you present a proof?

Uh...can you explain this please?

I think the question is:

Given that in each school in a district the boys are averaging higher than the girls, is it true that boys average higher than girls overall?

In answer to Art, no I cannot give you a proof since one doesn't exist.

Suppose school A has two students, b1 and g1. b1 (a boy) scores 101, g1 (a girl) score- 100. Suppose school B has four students, b2, g2, g3, g4. b2 (a boy) scores 111, g2, g3 and g4 (all girls) all score 110. The statistics are:

School A: mean boys = 101 > mean girls = 100

School B: mean boys = 111 > mean girls = 110

District wide: mean boys = 106 < mean girls = 107.5

It is left as an exercise for the reader to extend this proof to N schools

 

[slimshady2357]

Uh...can you explain this please?

Do you mean, "How can it be that the boys district wide average is NOT larger than the girls district wide average?"?

If so, here is a quick example: (if not, well what did you mean? )

Say there are only 2 schools in the district, X and Y.

In school X, there are three boys, their average grade is 80%. There are 10 girls, their average grade is 79%

In school Y, there are 10 boys, their average is 65%. There are 3 girls, their average is 64%.

The total number of boys = 13, their average score = 68.46 %
The total number of girls = 13, their average score = 75.54 %

So each school has the boys with a higher average and yet the district wide averages show the girls as better.
Adam