Math Problems

[mithrand1r]

Originally posted by Janny
Ok, but first tell me what's wrong with this.
('^2' is squared)

Let a = b
Therefore a^2 = ab
" a^2 - 2ab = - ab
" 2 a^2 - 2ab = a^2 - ab
" 2 (a^2 - ab) = a^2 - ab
" 2 = 1

Since a = b (from given)

as stated before by Radagast The Brown & Jonathan
(a^2-ab)=(ab-ab)=0

so
'2 (a^2 - ab) = a^2 - ab
becomes 2 (0) = (0)
which is 0 = 0 not 2 = 1

Or you would get an undefined value since division by zero (0) is undefined.

[Janny]

Dammit! 0.9 recurring was my next one! I'm disappointed both of you saw the division by zero immediately, you could have humoured me.
The 0.99 has always interested me, I'm only doing last year of GCSEs and we learn method not reason. It is logical, but the case of 0.9... was something I never got; is it something to do with varying degrees of infinity?

The 6 by 6 magic square: good grief! um... each line adds up to 111 (1+2+3+...36 = 666. 666/6) and it's number bonds to 37 right? I dunno, in short.

[mithrand1r]

Originally posted by Janny
Dammit! 0.9 recurring was my next one! I'm disappointed both of you saw the division by zero immediately, you could have humoured me.
The 0.99 has always interested me, I'm only doing last year of GCSEs and we learn method not reason. It is logical, but the case of 0.9... was something I never got; is it something to do with varying degrees of infinity?

The 6 by 6 magic square: good grief! um... each line adds up to 111 (1+2+3+...36 = 666. 666/6) and it's number bonds to 37 right? I dunno, in short.

Ironically it is easier to solve a 7x7 and 8x8 magic squares than a 6x6. (which is partially why I choose it. ) That and the grief it gave me for solving it.

[Janny]

Hmm. I can understand why it is easier to solve a 7 by 7, (you get a central square?) but why is an 8 x 8 easier than a 6 x 6?

[mithrand1r]

Originally posted by Jonathan

Someone tell me what's wrong with this :

Theorem: 0.9999... (to infinity) equals 1.

lim(m --> infinity) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1


Thus
''''''x = 0.9999...
''''10x = 9.9999...
10x - x = 9.9999... - 0.9999...
'''''9x = 9
''''''x = 1.

X is not exactly 1 (even if it is close enough to 1 for all practical purposes)
Therefore x does not equal 1.

At least that is my argument for now.

[mithrand1r]

Originally posted by Janny
Hmm. I can understand why it is easier to solve a 7 by 7, (you get a central square?) but why is an 8 x 8 easier than a 6 x 6?

8x8 magic square is based on a 4x4 magic square.

[Janny]

But I set out using 3 x 3's and couldn't do it? What's the difference? It's just bigger.

[Jonathan]

Originally posted by mithrand1r
X is not exactly 1 (even if it is close enough to 1 for all practical purposes)
Therefore x does not equal 1.

C'mon, you have to do better than that
I believe Janny was on the right track when he mentioned "varying degrees of infinity" or something.

[Janny]

Once more I remind you of my nil qualifications, but I believe it's the same principle as why 'infinity green bottles sitting on a wall' doesn't work as a maths song.
Ok, you're diplacing the 0.9... by one d.p.
I see it right at the end of infinity (here at 3 dp):
10x = 9.99
x = 0.999

Is 9x really 8.9999999(to infinity) and then 1?

But you can't draw a line on infinity and the problem arises because 10x reaches infinity one dp before 1x does?

[mithrand1r]

Originally posted by Janny
But I set out using 3 x 3's and couldn't do it? What's the difference? It's just bigger.

example of 4 x 4

Step one


01 02 03 04
05 06 07 08
09 10 11 12
13 14 15 16


Step two (switch the numbers as indicated)


01 15 14 04
12 06 07 09
08 10 11 05
13 03 02 16


Step three (add rows and colums to make sure everything totals out correctly)

an 8x8 square can be formed by a similar methodology.
(may take a little trial and error )

I think the reason it works is that 4 and 8 are both powers of 2. 3 is not a power of 2 and is not "related" to 8. (I cannot think of a good term to use here. )

3 is also an odd number as opposed to 4 or 8.

[Radagast The Brown]

About the magic square:

8 numbers were left - but didn't fit.
See? 1 _ _ _ _ 6
30 8 28 27 11 7
19 23 15 16 14 24
18 17 21 22 20 13
12 26 10 9 29 25
31 _ _ _ _ 36

It's lamost done - but probably wrong - because my last number - 2, 3, 4, 5, 32, 33, 34, and 35 doesn't fit. Was I close?

[mithrand1r]

Originally posted by Jonathan
Someone tell me what's wrong with this :

Theorem: 0.9999... (to infinity) equals 1.

lim(m --> infinity) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1

Thus'''''''x = 0.9999...
'''''''''10x = 9.9999...
'''''10x - x = 9.9999... - 0.9999...
''''''''''9x = 9
'''''''''''x = 1.

After further review the only thing, that I can find wrong with this may be the wording of the the text above "Thus x=09.9999...".
(I am assuming that 0.9999... means the same as a 9 that repeats forever.)

I do not see you using "x" anywhere in the limit.

[mithrand1r]

Originally posted by Radagast The Brown
About the magic square:

8 numbers were left - but didn't fit.
See?
01 __ __ __ __ 06
30 08 28 27 11 07
19 23 15 16 14 24
18 17 21 22 20 13
12 26 10 09 29 25
31 __ __ __ __ 36

It's lamost done - but probably wrong - because my last number - 2, 3, 4, 5, 32, 33, 34, and 35 doesn't fit. Was I close?

Here is a hint. (p.s. there may be more than one soln.)

row 1
01 04 13 30 31 32

row 6
05 33 24 07 06 36


Also, I am not sure if a 6x6 magic square can be solved in a similar fashion as a 4x4 square or a 8x8 square.

[Jonathan]

Here is another interesting thing. Imagine a trumpet in a coordinate system. You get the trumpet by drawing the curve y = f(x) = 1/x, where x>1.
Then you rotate the curve around the x axis, into the z axes, and you will get a 3-dimensional curved cone-like object: A trumpet.
The curve 1/x --- A picture of the trumpet --- A picture of the trumpet in 3D --- And another 3D pic

Now the trumpet will be infinitely long (because f(x) = 1/x will keep getting closer and closer to the x axis but never cross it). You can draw the conclusion that the outer area of the trumpet must be infinite as well. However, if you calculate its volume, it shows that the volume is not infinite!
In theory that means that if you want to paint the trumpet, you will have to use endless amounts of paint to colour the outside. But if you want to fill the inside with paint, you can actually do that since the trumpet’s volume is finite (with the volume pi ~ 3.14, if you do some calculations). How strange isn’t that? A paradox?

Mathematicians calls this paradoxal mathematical object “Gabriel’s trumpet” or “Gabriel’s horn”

Here is the evidence that the volume is indeed finite, even though the area is infinite. I just recently learnt how to rotate around axes (and will have a test on exactly that this week). Hopefully someone else understands it too. Since I don’t know how to write the integral thing symbol, I’ll a picture instead:



Now, this is not really a paradox. The explanation is that you can’t directly compare an infinite area with a finite volume. In practice, most of the length of the horn is inaccessible to paint, especially where the diameter of the horn is less than that of a paint molecule. So there
Well, at least I find this very interesting. I hope I’m not the only one.

Originally posted by Janny
Is 9x really 8.9999999(to infinity) and then 1?

But you can't draw a line on infinity and the problem arises because 10x reaches infinity one dp before 1x does?

Yes! That is at least the answer I came up with when I tried to solve the problem.

[Janny]

I don't really understand the working, but I can grasp that Gabriel's Trumpet concept. Cool!

I was right?! What? I'd like to see some algebraic proof it doesn't work tho... hell, I wouldn't understand but it would still be good...

[mithrand1r]

Can you think of a real world object with only one side?

[mithrand1r]

Do all triangles have angles equaling 180°?

[Janny]

Are those rhetorical questions?
A sphere only has one face.
I presumed that 180 d was a given with all triangles...

[mithrand1r]

Originally posted by Janny
Are those rhetorical questions?
A sphere only has one face.
I presumed that 180 d was a given with all triangles...

They are not retorical.

A sphere only has one face.

It still has 2 sides.

(think of a globe. You can put your hand on the outside. There is also the inside of a globe.[you cannot put your hand there unless you cut a hole in the globe, but it still is another side.])

I presumed that 180 d was a given with all triangles...

Not necessarily.

[Jonathan]

Originally posted by mithrand1r
Can you think of a real world object with only one side?

Are you thinking about a Klein bottle ? A bottle that where you cannot determine what is the inside and what is the outside?

[edit] Supposedly, such a bottle can only be created if you build it in four dimensions - or so I've read - so it's not really a "real world object". In these pictures, the Klein bottle might have determinable sides after all.

A colourful picture of a Klein bottle
a Klein bottle cut in two

Originally posted by mithrand1r
Do all triangles have angles equaling 180°?

A love triangle? The love between FF, Ruinel and SGH for instance is way hotter than 180°