Brain Teaser No 1

then 10x = 9.9999...

i.e 10x - x =9.9999... - 0.9999...

i.e 9x = 9

i.e. x = 1

what !!!???

enjoy

ken

It seems to me from a consideration of the angles, that lines touch circles at tangents, and that the normal to a tangent is a radius, that in general for our circles,

Where R is the radius of the encompassing circle, r is the radius of the internally disposed circles, x is the radius of the small centered circle and A is 180/n degrees, where 'n' is the count of small circles,

r = RsinA/(1 + sinA)
x = R - 2r

So, given R = 5, as in the original question,

n x
2 0
3 0.359
4 0.858
5 1.298

etc.

Paul

enjoy

ken

But I enjoy the stuff being knocked about here, however trivial in real terms, it's good for the mind to come back to after 20 years, and, like bashing away at the Times crossword, helps stave off something....

So, thanks, now, where's me washboard?

Paul

10x will always have one less decimal place than x. However, your equations assume it has the same number.

For example, consider a finite number of places:
x = 0.999999
10x = 9.999990
10x - x = 9x = 8.999991
x = 0.999999

BAM

sorry i wasnt clear. those decimals are recurring.
so x=0.99999.... means recurring forever.

enjoy

ken

1 - 0.99999.... == 10 - 10 * 0.99999...

which I think is the same as saying 1 == 10, which as any fule kno is unlikely.

Paul

sorry i wasnt clear. those decimals are recurring.
so x=0.99999.... means recurring forever.

I think the point (sorry!) that bam is making is that NO MATTER HOW MANY decimals you go, 10*x will ALWAYS have ONE decimal less than x. Even when the number of decimals in x is INFINITE, the the number in 10*x will be Infinity minus one!

But I will let Bam speak for himself, and allow ken c to tell me i've completely missed the point......

Cheers

Don

the point is that the expression

x=0.99999.... is NOT a direct assignment of x to a constant, but is actually short hand for the following:

x = SumToInfinity(0.9, 0.09, 0.009, ....)

this is a GP with first term 0.9 and common ratio 0.1

using the sum of GP to infinite number of terms, you can check that the assigment is actually to the number 1.

so, in fact there is nothing wrong with the conclusion. in fact, the logic used is the part of the trick used to sum a GP.

enjoy

ken

enjoy

ken

for a given ladder length L, there are are 2 possible values of <d> -- that much is obvious from the equation d + 1/d = x (a quadratic in <d> ). as i mentioned earlier, these 2 values actually represent d and 1/d, (which is sort of like rotating the whole diagram 90 degrees.)

there are no other solutions -- so for any other values of <d>, while the ladder is touching the floor and the box, the top of the ladder will not touch the wall of naim audio, so no use.

as you slide this fixed ladder up and down while touching the box and the ground, the top of the ladder will trace a curve which appears to actually break thru the wall (alarm bells!!) at the lower point and emerge out at the higher point.

i sort of suspected this curve was a hyperbola, but i am not sure where that leads me...

sorry about the waffle...

any thoughts anyone... we are trying to generate EXTRA information to finesse the correct equations that we have already...

what a bummer of a problem !!!

enjoy

ken

3 * 0.33333... = 0.99999....
3 * 1/3 = 1.0

so 1.0 = 0.99999....

But I prefer my contradiction, because I don't like the idea of 0.99999... not being not quite 1.0.

Paul

Paul

Ladder problems is still bugging me!

The two real solutions of Don's equations are the distance of the bottom of the ladder to the box and the distance from the top of the box to the top of the ladder, one is the reciprocal of the other, still trying to see if this gives any clues!

cheers

Matthew

1/7 can do strange things to the mind!

yes, wierd isnt it?? is this number the secret of designing good hifi? or is this the key to enlightenment? religion?

its intereting that integer multiples of 1/7 exhibit this repeating pattern which immediately breaks down as soon as you do something like (3.5/7) which is clearly 0.5.

apparently there are many other curiosities centred around 1/7? which makes me wonder whats so special about this rational number.

enjoy

ken

This gets stranger still.

try 1/17!

and 2/17....

freaky

Matthew

I have played around with the same equations, and got nowhere either. Bearing in mind BAM's comment about using less conventional maths, I've tried a different approach.

Use a spreadsheet to compute values of L for different values of X (distance of ladder bottom from box).

Observe how quickly increments of L approach 1, for each 1 increase in X.

Observe also that derivatives of these increments (ie differences between differences) never become constant.

Using calculus, one would expect "squared" derivatives to become constant after 2 iterations,
"cubed" derivatives after 3 iterations and so on.

My memory of calculus is very limited, but isn't this a sign of something like the exponential function "e"??

I'm playing about with the numbers, and getting pretty graphs, but still unable to find an elusive formula.

Hope this helps.

STEVE D

quote:

---

my opinion on the error in the puzzle is that it implies that
10 * 0.999... = 9 + 0.999...
Which clearly isn't true.

---

But it is true.

1 - 0.99999... = 0

I thought my demonstration with 1/3 showed it without recourse to infinite sums, but it took me a while to see it.

Paul

quote:

---

Observe how quickly increments of L approach 1, for each 1 increase in X.

---

I think this is a consequence of the angle between ladder and the ground tending to 0.

Paul

1 does not equal 0.999...

I agree that 1/3 can be expressed as 0.333...
But I do not agree that 1 can be expressed as 0.999...

Hence, if you agree with this, then it follows that the multiplication of a number with infinitely recurring digit does not follow the same rules as for exact numbers (I'm struggling to recal the proper terms for these things). This would not surprise me.

So 1/3 x 3 = 1 not 0.999... and everyone can sleep at night again.

BAM
(surd fractions? Irrational numbers?)

0.9999... IS equal to 1.

this is because 0.9999... is not a constant like 0.9999, but a FORMULA that says, start with 0.9 and keep adding 1/10 of the orevious number to infinity. this recipe, leads to 1.

similarly 0.3333... leads to 1/3

10x0.9999... = 10

10x0.3333... = 10/3

etc...

enjoy

ken

in fact, this happens for all prime numbers that i have tried, up to 29, although sometimes it may take a while to pick up the repeating pattern

i suspect this is a well known property, but i have never come across some "theorem" that specifically talks about it.

anyone?

enjoy

ken

quote:

---

I agree that 1/3 can be expressed as 0.333...
But I do not agree that 1 can be expressed as 0.999...

---

If you multiply 0.33333... by 3 you never get a carry so it must be 0.99999..., but it's also 1.

If a series runs a + ak + ak^2 +..+ ak^n,

the sum to n terms is,

Sn = a + ak + ak^2 +..+ ak^n

and

kSn = ak + ak^2 + ak^3 +..+ ak^n+1

Subtracting those,

Sn(1-k) = a + ak^n+1

so

Sn = a(1 + k^n+1)/(1-k)

In the case of 0.9999..... a is 0.9 and k is 0.1, so when n is infinity k^n+1 tends to 0, so the sum is 0.9 * 1 / 0.9 which is 1....

This 0 level maths stuff is fun!

Paul