h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

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Researcher 205946 Raymond the Curmudgeon

Welcome!

i have this problem in mind: if n=a(a+4)where a is an integer, then sqrt(n)=a+a/(a+a/a+...))). bit, i have found a different representation for this nuber n, and i have a book "elementary number theory" by burton, and it also has a different representation. isnt it a more convenient way to express continued fraction? by the way i am an amateur and i do math only in my spare time... thanks!!

combattant pour liberte

Can you tell me what 0.4, 1/3 or 1.74 are in binary?
I know 0.1 in binary = 1/10 in binary = a half
and 0.01bin = a quarter

But I have no idea about the first three

that is apart from these fractions, ahem, 100/1010, 1/11 and 1+(1001010/1100100) [i.e. 4/10, 1/3 and 1+(71/100) in binary]

what exactly does the repeated fraction sqrt(n)=a+a/(a+a/a+...))) .mean? Is it:

a+a
--------------------
sqrt(n) = (a+a)/(a+a)/(a+a)...

and isn't that just a+a (or did I misunderstand completely?


Also I'm interested in how you turn n = a(a+4) [which is the same as n= aa+4a] into that.

You can spell it encyclopaedia or encyclopedia--even in England both are correct. You can even put in good, old fashioned æ (ae liguture, or æ to you HTML writers) if you really want to!

Please sign me up, it'll ne fun (unless you have already). I'm gonna do Maths for A-level.

"Watch as I make this remainder disapear!"
(Maths wizard at Matin Prince's birthday party, just before isa points out that he's just divinding wrong)

.4 in binary....

.4 = 4/10 = 2/5 = 10/101 (BIN)

So, just set up the long division - 10 divided by 101 - and see what happens...

(doing long division in bases other than decimal is a good way to ensure that you *really* know how it works)

.4 (DEC) = .0110 0110 0110 (rep) (BIN)

Thanks. I tried doing that once, but I think I got it wrong. For some reason my computer's calculater can't be bothered to write binary or hexadecimal fractions/decimals.

I forgot the font isn't monospace. What I meant is:

Does it mean

sqrt(n) = (a+a)/[(a+a)/(a+a)/(a+a)...]

or have I completely misunderstood?

The most succint way to represent the given continued fraction is this:

if n = a(a+4)

sqrt(n) = X

where X = a+ (a/X)

...so the fraction contains copies of itself, like a fractal. What's cool is that you can then solve the second equation above for X.


Hmm.... doing that, I don't get that X=sqrt(n), but rather

X = (a + sqrt(n)) / 2

...so sqrt(n) = 2X - a = a + (2a / X). That actually works, if you calculate it.

(you do get two solutions for X, but the one with a minus sign is extraneous, 'cause it's negative, which makes no sense.)



Did I interpret the continued fraction correctly? That's what I've always understood continued fractions to represent...


2a
sqrt(n) = a + ---------------------------------
a
a + -------------------------
a
a + ---------
. . . . . . . . .


(a bit of patience, a lot of previews, and liberal use of the < space > smiley)

Can't you just use the [a_o; a_1, a_2....] thing? Isn't that the point of having it, so that you don't have to mess around with what the dirty thing actually looks like?

Please make me (combattant pour liberte) a member.

Oh you have. Thanks. You won't regret it.
I have memorised pi to FIVE WHOLE DP!
(3.14159)

'S more than I have.

On a site called http://www.foad.com/ they have pi to 400 million digits. However the page won't load sometimes because it's too big! (It won't load on my school network but it does on my computer and it is seriously big!)

:==R9==:

I think I can contribute to this bit and get something out. Give me time to see what's here already before I give my

Pimms