Hello, If you're referring to the manipulation at around 3:07, we're starting with 1/[ 2 + (1/x) ]. The numerator is left alone (it's only the denominator we're manipulating here). In the denominator we have: 2 + (1/x) = (2x + 1)/x There are a couple of ways to see this. In general, if we have two fractions (a/b) and (c/d), we can add them like this: (a/b) + (c/d) = [ (a x d) + (b x c) ] / (b x d) In this case 2 is the same as 2/1, so we have: (2/1) + (1/x) = (2x + 1)/x Alternatively we could multiply the numerator and denominator of '2' by 'x', giving us: 2 + (1/x) = (2x)/x + (1/x) = (2x + 1)/x Hope that helps!
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Loving this stuff. Thanks for posting it!
Amazing Help! Thanks!
A rigorous definition of the continued fracion [a(0); a(1), a(2), ...] is to let f(m, m) = a(m), and f(m, n) = a(n) + 1/f(m, n + 1)). Then, [a(0); a(1), a(2), ...] := lim f(m, 0) (m -> ∞). To see this concretely, notice that f(0, 0) = a(0). f(1, 0) = a(0) + 1/f(1, 1) = a(0) + 1/a(1). f(2, 0) = a(0) + 1/f(2, 1) = a(0) + 1/(a(1) + 1/f(2, 2)) = a(0) + 1/(a(1) + 1/a(2)). Etc. In fact, f(m, 0) = [a(0); a(1), ..., a(m)].
Infinite continued fractions? More like "Interesting information that saves us"...from boredom! 👍
I know I’m missing some basic algebra (I’ve checked the equality and obviously it works) but how is 1/2+1/x = 2x+1/x?
Cancel that. Duh it’s just flipping the flippin’ fraction lol. I need to listen to what’s he’s saying lol.
Hello,
If you're referring to the manipulation at around 3:07, we're starting with 1/[ 2 + (1/x) ]. The numerator is left alone (it's only the denominator we're manipulating here). In the denominator we have:
2 + (1/x) = (2x + 1)/x
There are a couple of ways to see this. In general, if we have two fractions (a/b) and (c/d), we can add them like this:
(a/b) + (c/d) = [ (a x d) + (b x c) ] / (b x d)
In this case 2 is the same as 2/1, so we have:
(2/1) + (1/x) = (2x + 1)/x
Alternatively we could multiply the numerator and denominator of '2' by 'x', giving us:
2 + (1/x) = (2x)/x + (1/x) = (2x + 1)/x
Hope that helps!
No it’s not working for me, it’s not just flipping the fraction where does the 2x come from? This is at 3mins 46secs
Got it. It’s seven year old maths stuff - how embarrassing!
Why is the circle the same thing as the whole of the x-term?
because the terms of the nested fraction is equal to x ; its given [1,2] = x