Infinite Continued Fractions, simple or not?

What Are Continued Fractions?
Continued Fractions,
Write sqrt(2) as a continued fraction, infinite simple continued fraction

Check out my site & social media

Thank you for supporting! You're awesome and I know it!
blackpenredpen | 曹老師

[danielpeter]

Its just the opposite of normal mathematics, instead of making equation simpler we just make it more complicated.

[vipulshukla]

7:34 That correctional 2️⃣ on the left side! ❤️

[alkankondo]

Ah, yes - continued fractions: the subject of my master's thesis, which I recently completed! I learned so much about simple continued fractions and their properties, and this video is a great introduction to the subject!

[BigDBrian]

when you take 2 = 1 + 2/2 you can also substitute for the 2 in the numerator, creating a fraction tower diagonally upwards.

No idea about the uses for this
, just seems cool

[CornishMiner]

The golden ratio is very elegant as a continued fraction.

[dammuraja]

Student from india. Lovely explaination sir, enjoyed.😄

[pauselab]

proving that they must converge at a fixed point is insanely difficult. made me learn banach's fixed point theorem.

[jumpman]

This was fun.
Especially since I about a week ago saw Virgin Rock's reaction video to Tool's song "Lateralis", whose musical structure is based on the Fibonacci sequence.
What I learned from that video is that the ratio between two consecutive Fibonacci numbers approximates the golden ratio and the further down the sequence we go the better the approximation will be.

What I managed to figure out is that the "Fibonacci ratio" can be written as the continued fraction 1 + 1/(1 + 1/(1 + ...
but I couldn't satisfactorily show that this continued fraction is equal to the golden ratio, i.e., (1 + √5)/2.


But then this video popped up in my feed and it painfully dawned on me that all I needed to do was to write (1 + √5)/2 as a continued fraction.

[axbs]

alright heres my work to solve for that continued fraction:

1+(1/(2+1/(2+....)

take out that initial (1+) term for now

set 1/(2+1/(2+.... = x

replace the repeating terms with x to get: x = 1/(2+x)

multiply both sides by (2+x) to get x^2 + 2x = 1

subtract by 1 on both sides to get a quadratic; x^2 + 2x - 1 = 0

quadratic formula: x = (-2 +/- sqrt(4+4))/2

simplfiy to get -1 +/- sqrt(8)/2

add that (+1) term back in to cancel out that (-1) term

x = sqrt(8)/2 = 2sqrt(2)/2 = *sqrt(2)*

[Bidisha_Nath]

Really useful... Thanks... From. Assam, India

[arnavtripathi]

I did it another way by adding 1 on both sides of eqn then we get x+1 = 2 +
If we write it like x+1 = 2+ 1/(x+1) ... solving for x we'll get x=±√2 and we can also conclude that the result will be positive since all stuff that's being added is positive.

[jagrutivispute]

Thank you so much before I wasn't knowing how to turn a fraction into the continuous fraction form but after watching video now I come to know thank you again

[hossamabdulsalam]

Sir,
Please solve this problem.
3^(cot(x))^2 = 9*sin(x)*cos(x)
Find the values of x.
Many time I wrote this problem 🙏🙏

[Gokuyen]

seems very similar to the thing you see when first introduced to series, where you add together the infinite sum of a repeated decimal or something of that sort :D

[borisburd]

i dont know why but the continued fraction of sqrt of 2 made me laugh haha its so elegant

[adamhrankowski]

How about changing the numerator 2's as well?

[jacoboribilik]

Great math channel, im so glad to have come across it.

[chriswebster]

In the past, apparently, this man proved that the square root of 2 is irrational. That's how I know he's very wise.

[yuvalbe]

I read more about continued fractions, especially infinite contined fractions. But I didn't understand how to find the value of a given infinite continued fraction with known sequence. For example: [1, 2, 3, 4...] or [1, 4, 9, 16, 25, 36...]. If there is a known sequence does it mean the value is algebric number? I understood that because "pi" is Transcendental number there is not a pattern for the continued fraction.

[ImKurono]

Before class started my Calc 3 teacher was talking about this on Wednesday