Start with a fraction close to √2, say 3/2. Thus a square of side 3 has area close to two squares of side 2. Line those up along a diagonal of a larger square. Fill in the lengths and argue that the area of the large square (side 7) must be close to the area of the two squares with sides 5.
1/2
Conclude that √2 is approximated by 7/5. Iterate. The next approximation you find is 17/12. Continue this and you’ll find a sequence that converges √2. This well-known sequence is what you can also find using Newton’s method, so it converges quadratically (interestingly fast).
2/2
Yeah, sorry, I was thinking about (3/2)^12 being close to 2^7 (which is related to the twelve-tone chromatic scale and the seven-tone diatonic scale).
I take back what I said: I don't know of any relevance of 17/12 being close to sqrt(2).
Comments
1/2
2/2
https://r-knott.surrey.ac.uk/fibonacci/CFintro.html#section6
I take back what I said: I don't know of any relevance of 17/12 being close to sqrt(2).
https://docs.google.com/document/d/1uWTn6CmmKJllhpZ0f7xXXTb_1dlZkp7H/edit?usp=drivesdk&ouid=117115613208898417545&rtpof=true&sd=true
It does happen to be one of the continued fraction approximations for √2:
1+1/(2+1/(2+1/2))=17/12.
1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169.
I quite like 99/70.
However completely get that I could be alone on this.
A square of area 288 is very close to a square of area 289.
A square of area 289 has side length 17
The square of area 288 has side length 12rt(2)
So rt(2)≈17/12
Here are some more - 41/29 is even better!
I was solving a problem, aimed at school students, last night and needed to know that a certain integer was very close to a multiple of sqrt(2).
Draw a line with 8 squares, find midpoint, draw a point 7 squares above it and join them up