On Wolfram Alpha, I see continued fractions being listed in the results. Although I understand continued fractions, and how they can be used for approximations, what is a better approximation than a decimal representation of the number?
For example,
31.999999999664
tells me so much more than
So, I do not get why continued fractions are important. Are they supposed to tell me something which I am missing all these days?
repeated fractions for prime square roots
You can give the exact value of the root of a non prime number. It is algorithmic. √17 = √16+1 = 4+1/8 with the denominator 8 having infinite fractions added to it. Squaring this number equals exactly 17. The square root of 18 is 4+2/8 with the 2/8 being added to every 8 in the denominator.
We took √18 to equal √16+2 because 16 is the nearest prime. The +2 becomes 2 over 8 repeated to infinity. We have to add 2/8 because we need a number bigger than 4 because 18 is bigger than the perfect square we used.
It works the other way too. The √3 is ≈ 1.73205. We can't use a perfect square less than 3 because non exist, they are all prime numbers. So we: √3 = √4-1 = 2-1/4 with the 1/4 subtracted to the denominator to infinity. We subtracted this time instead of adding because we took a perfect square of 4 that is larger than our targeted power of 3. 2-1/4...= √3 ≈ 1.73205.
Most people use calculators to find non prime numbers. People who MAKE calculators or software need to know number theory to program the logic. This isn't commonly taught because it is used for encryption of government and financial data. It is related to the study of large prime numbers and finding their roots and using it for encryption, programming logic for calculating software and so on. To make software for numerical analysis you need to understand what logic to program it with. This is a cornerstone of that logic.
The fraction is plus or minus the number of units away from the target prime over twice the root of the perfect square you used. 8 was double 4, and 1/x, and 2/x for 17 and 18 respectfully. Had to add the fractions to make a number slightly larger than 4^2. 4 was double 2 and 1/x. Had to minus the fraction since we had to make a number slightly less than 2^2.
There are many applications but mostly related to sensitive work.