Who came up with the first accurate method for determining the square root of a number?

Question

Believe it or not, the method incorporates guessing the answer yet in only 3 steps it is > 90% accurate no matter what number you guess if you use the resulting answer for the next step. In 6 steps it is > 99.99 % accurate (using worst case). Hint: it is 2000 years old.

Answer 1

The Babylonians had an accurate method 3900 years ago, but there was no guarantee it would take only 3 steps unless you had a fairly accurate first guess.

For example, if your first guess for the square root of 2 was 32565, it would take 61 guesses to get better than 90% accuracy (that's 30 steps). That's really stretching the limits, I know, but it does make a difference if you have to write a computer program to do this - you have to figure out a way to make sure your program's first guess is at least reasonably close. For example, I might decide the computer's first guess for the square root of 2 should be 2. It's obviously a wrong guess, but it's close enough to save processing steps (this takes 2 steps in a very liberal sense, but the first step has 3 sub-steps and the rest 2 sub-steps; a better option would be to have your computer use one half the number you're trying to find the square root of). You have to consider this for any computerized iterative method, so it's not something peculiar to the Babylonian method.

Whatever your first guess is, if it's wrong, divide by two and try again. If you're still not there, average your first and second guess and try again.

Still not as accurate as you want to be? Try whatever number you're trying to find the square root of divided by your third guess.

Not there? Average your third and fourth guess.

Still not there? Repeat the last two steps until you finally reach the desired level of accuracy.

Answer 2

That would be Hero (aka Heron) of Alexandria. He was a Greek engineer extrodinaire in the 1st century B.C. Besides developing this formula, he invented the first steam powered device, the first coin operated machine, the first "water into wine" jug, among many other things. Search for "Hero of Alexandria" on the net for more info.

His method of determining square root was as follows:
Pick a number to determine the square root of,
Guess the answer
Divide the number by the guess
Add the guess to the answer
Divide the total by 2

Use the resulting number as the guess for a second interation (or 2nd step) of the above

Repeat for a third time using resulting answer of 2nd step.

Repeat until answer repeats itself.

I'm not sure if it will end up 90% accurate if the first guess is astronomically erroneous as in the above babylonian example of the square root of 2 being 3209348 but for most guesses that are less than the number you are starting with it will be fairly accurate. If you start with the first guess being 1/2 of the starting number it works really well. Example:

Pick a number: 5
Guess the answer: 2.5 (1/2 of 5)
Divide the number by the guess: 2
Add the guess to the answer: 4.5
Divide Total by 2: 2.25

Using 2.25 as second step guess: 2.23611111

Using 2.3611111 as third step guess: 2.236067978

Last step using 2.36067978 as guess: 2.2360679774997900

Actual square root: 2.2360679774997900 Dead match.

Neat stuff. Worth the search to check this guy out!

Answer 3

I believe it was Simon Stevin from the Netherlands in the Golden Age

Answer 4

This space is to answer questions you have not to look for them. What would be the purpose to answer a question that you already know?