Approximate
The Babylonian Method works well here. It makes use of the bisection method and the fact that if you divide a number by its square root, you get the square root.
You need to make an initial guess.
Since that means .
Average these endpoints to get is a good starting guess.
.
So that means .
Average these out and you get a new guess.
Follow this process until you reach your desired level of accuracy.
razemsoft21, it helps if you tell us the full details in the first post. Saying Approximate isn't enough.
Also, there are many methods of solving squareroots. If you don't know how to solve it, then maybe asking for someone to tell you a method of solving squareroots.
Doing research would help too: Computing Square roots - Wikipedia. Here, it shows you many methods of how to solve a square root of a number.
Hello, razemsoft21!
Where did this problem come from?
It's unlikely that you would be assigned this problem
, , without being taught any approximation methods.
Approximate
The very worst you could do is guess-and-adjust.
Since is between 1 and 2 . . .
We have the two decimal places.
Which is the "better" answer: 1.41 or 1.42 ?
1.41 gives an error of: .
1.42 gives an error of: .
Since 1.41 has the smaller error: .
refer to this page Square root of 2 - Wikipedia, the free encyclopedia
. . . An elementary approximation of
Let be your first approximation to
Substitute into: . .[1]
This produces , a better approximation.
For more accuracy, substitute into [1] . . . and so on.
Example: .
Let
We have: .
. . Check: .
Then: .
. . Check: .
Then: .
. . Check: .
Therefore: . .to 9 decimal places.
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Why does [1] provide better and better approximations?
We want an approximation of
Our first guess is
We have factored into two factors: .
If we are extremely lucky, the two factors will be equal.
. . (Think about it!)
Since it is unlikely that the two factors are equal,
. . a better approximation is the average of the two factors.
. . (The number exactly halfway between them.)
So we use: .
. . which simplfies to: . . . . . . ta-DAA!
we must appreciate the works of Robert Nemiroff he calculated to 1 million digits just check this out!
The Square Root of Two to 1 Million Digits