Ask Uncle Colin: Computing
Dear Uncle Colin,
If I didn’t have a calculator and wanted to know the decimal expansion of
, how would I be best to go about it?
Roots As Decimals - Irrational Constant At Length
Hi, RADICAL, and thanks for your message!
There are several options for finding
One is the ‘long division’ method that @colinthemathmo has explained in more detail than I would ever care to, so I’ll just send you there if you’re interested in that.
But there are two other methods I would consider.
The binomial expansion
If you have a reasonable guess for the value of
You can repeat this process as often as you like: with
You can either work with fractions and do a great big long division at the end, or you can say (for example) “15 is my estimate for
Continued fractions.
With a litte bit of surd work, you can show that:
But, you know that the
In fact, if you know that the
Again, you can use this to generate an approximation as close to the answer as you would like!
Hope that helps,
- Uncle Colin
* Updated 2019-02-06 to correct LaTeX in title.