# Re: Binary128 - PI and E

```On May 11, 2011, at 22:09, Eric Postpischil wrote:

```
```Now, suppose x rounded to 35 decimal places is 2.000...00116. This tells you
x is between 2.000...00115,5 and 2.000...00116,5. Some of the numbers in that
interval are below 2.000...00115,55 and some are above it. So knowing x to 35
decimal places does not give you enough information to know which 128-bit
floating-point number is closest to x.

Some number theory can be used to find where the representable numbers most
closely approach decimal values, and that would tell you how many decimal
digits you need in the worst case. However, I have run out of time for
tonight.
```
```
It was definitely time to sleep rather than post. Those points are the
representable numbers. The actual decision points are midway between
representable numbers, since we want to know whether x is closer to one
representable number or another. However, the idea is the same: some of the
decision points are within the interval that funnels into a particular
35-decimal-place numeral, so the numeral alone is inconclusive about which
representable number is closest.

— edp (Eric Postpischil)
http://edp.org

... when life ceases to be a fraction and becomes an integer. — Harry Emerson
Fosdick

```