**Notes on rounding and rounding error with fractions.**

**Notes on binary and decimal representation of numbers.**

**Notes on rounding to significant digits.**

The link to rounding to significant digits deals with rounding a number great than 1. For example 2^16 = 65,536.

65,536 rounded to one significant digit = 70,000

65,536 rounded to two significant digits = 66,000

65,536 rounded to three significant digits = 65,500

Let's consider rounding a number less than 1.

2^(-16) = 0.000015258789...

If we were asked to round this to the nearest thousandth, we would get 0.000. It's never a good idea to round a number that isn't zero to zero. Doing this means we aren't thinking at the right scale. Rounding to significant digits ensures we will never round a non-zero number to zero.

0.000015258789... rounded to one significant digit = 0.00002

0.000015258789... rounded to two significant digits = 0.000015

0.000015258789... rounded to three significant digits = 0.0000153

In most cases except scientific papers, rounding to three significant digits is considered sufficient.

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