__Fixed-Point__Numbers are represented by a fixed number of decimal places. Examples are

which are all expressed correct to 3 decimal places.__Floating-Point__Numbers have a fixed number of significant places. Examples are

which are all given as four significant figures. The position of the decimal point is determined by the powers of .__Significant Digit__This is any given digit, except possibly for 0 to the left of the first non-zero digit that fixes the position of the decimal point. The following numbers all have 4 significant figures.

In the last number the first three zeros simply fix the decimal point. Computers can only store a fixed nuber of significant figures, normally in the range 8 to 16 significant figures, so that numbers are NOT stored exactly. In decimal form

is truncated after a certain number of significant figures. This truncation process is called__rounding__.__Rounding__To round to decimal places (or significant figures) means that we must discard the plus decimals. The rules for rounding are- Rule (a) If the discarded number is less than
, so
that the first discarded digit is less than 5, leave the
decimal unchanged. This is rounding down.
- Rule (b) If the discarded number is greater than
,
so that the first discarded digit is greater than 5 add one to the
decimal. This is rounding up.
- Rule (c) If the discarded number is exactly then round to the nearest even decimal.

__Example 1. .3__- Round to 3, 2 and 1 decimals.

- Round to 3, 2 and 1 decimals.

__rounding errors__.- Rule (a) If the discarded number is less than
, so
that the first discarded digit is less than 5, leave the
decimal unchanged. This is rounding down.
__Algorithm Stability__An*algorithm*is a set of rules, describing the numerical method. This set of rules must include a*stopping criterion*. The algorithm is*stable*if small changes in the initial data only generates small changes in the final result. If the final result exhibits large changes, then the algorithm is*unstable*.__Truncation Errors__These errors occur, for example, from stopping a Taylor series after only a few terms.

can be truncated and only the first few terms used, so that

In this case the truncation error is the error term.__Absolute Error__To measure the importance of the error we introduce two different measures for the error. Firstly, the*absolute error*is defined in the following way. If is the exact value of a quantity and is the approximation, then the absolute error, , is

__Relative Error__The second measure of the error is the*relative error*is defined by