# Error intervals

## In a nutshell

Rounding errors involves looking at a rounded measurement and finding the interval within which the measurement could lie. The range of numbers is called the error interval. An error interval can be found from rounded numbers or truncated numbers. A truncated number is a number where the digits after a certain decimal place have been removed.

## Rounded measurements

To find an error interval, look at how the number has been rounded, and think about what range of numbers would round to the rounded answer.

#### Procedure

1. | Half the unit that the number is rounded to. |

2. | Add this to the number to find the highest value for the rounded number and subtract this number to find the lowest value for the rounded number. |

3. | Write the answer as an error interval using inequality notation. |

The inequality is written as (lower bound) $\leq x \lt$ (upper bound).

**Example 1**

*A pen has a length, $x$, of $14cm$ to the nearest $1cm$. Find the error interval for the length of the pen.*

*Half *the unit $1cm$.

*$1 \div 2 = 0.5$*

*Add and subtract this number from $14cm$*

*$14-0.5 = 13.5cm \\14 + 0.5 = 14.5cm$*

**

*Write the answer using inequality notation*

*The error interval is* $\underline{13.5\le x < 14.5}$.

## Truncated measurements

Truncation is removing all digits past a certain decimal point. It is like always rounding down. It is possible to find the error interval for truncated measurements.

##### Example 2

*The mass of an object, $m$, is truncated to $5.6g$ correct to $1$ decimal place. Find the error interval for the mass of the object.*

*The object could have a mass ranging from $5.6g$ up to, but not including, $5.7g$. The error interval is given as*

*The mass of the object lies in the interval *$\underline{5.6 \le m \lt 5.7}$.