### Numbers: Ratios

### Rounding numbers

Decimal numbers may have many digits, sometimes even infinitely many. We are not always interested in all those digits. In that case, we can round the number. We have a rule for the rounding of numbers.

We can round a decimal number to a certain number of decimal places.

To determine whether we round up or round down, we look at the value of the next digit. For example if we want to round to two decimal places, we look at the value of the third decimal place.

When the value of the next decimal is:

\[\begin{cases}\phantom{x} \lt 5 & \phantom{x} \text{we should round down} \\\phantom{x} \geq5 & \phantom{x} \text{we should round up}\end{cases}\]

Here, rounding down means that the digit we are rounding remains the same. Rounding up means that we add #1# to the digit we are rounding. If a digit equals #9#, it becomes a #0# and the digit in front of it is increased by #1#.

To indicate that a number is rounded and not an exact number, we use the approximation sign (#\approx#) instead of the equal sign (#=#).

**Example**

#16.184598#

Rounding to

integers: #16#,

because #1 \lt 5#

#1# decimal place: #16.2#,

because #8 \geq 5#

#2# decimal places: #16.18#,

because #4 \lt 5#

#3# decimal places: #16.185#,

because #5 \geq 5#

#4# decimal places: #16.1846#,

because #9 \geq 5#

#5# decimal places: #16.18460#,

because #8 \geq 5#

To decide whether we round #69.7688# up or down, we look at the value of the first decimal place. In this case this is #7#. Because this is greater than or equal to #5#, we round up. This means that we add #1# to the ones.

The answer is #70#.

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