Rounding numbers

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.

Rounding

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 adding #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#

Context

In practice, there are contexts in which these rules do not apply. Suppose we do a calculation to determine how many parking spots we can fit in a certain area, and we find the answer #180.9#. According to the above rules, we would round this to #181# parking spots. However, regardless of how large the remaining part of the last parking spot is, the parking spot will not fit in this area. Therefore, we should round down in this case.

Therefore, it is important to consider the context when rounding decimal numbers. If the context gives no reason to round differently, we will use the rule above.

Negative numbers

Rounding of negative numbers works the same way. We look at the value of the decimal place after the digit we have to round.

When the value of the next decimal is:

\[\begin{cases}\phantom{x} \lt 5 & \phantom{x} \text{the digit we are rounding remains the same} \\\phantom{x} \geq5 & \phantom{x} \text{we add }1 \text{ to the digit we are rounding}\end{cases}\]

This means that when rounding a negative decimal number down, the number increases; when rounding a negative decimal number up, the number decreases. Yet, we continue to use these words.

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#

Round #78.6016# to the nearest integer.

#79#

To decide whether we round #78.6016# up or down, we look at the value of the first decimal place. In this case, this is #6#. Because this is greater than or equal to #5#, we round up. This means that we add #1# to the ones.
The answer is #79#.

New example