Decimals and Significant Figures
Decimal Numbers
Decimals are the numbers after the decimal separator. In Finland, a
comma "," is used as a decimal separator, but in many other countries
is also used a point ".". Because of this, many computer programs use
a decimal point as a decimal separator. In this document, I use a decimal comma
as the decimal separator.
Let's look at examples of decimal numbers. This number has one decimal
place, 0,3 m. The trailing m is a unit of meters. Also this next number has one
decimal 2,5 m. These numbers 5,67 m and 0,52 m have two decimals. These numbers
3,152 m and 0,543 m have three decimal places. A decimal number can also be
represented as a fraction, for example 0,4 is as fraction 4/10 .
However, we use decimal numbers in physics.
Picture Pixabay.
Picture Pixabay.
Significant Figures
Significant figures are all other than zeros. Significant figures are not zeros before the
decimal number and after an integer number. However, without additional
information, it is not known whether the zero is significant at the end of the
integer number.
Let's look at examples of significant numbers. This number has one significant figure, 0,05
m, so those zeros in front of the decimal number are not significant figures.
The trailing m is the unit meter. This next number has two significant figures 2,5
m. The number 5,67 m has three significant figures and the number 0,52 m has
two significant figures. The number 3,152 m has four significant figures and 0,543
m has three.
Trailing zeros in an integer may or may not be significant. For example, the number 1000 m,
this can have four, three, two or only one significant number. If the wording
were, "the length is 1000 m with accuracy of one meter", then we know
that there are 4 significant figures in the number. 1000,0 m would have five
significant figures.
Picture Pixabay.
Picture Pixabay.
Rounding Addition and Subtraction
Addition and subtraction can only be performed if the numbers are with
exactly the same units. After the calculation, the final result is rounded
according to the most imprecise initial number. Below is an example of rounding
the final result with addition and subtraction.
3,14 m + 1,4 m = 4,54 m » 4,5 m
Here we rounded down, because 4,5 m is closer than 4,6 m. If we were
equally far away, e.g. if the number was 2,25 m, then according to the rounding
rule used in Finland, we round up to the number 2,3 m.
Then why must the result be rounded in the first place. In this example,
the number 1,4 m is less accurate. It is therefore measured more imprecisely
than the number 3,14 m, perhaps with a more imprecise measuring device, but we
do not know from the number 1,4 m what the next decimal of this measure would
be. If the measurement were "exactly" 1,4 m, then it should be
written as 1,40 m. Now, this method of presentation tells us that we know the
number more precisely and then the final result of the addition could be presented
with two decimal places.
3,14 m + 1,40 m = 4,54 m
Picture Pixabay.
Picture Pixabay.
Rounding Multiplication and Division
After multiplication and division, the final result is rounded according to the least
accurate number, i.e. according to which of the numbers had the least
significant figures. The final result will have the same number or one more
significant figure than what was in the most imprecise one. Below is an example
of rounding the final result in multiplication and division.
Example 1.
2,65 m / 0,38 s ≈ 6,9736842 m/s ≈ 7,0 m/s
Rounded up here because 7,0 m/s is closer than 6,9 m/s. If we were at the same distance,
e.g. if the number was 6,45 m/s, then according to the rounding rule we use
here in Finland, we round up to the number 6,5 m/s. The final result may have
one more significant figures than the initial values, i.e. in the example
above, the answer may also be 6,97 m/s. The result must not be rounded to 7 m/s
because there would be only one significant figure and
thus the result would be wrong.
Example 2.
