FHSST Physics/Units/Scientific Notation, Significant Figures, and Rounding

Scientific Notation, Significant Figures and Rounding

If you are only sure of say, both digits of a two-digit number, and put it in a formula and get a long series of numbers to the right of the decimal place, then those digits are probably not very accurate. This is the idea of significant figures.

Take 10 and divide by 3. If you are not sure that the number 10 is perfectly accurate, then you do not need to write down 3.333... and can get away with something like 3.3 or 3.33

(NOTE TO SELF: still to be written)

The accuracy of a measurement using significant figures is represented by the number of digits that it contains. A number is said to have the number of significant figures equal to the number of digits in the number not including leading 0s or trailing 0s unless there is a decimal point. The table below contains a list of numbers and how many significant digits each contains.

Table ?: Significant Digits

Number	Significant Digits
1000	1
1000.	4
10.0	3
010	2
232	3
23.2	3
$1\times {10^{3}}$	1
$1.00\times {10^{3}}$	3

As you may have noted, some numbers cannot be shown in proper significant figure notation without the use of scientific notation. For example, the number 1000 can only be shown to have 1 or 4 or more significant digits by the inclusion of a decimal point. However, by rewriting 1000 as $1.\times {10^{3}}$ any number of significant digits may be added by simply add additional 0s after the decimal point.

Sometimes you may be asked to determine the number of significant figures in a given number. There are three rules to determine what numerals are significant.

1. Leading zeros are never significant. Leading zeros are zeros that appear on the left end of the number.
2. All non-zero digits are significant. Trapped zeros (zeros between non-zero digits) are also significant.
3. Trailing zeros are never significant unless there is a decimal point. Trailing zeros are zeros that appear on the right end of the number.

E notation

Very large numbers such as the speed of light (the C part of Einstein's famous $E=MC^{2}$) are difficult to write accurately.

We could write 300,000,000 m/sec, $3x10^{8}$, 300 million meters per second or some such. There is a much better way!

We simply separate the number (coefficient) part 3 from its multiplier 00000000 base!

But be careful, there is an Elephant trap here, and it is that in scientific notation the number is always expressed as a decimal fraction with a maximum value of 1.0, (in this case =0.3) so the multiplier part is one bigger than you might expect! $3x10^{8}$ is the same as 0.3E9. (Because there are a total of nine digits after the decimal point)

A tiny dust particle might weigh as little as 0.000 000 000 678 kg.! This time we shift the decimal point 9 places to the right so the number (678) has a negative base so our weight is written as 0.678E-9 kg.

Now is that not a whole lot easier to write and understand? That is why many scientific calculators and most spread-sheets allow input and display in E notation format.

Commas and points

Most English speaking people use the comma [ , ] to separate thousands and the dot (ful-stop, or point) [ . ] for the decimal indicator. Europeans often use these the other way around. Many spread-sheets allow both, but it is just one more complexity you need to know about!