Presenting Numbers

© Copyright 2004 Herbert J. Bernstein

When we perform an experiment we are likely to deal with numbers representing measured data. Clear comprehensible representations of numbers are important. In most cases, we base this presentation on decimal positional notation in which numbers are represented by strings of decimal digits (digits drawn from 0, 1, 2, 3, 4 5, 6, 7, 8 or 9) in which the digits are implicitly weighted by powers of 10. Thus 325.15 stands for 3 hundreds plus 2 tens plus 5 ones plus 1 tenth plus 5 hundredths.

It is important to be aware that even this commonly accepted notation has important variations. In parts of the world digits are grouped by commas and the decimal fraction is separated by a period (a decimal point). In other parts of the world digits are groups with periods and the deciaml fraction is separated by a comma, e.g.

instead of

There has also been disagreement of the proper names for some powers of ten. In the United States, 1,000,000,000 is called a billion, while in England, 1,000,000,000 is called a thousand million and 1,000,000,000,000 is called a billion. To avoid such confusion, there is an international standard set of names for the powers of 10. They are given as perfixes (e.g. as in gigabyte) from the following table:

Prefix Power of ten
yocto 10-24
zepto 10-21
atto 10-18
femto 10-15
pico 10-12
nano 10-9
micro 10-6
milli 10-3
centi 10-2
deci 10-1
or deka
hecto 102
kilo 103
mega 106
giga 109
tera 1012
peta 1015
exa 1018
zetta 1021
yotta 1024
However, the greatest clarify is achieve by scientific notation, in which a number is simply followed by a power of 10, as in
321.45 1028

When working with the results of experiments, we often have reason to belief that there is some uncertainty in that value, which can be expressed in may ways. For eaxmple, if we have the number 3.14159, but believe that we could have gotten a number ranging from 3.1385 through 3.1595, we could present that information as an interval

[ 3.1385, 3.1595 ]

with square brackets to include the endpoints, or

( 3.1385, 3.15950 ) for by the measured number and indicators of uncertainty
3.14159 (+.01791 -.00309) or by the center of the interval and half its width

An approximation to this notation is to simply say that we will retain only those digits that are reasonably certain, as in 3.14 for 3.149+/-.0105, but that makes looses a lot of information. A more precise, commonly used notation is to include the digits of the uncertainty in parentheses without the decimal point, so that 3.149+/-.0105 would be written as 3.1490(105), meaning the number is measured as 3.1490 with an uncertanty of 105 in the last three digits, or approximated by 3.14(1), meaning that the value is measured as 3.14 with an uncertainty of 1 in the last digit.