often involve really big or really small numbers. Physics has a way of dealing with very large and very small numbers; to help reduce clutter and make them easier to digest, it uses scientific notation.
For example, say you’re dealing with the average distance between the sun and Pluto, which is about 5,890,000,000,000 meters. You have a lot of meters on your hands, accompanied by a lot of zeroes. You can write the distance between the sun and Pluto as follows:
The exponent is 12 because you count 12 places between the end of 5,890,000,000,000 (where a decimal would appear in the whole number) and the decimal’s new place after the 5.
Scientific notation also works for very small numbers, such as the one that follows, where the power of ten is negative. You count the number of places, moving left to right, from the decimal point to just after the first nonzero digit (again leaving the result with just one digit in front of the decimal):
Here’s a simple example: How does the number 1,000 look in scientific notation? You’d like to write 1,000 as 1.0 times ten to a power, but what is the power? You’d have to move the decimal point of 1.0 three places to the right to get 1,000, so the power is three:
USING UNIT PREFIXES
Scientists have come up with a handy notation that helps take care of variables that have very large or very small values in their standard units. Say you’re measuring the thickness of a human hair and find it to be 0.00002 meters thick. You could use scientific notation to write this as
A more familiar unit prefix is k, as in kilo, which represents 103 times the unit. For example, the kilometer, km, is 103 meters, which equals 1,000 meters. The following table shows other common unit prefixes that you may see.
| Unit Prefix | Exponent |
|---|---|
| mega (M) | 106 |
| kilo (k) | 103 |
| centi (c) | 10–2 |
| milli (m) | 10–3 |
micro ( |
10–6 |
| nano (n) | 10–9 |
| pico (p) | 10–12 |
Checking the Accuracy and Precision of Measurements
Accuracy and precision are important when making (and analyzing) measurements in physics. You can’t imply that your measurement is more precise than you know it to be by adding too many significant digits, and you have to account for the possibility of error in your measurement system by adding a
Knowing which digits are significant
This section is all about how to properly account for the known precision of the measurements and carry that through the calculations, how to represent numbers in a way that is consistent with their known precision, and what to do with calculations that involve measurements with different levels of precision.
Finding the number of significant digits
In a measurement, significant digits (or significant figures) are those that were actually measured. Say you measure a distance with your ruler, which has millimeter markings. You can get a measurement of 10.42 centimeters, which has four significant digits (you estimate the distance between markings to get the last digit). But if you have a very precise micrometer gauge, then you can measure the distance to within one-hundredth of that, so you may measure the same thing to be 10.4213 centimeters, which has six significant digits.
By convention, zeroes that simply fill out values down to (or up to) the decimal point aren’t considered significant. When you see a number given as 3,600, you know that the 3 and 6 are included because they’re significant. However, knowing which, if any, of the zeros are significant can be tricky.
Rounding answers to the correct number of digits
When you do calculations, you often need to round your answer to the correct number of significant digits. If you include any more digits, you claim a precision that you don’t really have and haven’t measured.
For example,
