Physics I For Dummies. Steven Holzner
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Название: Physics I For Dummies

Автор: Steven Holzner

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119872245

isbn:

СКАЧАТЬ 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.

      

In scientific notation, you write a number as a decimal (with only one digit before the decimal point) multiplied by a power of ten. The power of ten (10 with an exponent) expresses the number of zeroes. To get the right power of ten for a vary large number, count all the places in front of the decimal point, from right to left, up to the place just to the right of the first digit (you don’t include the first digit because you leave it in front of the decimal point in the result).

      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):

      

If the number you’re working with is larger than ten, you have a positive exponent in scientific notation; if it’s smaller than one, you have a negative exponent. As you can see, handling super large or super small numbers with scientific notation is easier than writing them all out, which is why calculators come with this kind of functionality already built in.

      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:

      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

meters (
meters), or you could use the unit prefix
, which stands for micro:
. When you put
in front of any unit, it represents 10–6 times that unit.

      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

      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

when necessary. This section delves deeper into the topics of significant digits, precision, and accuracy.

      Knowing which digits are significant

      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.

      

The best way to write a number so that you leave no doubt about how many significant digits there are is to use scientific notation. For example, if you read a measurement of 1,000 meters, you don’t know if there are one, two, three, or four significant figures. But if it were written as
meters, you would know that there are two significant figures. If the measurement were written as
meters, then you would know that there are four significant figures.

      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, СКАЧАТЬ