5.2 The Metric System and Scientific Notation

Base UnitsThere is a 'base' unit for each measurement in the metric system, to which we will add prefixes to show how big or small our measurement is:

Base units:

Distance: meter (m)

Mass: gram (g)

Time: second (s)

Volume*: liter/litre (L) spelling depends on how British you are

* Volume could be done in distance cubed (m3), but a cubic meter is very large, so it's typical to use liters instead. This will be true for some other units along the way.

PrefixesIf you want units bigger or smaller than the base units (measuring ants in meters would be a pain), you need to add a prefix:

Bigger Smaller kilo (k-) 103 milli (m-) 10-3

Mega (M-) 106 micro (μ-) 10-6

Giga (G-) 109 nano (n-) 10-9

There are ones for essentially every power of ten (centi=10-2), but only the multiples of three are commonly used. You must know kilo and milli.

Putting Them TogetherTHIS IS THE PART STUDENTS TEND TO SCREW UP ALL YEAR LONG. PAY ATTENTION!

1 kilometer is bigger than a meter. How much bigger? 103 times bigger. Therefore, there are 1000 of the small ones in one of the big one:

1 km = 1000 m 1 kg = 1000 g ← GOOD

NOT1000 km = 1 m 1000 kg = 1 g ←

WRONG!

Stop to think when you do these. One km is a pretty long way. One meter is not. If the WRONG one were true, 5k races would end with a small twitch forward.

Putting Them TogetherSimilarly, one millimeter is one thousandth of a meter (10-3), so there a thousand of them in a meter:

1000 mm = 1 m or 1 mm = 0.001 m

If I want to go through multiple prefixes, I can just do it stepwise:

1 km = 1000 m = 1,000,000 mm (a thousand thousand)

Or

1 mm = 0.001 m = 0.000001 km (one thousandth of one thousandth)

Scientific NotationSome numbers are just so big or so small that there is no common prefix for them, so we use scientific notation instead, breaking the number into two parts:

1. value 2. size (aka power of ten)

For example, 320 grams could be written as:

32 x 101 g or 3.2 x 102 g

Both of which equal 320 if multiplied out. Some places will tell you that only the second is 'correct' (only one number before the decimal point), but people do the first all the time.

Now, it's not very useful for 320 g, but it is useful for...

Scientific Notation30000000000000000000000

(approximate number of stars in the universe)

Or

0.0000000000002(number of seconds it takes the pigment in your eye to react to light)

These are much easier to work with as:

3 X 1022 and 2 x 10-13

Scientific NotationA dire warning about calculators:

If you put scientific notation into your calculator as “3*10^22”, you will sometimes run into problems where it doesn't do the math you want it to.

Instead, look for the little button that says either “EE” or “EXP”, and use it to enter it as “3E23”, which the calculator understands to be scientific notation

SummaryIf I have a measurement like 30000 m, I have two ways to alter it to use a smaller number:

* Change the unit by using a prefix (30 km)* Use scientific notation (3x104 m)

Either way, the value gets smaller (becomes 30 or 3), so the size gets bigger (kilometers, or x104).

Value

Size

Summary

Likewise, 0.004 grams could become:* 4 mg* 4x10-3 g

(value got bigger, so unit/size became smaller)

Value

Size