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Metrics

Metric Unit Prefixes | ||||

Prefix | Symbol | Scientific Notation | Number of
Base Units |
These
prefixes are what makes the metric system so easy to use. Doing
conversions between units with different prefixes is as easy as
moving the decimal point.
Base units include the meter (m), the gram (g), the liter (L), the second (s), the kelvin (K), and the mole (mol). The SI unit of mass is the kilogram but the base unit for the purposes of this system of prefixes is the gram. The prefix centi- does not follow the pattern of the
other prefixes: it is 10x bigger than milli- and 100x
smaller than the base unit. All other prefixes are 1,000x bigger
or smaller than adjacent prefixes. It is only used for
centimeters (cm). The prefixes hecto- (h), deka- (da), and deci- (d) have been left
out because they are so seldom used. |

Giga- | G | 10^{9} |
1,000,000,000 | |

Mega- | M | 10^{6} |
1,000,000 | |

kilo- | k | 10^{3} |
1,000 | |

base
unit |
meter
gram Liter |
10^{0} |
1 | |

centi- | c | 10^{-2} |
0.01 | |

milli- | m | 10^{-3} |
0.001 | |

micro- | μ | 10^{-6} |
0.000 001 | |

nano- | n | 10^{-9} |
0.000 000 001 | |

pico- | p | 10^{-12} |
0.000 000 000 001 | |

femto- | f | 10^{-15} |
0.000 000 000 000 001 |

We will concentrate on using the metric units chart (above) to create conversion factors. Conversion factors can be used as you learned to do for dimensional analysis. For example:

There are 1,000 mm (millimeters) in 1 m (meter) 1 m 4,350 mm × --------- = 4.35 m 1,000 mm

The old unit cancels out, giving a new unit.

Unit | Abbrev. | Measures |

meter | m | length |

gram | g | mass |

liter | L | volume |

First, we should review the metric units and how they are put together with the prefixes. See the chart at right for the commonly used units and what they measure. Prefixes are put in front of the names or abbreviations to create new units that measure either larger or smaller amounts. For example, a milliliter (mL) is smaller than a liter (L) and a kilometer (km) is larger than a meter.

To make a conversion factor from the metric units chart is easy. On the chart you will see that the unit prefixes all have a power of ten next to them. By counting the number of powers of ten between a pair of units on the chart you can tell how many of the smaller unit fit in one of the bigger unit. For example:

There are three powers of ten between the base unit (meter) and milli- (millimeter). meter: 10^{0}millimeter: 10^{-3}0 — (-3) = +3 10^{3}= 1,000 so 1 m = 1,000 mm

**Here’s how you do it, step-by-step:**

- Identify the pair of units you need a conversion factor for and decide which one is the bigger unit. The bigger unit is always the one closer to the top of the chart.
- Write down the bigger unit’s abbreviation next to the number 1. This is how many of that unit will be in your conversion factor.
- Figure out how many powers of ten separate the two units.
- Write down 1 × 10
^{x}next to the smaller unit’s abbreviation, where the x is the difference in powers of ten.

Here are two examples:

Convert kilograms to milligrams kg: 10^{+3}mg: 10^{-3}kg is the bigger unit so write: 1 kg = _____ mg do this: +3 — (-3) = +6 which means 1 × 10^{6}mg so the conversion factor is 1 kg = 1 × 10^{6}mg (1,000,000 mg) Convert millimeters to micrometers mm: 10^{-3}μm: 10^{-6}mm is the bigger unit so write: 1 mm = _____ μm do this: -3 — (-6) = +3 which means 1 × 10^{3}μm so the conversion factor is 1 mm = 1 × 10^{3}μm (1,000 μm)

The answers should always give you a positive power of ten when you do it this way because you should always expect to have a large number of small units for each big unit.

One thing that is helpful to notice is that most of the prefixes are three powers of ten away from their two nearest neighbors. It becomes as easy as counting by 3s: 3, 6, 9, 12, etc. The only exception in the chart provided here is *centi-* which is 10^{-2} and is one power of ten away from *milli-* and two powers of ten away from the base unit. So skip *centi-* most of the time (it’s only used for meters, anyway) and count by 3s.

Here is something that will help you decide whether your answers are correct after you do a conversion. If you are changing from a bigger unit to a smaller unit then the numerical answer will be bigger. If you are changing from a smaller unit to a bigger unit then the numerical answer will be smaller. For example:

1 × 10^{6}mg 2.9 kg × --------- = 2.9 × 10^{6}mg (or 2,900,000 mg) 1 kg So the number of mg equivalent to a certain numer of kg is a much larger number. 1 kg 510,000 mg × --------- = 0.51 kg 1 × 10^{6}mg So the number of kg equivalent to a certain numer of mg is a much smaller number.

Do the following exercises on a separate piece of paper. Number each problem carefully and clearly.

Figure out how many of the first unit there are in the second
unit then write a unit equality in the form 1 × 10^{–2} mentals = 1 centimental or 10^{12} picolos = 1 los

- How many g in a kg?
- How many cm in a m?
- How many μm in a m?
- How many mm in a cm?

- How many nL in a mL?
- How many L in a GL?
- How many m in a km?
- How many nm in a μm?

Convert each quantity into the units shown. For each calculation create the correct conversion factor using the metric units chart. Hint: you may have made a few of these conversion factors in the exercise above. **Show work for the dimensional analysis calculation**.

- 2.3 m convert to cm and mm
- 4.89 mm convert to cm and μm
- 4.35 × 10
^{2}mL convert to L and μL - 5.23 × 10
^{2}μm to mm and nm - 47 g to kg and mg
- 1.0 × 10
^{3}mg to g and kg

- 4.3 × 10
^{4}kg to Mg and g - 5.30 × 10
^{5}g to kg and Mg - 4.56 × 10
^{3}cm to m and mm - 7.82 × 10
^{9}mm to cm and m - 9.94 mL to nL and μL
- 3.50 × 10
^{2}μL to mL and L