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Metrics Essentials

Unit | Abbrev. | Measures |

meter | m | length |

liter | L | volume |

gram | g | mass |

second | s | time |

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 |

Memorize the following metrics conversions. You should see a pattern that makes this relatively simple. In the end, you will memorize the list of metrics prefixes and a simple relationship between most neighboring pairs of units.

Length | Volume | Mass | Time |

1 Gm = 1,000 Mm | 1 GL = 1,000 ML | 1 Gg = 1,000 Mg | — |

1 Mm = 1,000 km | 1 ML = 1,000 kL | 1 Mg = 1,000 kg | — |

1 km = 1,000 m | 1 kL = 1,000 L | 1 kg = 1,000 g | — |

1 m = 1,000 mm | 1 L = 1,000 mL | 1 g = 1,000 mg | 1 s = 1,000 ms |

1 mm = 1,000 µm | 1 mL = 1,000 µL | 1 mg = 1,000 µg | 1 ms = 1,000 µs |

1 µm = 1,000 nm | 1 µL = 1,000 nL | 1 µg = 1,000 ng | 1 µs = 1,000 ns |

1 nm = 1,000 pm | 1 nL = 1,000 pL | 1 ng = 1,000 pg | 1 ns = 1,000 ps |

Notes: 1 Mg is a metric ton The most commonly used unit conversions are in bold Metric prefixes are not used for time units greater than a second (instead, hours, minutes, days, and years are used) The centi- prefix is generally only used for meters: | |||

1 m = 100 cm | 1 cm = 10 mm |

Here is how to set up a conversion between two metric units:

Memorize that there are 1,000 mm (millimeters) in 1 m (meter) and create a conversion factor; then use it: 1 m 4,350 mm × --------- = 4.35 m 1,000 mm

Here are two examples of conversions between two metric units. These conversions require multiple steps.

Convert micrometers to meters: 1 mm 1 m 650,000 µm × --------- × --------- = 0.65 m 1,000 µm 1,000 mm |
Convert liters to microliters 1,000 mL 1,000 µL 0.00357 L × --------- × --------- = 3,570 µL 1 L 1 mL |

A helpful tip: When you convert *from* a larger unit (which is higher up on the chart) *to* a smaller unit then the answer should be a *larger* number. For example, 1.3 km is equal to 1,300 m. Also, when you convert *from* a smaller unit *to* a larger unit then the answer should be a *smaller* number. For example, 385 mm is equal to 0.385 m.

If you memorize the information in the chart on the previous page, which is meant to be easy because nearly all of the conversions are based on the numbers 1 and 1,000, then you should be able to convert between any metrics prefixes. You will have to do so using multiple steps, however, if the units you are converting between are not right next to each other on the chart. It’s possible to create single conversions between units far apart on the chart. The text and examples below show you how.

Note: this section of the background information is optional. You can be completely successful in all metrics conversions

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:

Create a conversion factor for 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) Create a conversion factor for 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.

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

Convert each quantity into the unit shown. Each calculation will require only one conversion factor. **Show work for the dimensional analysis calculation**.

Show your work like this:

1,000 μm 42 mm x --------- = 42,000 μm 1 mm

- 2.3 m to cm
- 4.89 mm to cm
- 0.435 L to mL
- 523 μm to mm
- 0.047 kg to g
- 43,000 kg to Mg

- 1.500 g to mg
- 530,000 g to kg
- 4.560 µm to nm
- 250 pm to nm
- 9.94 mL to μL
- 350 μg to mg

Convert each quantity into the unit shown. Each calculation will require two or more conversion factors. **Show work for the dimensional analysis calculation**.

*Optional:* You may, if you wish, study the material in this document to learn how to make a *single conversion factor* between any pair of metric prefixes. If you do this, make sure the conversion factor is correct and write down the dimensional analysis conversion using your constructed conversion factor.

Show your work like this:

1,000 μL 1,000 nL 0.054 mL x --------- x --------- = 54,000 nL 1 mL 1 μL

- 4.5 cm to μm
- 5,360,000 mm to km
- 0.00258 L to μL
- 62,400 μm to m
- 0.000072 kg to mg
- 75,000,000 g to Mg

- 0.00095 g to ng
- 2,630,000 g to Gg
- 0.000258 mm to pm
- 750,000 pL to μL
- 0.026 mL to nL
- 97,000,000,000 μg to Mg

It is nearly always the case that when people learn the metric system they are taught how to count decimal places and how to move the decimal to convert from one metric prefix to another. This lesson does not do that because it is important for you to also know the actual conversions between units. Never again should you be unable to say how many cm are in a meter or how many mL there are in a liter.

Now you will do some conversions without writing them out but instead by ‘moving the decimal’ in your head. There is no explicit lesson for these exercises. The idea is for you to figure it out for yourself so that you truly understand it. For this set of problems write the given value and unit and an equal sign followed by the converted number with its new unit. The conversions will be between the same units as in the first section above with different numbers. Use this fact to help you to make sure you’re on the right track with your answers.

Show your work like this:

32 mm = 0.032 m

- 42 m to cm
- 15 mm to cm
- 2.3 L to mL
- 61 μm to mm
- 0.54 kg to g
- 698 kg to Mg

- 95.3 g to mg
- 6,420 g to kg
- 0.258 µm to nm
- 56 pm to nm
- 0.57 mL to μL
- 5,100 μg to mg