Units are a valuable tool; careful attention to the units involved at each stage of a calculation can help you catch and fix mistakes. Much confusion can be avoided if you work with units as though they were symbols like those in algebra. For example:

- Multiply units along with numbers:
(5

**m**) × (2**sec**) = (5 × 2) × (**m**×**sec**) = 10**m****sec**.The units in this example are meters times seconds, pronounced as `meter seconds' and written as `

**m****sec**'. - Divide units along with numbers:
(10

**m**) ÷ (5**sec**) = (10 ÷ 5) × (**m**÷**sec**) = 2**m**/**sec**.The units in this example are meters divided by seconds, pronounced as `meters per second' and written as `

**m**/**sec**'; these are units of*speed*. - Cancel when you have the same units on top and bottom:
(15

**m**) ÷ (5**m**) = (15 ÷ 5) × (**m**÷**m**) = 3.In this example the units (meters) have canceled out, and the result has

*no units*of any kind! This is what we call a `pure' number. It would be the same regardless what system of units were used. - When adding or subtracting, convert both numbers to the
*same*units before doing the arithmetic:(5

**m**) + (2**cm**) = (5**m**) + (0.02**m**) = (5 + 0.02)**m**= 5.02**m**.Recall that a `

**cm**', or centimeter, is one hundredth of a meter. So 2**cm**= (2 ÷ 100)**m**= 0.02**m**. - You can't add or subtract two numbers
*unless*you can convert them both to the same units:(5

**m**) + (2**sec**) = ???Meters and seconds are different

*kinds*of quantities; one is a length, and the other is a time. As a rule, we can't convert a length to a time, or a time to a length, so there is no way to add these quantities. (Experts in Relativity Theory know that this rule has exceptions -- in Relativity, we can convert between units of length and time. But that's an advanced topic.)

Astronomers use a mixture of units, and we often have to convert one to another. Converting between different units is easier if you remember to treat units like symbols; you simply replace the original unit with its equivalent in the unit desired, and do the necessary arithmetic. For example:

- Convert feet to meters using the equality 1
**ft**= 0.3045**m**:6

**ft**= 6 × (1**ft**) = 6 × (0.3045**m**) = (6 × 0.3045)**m**= 1.84**m**. - Convert pounds to kilograms using the equality 1
**lb**= 0.454**kg**:165

**lb**= 165 × (1**lb**) = 165 × (0.454**kg**) = (165 × 0.454)**kg**= 75**kg**. - Convert years to seconds using the equality 1
**yr**= 3.15 × 10^{7}**sec**:43

**yr**= 43 × (1**yr**) = 43 × (3.15 × 10^{7}**sec**) = (43 × 3.15 × 10^{7})**sec**= 1.35 × 10^{9}**sec**.

What about converting the other way? Again, treating units as symbols simplifies the problem:

- Convert meters to feet; if 0.3045
**m**= 1**ft**then 1**m**= (1 ÷ 0.3045)**ft**, so:5

**m**= 5 × (1**m**) = 5 × (1 ÷ 0.3045)**ft**= (5 ÷ 0.3045)**ft**= 16.4**ft**. - Convert kilograms to pounds; if 0.454
**kg**= 1**lb**then 1**kg**= (1 ÷ 0.454)**lb**, so120

**kg**= 120 × (1**kg**) = 120 × (1 ÷ 0.454)**lb**= (120 ÷ 0.454)**lb**= 264**lb**.

Here are the factors needed to convert between the different systems of units used in astronomy. Notice that there are many different length units:

Years |
1 yr = 3.15 × 10^{7} sec |

Astronomical Units |
1 AU = 1.496 × 10^{11} m |

Light Years |
1 ly = 9.461 × 10^{15} m |

Parsecs |
1 pc = 2.062 × 10^{5} AU = 3.086
× 10^{16} m |

Kiloparsecs |
1 kpc = 10^{3} pc = 3.086 ×
10^{19} m |

Megaparsecs |
1 Mpc = 10^{6} pc = 3.086 ×
10^{22} m |

Earth Masses |
1 M_{} =
5.967 × 10^{24} kg |

Solar Masses |
1 M_{} =
1.989 × 10^{30} kg |

Joshua E. Barnes (barnes@ifa.hawaii.edu)

Last modified: August 22, 2001

`http://www.ifa.hawaii.edu/~barnes/ast110/units.html`