Working With Units

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 × 10⁷ sec:

  43 yr = 43 × (1 yr) = 43 × (3.15 × 10⁷ sec) = (43 × 3.15 × 10⁷) sec = 1.35 × 10⁹ 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, so

  120 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 × 107 sec
Astronomical Units	1 AU = 1.496 × 1011 m
Light Years	1 ly = 9.461 × 1015 m
Parsecs	1 pc = 2.062 × 105 AU = 3.086 × 1016 m
Kiloparsecs	1 kpc = 103 pc = 3.086 × 1019 m
Megaparsecs	1 Mpc = 106 pc = 3.086 × 1022 m
Earth Masses	1 M = 5.967 × 1024 kg
Solar Masses	1 M = 1.989 × 1030 kg

Last modified: August 22, 2001
http://www.ifa.hawaii.edu/~barnes/ast110/units.html