Dimensioned quantities always have two parts: the numeric part, and the unit part. The "unit" is the dimension — years or seconds in the examples above. Dimensioned quantities obey the rules of arithmetic, with a couple of modifications:

• Addition (or subtraction) of two dimensioned quantities is possible only if their unit parts agree. The unit part of their sum (difference) is the same as the unit part of each term of the sum (difference). The number part of their sum (difference) is the sum (difference) of their number parts.
• A pure number can be viewed as a dimensioned quantity with a unit part of 1.
• The ratio of two identical unit parts is 1.
• The product (ratio) of two dimensioned quantities is a dimensioned quantity with a number part that is the product (ratio) of the two number parts, and a unit part that is the product (ratio) of the two unit parts.
• Only pure numbers can appear in an exponent.

We usually have one of two problems when trying to compute with dimensioned quantities: either we're trying to add or subtract quantities that have different dimensions, or we're trying to convert from one unit to another. In the former case, the best approach is to convert one of the two quantities' units to that of the other, so in either case we have to do a conversion. The fundamental trick is: convert by multiplying by 1.

Sounds dumb, but it really is pretty clever. Let's say we want to convert kilos to ounces. There are 2.2 pounds in a kilo, and 16 ounces in a pound. How many ounces in a kilo?.

Well, 16 ounce = 1 pound. Therefore, 1 = 16 ounce/pound. And since 2.2 pound = 1 kilo,

1 kilo = 2.2 pound

 = 2.2 pound * 1

 = 2.2 pound * (16 ounce / pound)
 = 2.2 * 16 ounce = 35.2 ounce

As you can see, we can manipulate the unit names just as if they were algebraic quantities.

At the end of the fiscal year, we plan to lease the offices next door. We now have 1000 square feet of space. The space next door is an additional 800 square feet. When we lease the new space we'll have 1800 square feet.

Sounds easy enough. Viewed as dimensional quantities, we can express this as:

Total space = 1000 ft²+ 800 ft² = 1800 ft²

If we were to double our space instead, we would multiply our current space by the "pure number" 2:

Total space = 2 * 1000 ft² = 2000 ft²

We want to carpet the combined space. Carpeting is sold by the square yard. There are 3 feet in a yard. How much carpeting will we need?.

Total yardage = Total space = 1800 ft² * (1 yd/3 ft)²
= 200 yd²

Here we used the fact that multiplying by 1, in the form

1 = (1 yard/3 ft)²,

doesn't change the value of the dimensioned quantity.