Quantities

When we use mathematics to model the physical world, we need additional tools to represent physical quantities. A number only gives a magnitude. When we combine that number with a unit (length, mass) it becomes a quantity that can represent something physical.

Dimensions, units, quantities

A quantity represents a physical measurement like mass, length or
amount of energy
We represent a quantity with a number and a unit
The dimension of a quantity is different than the unit
For example 1 inch is the same as 2.54 centimeters even though 1 and
2.54 are not the same number

Physical Quantities

Our numbers are often helping us represent physical quantities
Examples:
- The length of a tree
- The number of animals observed
- The number of molecules of mercury in a fish
A physical quantity is expressed as the product of a unit and a
numerical factor

Dimensions

These physical quantities often have a dimension
Examples:
- Length
- Time
- Mass

In our estimations, we may have quantities that are not in fundamental units such as length or time. For example, we may be counting a population of animals, or the number of power plants needed.

Units

To quantify dimensions, we use units
One dimension may have multiple units
Length: inches, miles, kilometers, light-years
Mass: grams, pounds, kilograms
There are also systems of units like SI or English

Measurement

Each measurement we make is an estimation of the physical quantity

Consequences

NASA Mars Climate Orbiter destroyed because of newton vs pounds of
force
A cargo flight was lost in 1999 when crew confused meters and feet

Unit conversion factors

These factors are equivalent to one or unity and are dimensionless
They are not numerically equal to one in most cases.
Units can be crossed out

Unit conversion factors

These factors are equivalent to one or unity and are dimensionless
They are not numerically equal to one in most cases.
Units can be crossed out
They can have units but be dimensionless

Unit conversion factors

These two quantities are equal.

$3 \textrm{feet} = 1 \textrm{yard}$

If we divide both sides by 3 feet, we get

$1 = \frac{1\ \textrm{yard}}{3\ \textrm{feet}}$

Note that this quantity has units of yard per feet, but has no dimensions since it is a length divided by a length. If you multiply a quantity of feet by this, you won't change the quantity but you will change the units.