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.

Two units in the denominator

• Births per capita per year

• Hours per week per unit

Combinations of units

• We often combine units to express new quantities

Student density

• Students per acre

• Number per area

• Spans two orders of magnitude

Example: Units of students per acre

• Cal State densities

  

Dimensional analysis example

We can use the dimensions of the relevant quantities to deduce the form of an equation