# 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

    &#x20; 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

    &#x20; 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

![](https://3104074691-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmGGOR79oy-_JywJoK5%2F-LmGGPVIa9dVx26dpNEz%2F-LmGGRAGGFBnvQTa2YNH%2Finch-cm-conversion-1.png?generation=1565805365804164\&alt=media)

![](https://3104074691-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmGGOR79oy-_JywJoK5%2F-LmGGPVIa9dVx26dpNEz%2F-LmGGRAIxsdd5ZYRWJTP%2Finch-cm-conversion-2.png?generation=1565805384452880\&alt=media)

## 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](https://twitter.com/calpolypomona/status/431937140457349120/photo/1)

    ![](https://3104074691-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmGGOR79oy-_JywJoK5%2F-LmGGPVIa9dVx26dpNEz%2F-LmGGRANGh8PaJtZWSrt%2Fcal-state-density.jpg?generation=1565805370932165\&alt=media)

## Dimensional analysis example

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

![](https://3104074691-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmGGOR79oy-_JywJoK5%2F-LmGGPVIa9dVx26dpNEz%2F-LmGGRAPOhO6w_WzMd7c%2Fdimensional-analysis-1.png?generation=1565805384876595\&alt=media)

![](https://3104074691-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmGGOR79oy-_JywJoK5%2F-LmGGPVIa9dVx26dpNEz%2F-LmGGRARlFj1Gi3mcVtp%2Fdimensional-analysis-2.png?generation=1565805367633716\&alt=media)
