# Introduction to Theoretical Physics/Mathematical Foundations/Arithmetic, Physics, and Mathematics

Tentative outline for Arithmetic, Physics, and Mathematics; or the Units of Measurement

## Units of Measurement as Mathematical Constants

* Physics and Mathematics begin with counting
** 1 apple, 2 apples, etc.
** Thus an "apple" in our records is a unit of measurement, the quantity in question being "number of apples"
* This evolves into simple arithmetic
** 1 apple added to 1 apple is 2 apples
** 10 apples subtracted from 30 apples is 20 apples
* Introduction of shorthand notation
** $1\;apple+1\;apple=2\;apples$
** $30\;apples-10\;apples=20\;apples$
* Mathematics can discard the physical objects in question and has the luxury of concerning itself with abstract concepts
** $1+1=2$
** $(1+1)\times a=2\times a$
** $1\times a+1\times a=2\times a$
* Whereas in mathematics the constant $a$ represents a numerical constant, in physics this constant can represent a physical constant, thereby allowing physical objects to behave as mathematical entities in mathematical equations
** $1\times apple+1\times apple=2\times apple$
* Units of measurements are important in mathematical equations since they represent critical information and errors in calculations will result if these physical constants are neglected
** $1+1=2$ is wrong in the sense that $1\times apple+1\times orange=1\times apple+1\times orange$ is the only answer allowed under the rules of mathematics
* Also, care must be taken when we perform mathematical operations
** $(3\times apples)\times (3\times apples)=9\times apples^{2}$ represents 9 apples arranged in a square
** $(3\times apples)\times (3\times oranges)=9\times apples\times oranges$ creates a new physical quantity apple(orange) that is neither apples nor oranges! This is how new physical quantities, i.e. energy are created from lengths, times, and masses.

## Basic Units of Measurement

* Time
** Usually measured in seconds
*** Shorthand is s
**** 10 seconds
**** 10 s
** Only unit of measurement not to be decimalized (although such a system does exist)
* Distance
** Usually measured in meters
*** Shorthand is m
**** 10 meters
**** 10 m
* Mass
** Base unit is the kilogram
*** Shorthand is kg
**** 10 kilograms
**** 10 kg
** Sometimes measured in grams
*** Shorthand is g
**** 10 grams
**** 10 g

## Derived Units of Measurement

* Area
** Usually measured in meters squared
*** $10\;meters\times meters$
*** $10\;square\ meters$
*** $10\;{\mbox{m}}^{2}$
* Volume
** Usually measured in meters cubed
*** $10\;meters\times meters\times meters$
*** $10\;cubic\ meters$
*** $10\;{\mbox{m}}^{3}$
* Density
** Linear density
*** Usually measured in kilograms per meter
**** $10\;kilograms\ per\ meter$
**** $10\;{\mbox{kg}}/{\mbox{m}}$
** Area density
*** Usually measured in kilograms per meter squared
**** $10\;kilograms\ per\ square\ meter$
**** $10\;{\mbox{kg}}/{\mbox{m}}^{2}$
** Volumetric density
*** Usually measured in kilograms per meters cubed
**** $10\;kilograms\ per\ cubic\ meter$
**** $10\;{\mbox{kg}}/{\mbox{m}}^{3}$

## Scientific Notation

* Shorthand notation for large or tiny numbers based on powers of 10
* Large
** $1,000,000=10^{6}=1\times 10^{6}$
** $2,500,000=2.5\times 10^{6}$
* Small
** $0.001=10^{-3}=1\times 10^{-3}$
** $0.000234=2.34\times 10^{-4}$

## Système International d'Unités (International System of Units, aka SI)

* Further simplification of written numbers
** $4,430{\mbox{ meters}}=4.43\times 10^{3}{\mbox{ meters}}=4.43{\mbox{ kilometers}}$
** $4,430{\mbox{ m}}=4.43\times 10^{3}{\mbox{ m}}=4.43{\mbox{ km}}$
| $10^{-24}$ | $=$ | $yocto$ | $=$ | y |
| $10^{-21}$ | $=$ | $zepto$ | $=$ | z |
| $10^{-18}$ | $=$ | $atto$ | $=$ | a |
| $10^{-15}$ | $=$ | $femto$ | $=$ | f |
| $10^{-12}$ | $=$ | $pico$ | $=$ | p |
| $10^{-9}$ | $=$ | $nano$ | $=$ | n |
| $10^{-6}$ | $=$ | $micro$ | $=$ | µ |
| $10^{-3}$ | $=$ | $milli$ | $=$ | m |
| $10^{-2}$ | $=$ | $centi$ | $=$ | c |
| $10^{-1}$ | $=$ | $deci$ | $=$ | d |
| $10^{1}$ | $=$ | $deka$ | $=$ | da |
| $10^{2}$ | $=$ | $hecto$ | $=$ | h |
| $10^{3}$ | $=$ | $kilo$ | $=$ | k |
| $10^{6}$ | $=$ | $mega$ | $=$ | M |
| $10^{9}$ | $=$ | $giga$ | $=$ | G |
| $10^{12}$ | $=$ | $tera$ | $=$ | T |
| $10^{15}$ | $=$ | $peta$ | $=$ | P |
| $10^{18}$ | $=$ | $exa$ | $=$ | E |
| $10^{21}$ | $=$ | $zetta$ | $=$ | Z |
| $10^{24}$ | $=$ | $yotta$ | $=$ | Y |

## The Mathematics of Conversion Between Units

* In mathematical equations, units of measurement behave as constants
** $(1{\mbox{ m}}+2{\mbox{ m}})\times 4{\mbox{ m}}=12{\mbox{ m}}^{2}$
* To convert from one unit of to another, we utilize an equation relating the two measurements
** $1{\mbox{ km}}=1000{\mbox{ m}}\,$
* We can solve and substitute for the constant $m$
** ${\frac {1}{1000}}{\mbox{ km}}={\mbox{ m}}$
** $\left[1\left({\frac {1}{1000}}{\mbox{ km}}\right)+2\left({\frac {1}{1000}}{\mbox{ km}}\right)\right]\times 4\left({\frac {1}{1000}}{\mbox{ km}}\right)=12\left({\frac {1}{1000}}{\mbox{ km}}\right)^{2}$
** $\left(1\times 10^{-3}{\mbox{ km}}+2\times 10^{-3}{\mbox{ km}}\right)\times 4\times 10^{-3}{\mbox{ km}}=12\times 10^{-6}{\mbox{ km}}^{2}$
The Mathematics of Conversion Between Units

## A Physicists' View of Calculus

* The derivative and small quantities
* The integral and summation of infinite quantities