# 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

1. 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"
2. This evolves into simple arithmetic
• 1 apple added to 1 apple is 2 apples
• 10 apples subtracted from 30 apples is 20 apples
3. Introduction of shorthand notation
• $1\;apple+1\;apple=2\;apples$
• $30\;apples-10\;apples=20\;apples$
4. Mathematics can discard the physical objects in question and has the luxury of concerning itself with abstract concepts
• $1+1=2$
• $(1+1)\times a=2\times a$
• $1\times a+1\times a=2\times a$
5. 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
• $1\times apple+1\times apple=2\times apple$
6. 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
7. 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

1. 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)
2. Distance
• Usually measured in meters
• Shorthand is m
• 10 meters
• 10 m
3. 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

1. Area
• Usually measured in meters squared
• $10\;meters\times meters$
• $10\;square\ meters$
• $10\;{\mbox{m}}^{2}$
2. Volume
• Usually measured in meters cubed
• $10\;meters\times meters\times meters$
• $10\;cubic\ meters$
• $10\;{\mbox{m}}^{3}$
3. Density
1. Linear density
• Usually measured in kilograms per meter
• $10\;kilograms\ per\ meter$
• $10\;{\mbox{kg}}/{\mbox{m}}$
2. Area density
• Usually measured in kilograms per meter squared
• $10\;kilograms\ per\ square\ meter$
• $10\;{\mbox{kg}}/{\mbox{m}}^{2}$
3. 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
1. Large
• $1,000,000=10^{6}=1\times 10^{6}$
• $2,500,000=2.5\times 10^{6}$
2. 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

1. In mathematical equations, units of measurement behave as constants
• $(1{\mbox{ m}}+2{\mbox{ m}})\times 4{\mbox{ m}}=12{\mbox{ m}}^{2}$
2. To convert from one unit of to another, we utilize an equation relating the two measurements
• $1{\mbox{ km}}=1000{\mbox{ m}}\,$
3. 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

  1. In mathematical equations, units of measurement behave as constants
* (1\mbox{ m} + 2\mbox{ m})\times 4\mbox{ m} = 12\mbox{ m}^2
2. To convert from one unit of to another, we utilize an equation relating the two measurements
* 1\mbox{ km} = 1000\mbox{ m} \,
3. 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


## A Physicists' View of Calculus

1. The derivative and small quantities
2. The integral and summation of infinite quantities