# 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
• ${\displaystyle 1\;apple+1\;apple=2\;apples}$
• ${\displaystyle 30\;apples-10\;apples=20\;apples}$
4. Mathematics can discard the physical objects in question and has the luxury of concerning itself with abstract concepts
• ${\displaystyle 1+1=2}$
• ${\displaystyle (1+1)\times a=2\times a}$
• ${\displaystyle 1\times a+1\times a=2\times a}$
5. Whereas in mathematics the constant ${\displaystyle a}$  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
• ${\displaystyle 1\times apple+1\times apple=2\times apple}$
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
• ${\displaystyle 1+1=2}$
is wrong in the sense that
${\displaystyle 1\times apple+1\times orange=1\times apple+1\times orange}$
is the only answer allowed under the rules of mathematics
7. Also, care must be taken when we perform mathematical operations
• ${\displaystyle (3\times apples)\times (3\times apples)=9\times apples^{2}}$
represents 9 apples arranged in a square
• ${\displaystyle (3\times apples)\times (3\times oranges)=9\times apples\times oranges}$
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
• ${\displaystyle 10\;meters\times meters}$
• ${\displaystyle 10\;square\ meters}$
• ${\displaystyle 10\;{\mbox{m}}^{2}}$
2. Volume
• Usually measured in meters cubed
• ${\displaystyle 10\;meters\times meters\times meters}$
• ${\displaystyle 10\;cubic\ meters}$
• ${\displaystyle 10\;{\mbox{m}}^{3}}$
3. Density
1. Linear density
• Usually measured in kilograms per meter
• ${\displaystyle 10\;kilograms\ per\ meter}$
• ${\displaystyle 10\;{\mbox{kg}}/{\mbox{m}}}$
2. Area density
• Usually measured in kilograms per meter squared
• ${\displaystyle 10\;kilograms\ per\ square\ meter}$
• ${\displaystyle 10\;{\mbox{kg}}/{\mbox{m}}^{2}}$
3. Volumetric density
• Usually measured in kilograms per meters cubed
• ${\displaystyle 10\;kilograms\ per\ cubic\ meter}$
• ${\displaystyle 10\;{\mbox{kg}}/{\mbox{m}}^{3}}$

## Scientific Notation

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

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

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

## The Mathematics of Conversion Between Units

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

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