# 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 ConstantsEdit

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 MeasurementEdit

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 MeasurementEdit

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 NotationEdit

• 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)Edit

• 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 $10^{-15}$ $=$ $temto$ $=$ 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 UnitsEdit

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 CalculusEdit

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