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