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

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

Contents 1 Units of Measurement as Mathematical Constants 2 Basic Units of Measurement 3 Derived Units of Measurement 4 Scientific Notation 5 Système International d'Unités (International System of Units, aka SI) 6 The Mathematics of Conversion Between Units 7 A Physicists' View of Calculus

[edit] 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
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   • 
4. Mathematics can discard the physical objects in question and has the luxury of concerning itself with abstract concepts
   • 1 + 1 = 2
   • 
   • 
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
   • 
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
     
     is the only answer allowed under the rules of mathematics
7. Also, care must be taken when we perform mathematical operations
   • 
     represents 9 apples arranged in a square
   • 
     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.

[edit] 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

[edit] Derived Units of Measurement

1. Area
   • Usually measured in meters squared
     • 
     • 
     • 
2. Volume
   • Usually measured in meters cubed
     • 
     • 
     • 
3. Density
   1. Linear density
      • Usually measured in kilograms per meter
        • 
        • 
   2. Area density
      • Usually measured in kilograms per meter squared
        • 
        • 
   3. Volumetric density
      • Usually measured in kilograms per meters cubed
        • 
        • 

[edit] Scientific Notation

• Shorthand notation for large or tiny numbers based on powers of 10

1. Large
   • 
   • 
2. Small
   • 
   • 

[edit] Système International d'Unités (International System of Units, aka SI)

• Further simplification of written numbers
  • 
  • 

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

101 = deka = da 102 = hecto = h 103 = kilo = k 106 = mega = M 109 = giga = G 1012 = tera = T 1015 = peta = P 1018 = exa = E 1021 = zetta = Z 1024 = yotta = Y

[edit] The Mathematics of Conversion Between Units

1. In mathematical equations, units of measurement behave as constants
   • 
2. To convert from one unit of to another, we utilize an equation relating the two measurements
   • 
3. We can solve and substitute for the constant m
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   • 
   • 

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

[edit] A Physicists' View of Calculus

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