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

Units of Measurement as Mathematical Constants[edit]

  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[edit]

  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[edit]

  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[edit]

  • 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 Units[edit]

  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[edit]

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