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
 * 2) *1 apple, 2 apples, etc.
 * 3) *Thus an "apple" in our records is a unit of measurement, the quantity in question being "number of apples"
 * 4) This evolves into simple arithmetic
 * 5) *1 apple added to 1 apple is 2 apples
 * 6) *10 apples subtracted from 30 apples is 20 apples
 * 7) Introduction of shorthand notation
 * 8) *$$1\; apple + 1\; apple = 2\; apples$$
 * 9) *$$30\; apples - 10\; apples = 20\; apples$$
 * 10) Mathematics can discard the physical objects in question and has the luxury of concerning itself with abstract concepts
 * 11) *$$1 + 1 = 2$$
 * 12) *$$(1 + 1) \times a = 2 \times a$$
 * 13) *$$1 \times a + 1 \times a = 2 \times a$$
 * 14) 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
 * 15) *$$1 \times apple + 1 \times apple = 2 \times apple$$
 * 16) 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
 * 17) *$$1 + 1 = 2$$ 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
 * 18) Also, care must be taken when we perform mathematical operations
 * 19) *$$(3 \times apples) \times (3 \times apples) = 9 \times apples^2$$ represents 9 apples arranged in a square
 * 20) *$$(3 \times apples) \times (3 \times oranges) = 9 \times apples \times oranges$$ 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
 * 2) *Usually measured in seconds
 * 3) **Shorthand is s
 * 4) ***10 seconds
 * 5) ***10 s
 * 6) *Only unit of measurement not to be decimalized (although such a system does exist)
 * 7) Distance
 * 8) *Usually measured in meters
 * 9) **Shorthand is m
 * 10) ***10 meters
 * 11) ***10 m
 * 12) Mass
 * 13) *Base unit is the kilogram
 * 14) **Shorthand is kg
 * 15) ***10 kilograms
 * 16) ***10 kg
 * 17) *Sometimes measured in grams
 * 18) **Shorthand is g
 * 19) ***10 grams
 * 20) ***10 g

Derived Units of Measurement

 * 1) Area
 * 2) *Usually measured in meters squared
 * 3) **$$10\; meters\times meters$$
 * 4) **$$10\; square \ meters$$
 * 5) **$$10\; \mbox{m}^2$$
 * 6) Volume
 * 7) *Usually measured in meters cubed
 * 8) **$$10\; meters\times meters\times meters$$
 * 9) **$$10\; cubic \ meters$$
 * 10) **$$10\; \mbox{m}^3$$
 * 11) Density
 * 12) Linear density
 * 13) *Usually measured in kilograms per meter
 * 14) **$$10\; kilograms \ per \ meter$$
 * 15) **$$10\; \mbox{kg}/\mbox{m}$$
 * 16) Area density
 * 17) *Usually measured in kilograms per meter squared
 * 18) **$$10\; kilograms \ per \ square \ meter$$
 * 19) **$$10\; \mbox{kg}/\mbox{m}^2$$
 * 20) Volumetric density
 * 21) *Usually measured in kilograms per meters cubed
 * 22) **$$10\; kilograms \ per \ cubic \ meter$$
 * 23) **$$10\; \mbox{kg}/\mbox{m}^3$$

Scientific Notation

 * Shorthand notation for large or tiny numbers based on powers of 10
 * 1) Large
 * 2) *$$1,000,000 = 10^6 = 1 \times 10^6$$
 * 3) *$$2,500,000 = 2.5 \times 10^6$$
 * 4) Small
 * 5) *$$0.001 = 10^{-3} = 1 \times 10^{-3}$$
 * 6) *$$0.000234 = 2.34 \times 10^{-4}$$

Syst&egrave;me International d'Unit&eacute;s (International System of Units, aka SI)

 * Further simplification of written numbers
 * $$4,430 \mbox{ meters} = 4.43 \times 10^3 \mbox{ meters} = 4.43 \mbox{ kilometers}$$
 * $$4,430 \mbox{ m} = 4.43 \times 10^3 \mbox{ m} = 4.43 \mbox{ km}$$

The Mathematics of Conversion Between Units

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

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