Mathematics Background
I) The number system: Be able to perform all + - * / ^ operations with all types:
A) Integers
1) Positive integers (counting) 1, 2, 3,… Know + - * / ab = a^b
2) Negative integers (inverse addition) -1, -2, -3… from 3 + x =0 or x = -3
3) Zero – for a long time this was not a number
B) Rational numbers = a/b (ratios of integers from inverse division) a * x =1 or x=1/a
C) Irrational numbers (from inverse exponentiation) ab such as (2)1/2, also e, 
D) Complex numbers (also from inverse exponentiation)(-1)1/2 = i
1) imaginary numbers and complex values = a + ib
2) With infinity, the complex numbers close under all operations.
E) Infinity: Cantor – concept of 1 to 1 matching
1) Infinity of counting 1,2,3,… Note same value as even integers
(a)Same as the infinity of rational numbers a/b
2) Infinity of real numbers
3) Infinity of functions
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F) ‘Scientific numbers’: 1.23456E3 = 1.23456*103 = 1234.56
G) ‘Uncertain numbers’ (numerical uncertainty or fuzzy numbers) 1.23 = 1.23???...
H) ‘Order of magnitude numbers’ 2E32 or maybe just 1E32 and calculations.
II) Some milestones of the last century:
A) 1900 Russell & Whitehead ‘Principles of Mathematics’ – laid the foundations of
mathematics with logic and arithmetic (using Boole’s logic of 1820).
B) Gödel theorem: There are some algebraic problems that are undecidable.
C) 1945 John von Neumann - stored program digital computer
D) 1948 Shannon – Information defined as log2(P)
III) Prefixes (see text): Kilo, Mega, Giga, Tera, Peta, Exa …Centi, Milli, Micro, Nano
IV) The Greek alphabet – useful to know and recognize:

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V) Supporting concepts in Logic
A) Special notations
1) There exists
2) Member of
3) Such that
4) Implies
5) For all
6) Isomorphic 1-1
7) Set {s}
8) Equality = and not equal
9) Greater than >, less than <
10) Includes
B) Logic
1) Elements 1, 0 or T, F
2) Operations AND, OR, NOT, NOR NAND, EQV, (16 operatons)
VI) Basic Algebra
A) Equations: Solve by doing the same thing to both sides of an equation
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B) Powers add xa * xb = x(a+b) (xa)b = x(a*b)
C) Factoring x2 – y2 = (x+y)*(x-y)
D) Quadratic Equation solutions ax2 +bx +c =0
E) Linear equations: y = mx+b gives b as intersection at x=0 and m=slope
F) Simultaneous equations - solution is intersection
G) Logarithms log a + log b = log (a*b) and log a - log b = log (a/b)
1) y = logax implies x = ay
2) socioeconomic variables (population, electric use) are exponential
3) ratios of socioeconomic variables are relatively constant
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VII) Geometry
A) Angular degrees & radians s/r
B) Area & volume
1) Rectangle & rectangular solids, parallelogram area
2) Triangle
3) Circle C=2 r A=  r2 Sphere A = 4  r2 V = (4/3)  r3
4) Cylinder base area * height
VIII) Trigonometry
A) Basic triangle x y r: sin y/r , cos x/r , tan y/x = sin cos 
B) sin2  cos2  & review your trig identities
C) ei = cos  i sin  also z = u + iv = rei = r cos  i r sin 
D) review your hypergeometric functions ch, sh, th, …
IX) Series expansions
A) ex = 1 + x + x2/2! + x3/3! + x4/4! …..
B) log(1+x) = x – x2/2 +x3/3 C) sin x3/3! +x5/5! and cos  = 1 - x2/2! + x4/4!
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D) Binomial series (a + b)n = an + n a(n-1)b + n(n-1) a(n-2) b2/2! + …
E) Taylor series
X) Calculus
A) Define velocity v(t) and acceleration a(t) from position r(t). (1 dim & 3 dim)
B) Use constant a = a0 to get standard equations for esp a = g acceleration of gravity
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XI) Scalars, Vectors, Matrices, Tensors
A) Scalar: Specified by a single real number: time, temperature, mass, volume, energy
B) Vector: An ordered n-tupe of real numbers: (x, y, z) or (x1, x2, x3) eg (1,-5,0)
1) The dimensionality of a space is the number of numbers needed to specify a
point.
2) A vector in that space has exactly that many ordered numbers in its specification
3) Examples are position, velocity, acceleration, force, momentum
C) Matrix: A two dimensional array of numbers Cij where i is the row and j is the
column
1) A matrix is often used to perform a linear transformation on a vector or to solve a
set of simultaneous linear equations. <example of rotations>
D) A scalar is a tensor of rank 0, a vector is a tensor of rank 1, a matrix is a tensor of
rank 2
E) Operations with vectors:
1) Graphical (as used in high school)
2) i, j, k unit vectors as used in some engineering texts (do not use this notation)
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3) r = (x, y, z) or (x1, x2, x3) or simply as xi or for example (3, -2, 5)
4) Addition, subtraction, multiplication by a scalar <examples>
5) Scalar product A*B = |A| |B| cos  <examples>
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XII) Units:
A) The value for a physical quantity is normally a quantity (real number) of
fundamental ‘units’ that constitute the quantity such as 12 ft, 4 m, 13.5 s, or 9.8
m/s2
B) The ‘fundamental’ units: length (meter = m), time (second = s), mass (kilogram =
kg).
1) Kilogram: The mass of a specific platinum-iridium alloy cylinder in Paris.
2) Second: 9,192,631,700 oscillations of radiation from cesium 133 (since 1967)
(a)Before 1960 was 1/86,400 of average solar day
3) Meter: The distance light travels in 1/299,792,458 s (since 1983)
(a)Originally 10-7 of the distance from the equator to the north pole. (1799)
(b)
Until 1960, the distance between two lines on a platinum iridium bar in
Paris
(c)In 1960 was defined as the 1,650,763.73 wavelengths of Krypton 86 light
C) English Units: foot, inch, hand, yard, cubit, fathom, mile, acre, (also, day, hour,
min..)
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D) Additional units are needed to measure: electrical current (Ampere = A),
temperature (degrees Kelvin = K), and brightness (candela = cd).
E) Dimensional analysis: only add & subtract units of the same type (apples to apples).
1) Examples of lengths, masses, times, velocity of light (3E8 m/s) & sound (1100
ft / sec)
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