[FMQM] W00 - Symbols and Algebra Notation

arithmetic symbols

• ’=’: Equals (left hand side is equal to the right hand side)
• ”+”: Addition or “plus”
• ”-“: Subtraction, “minus”
• “x” or “.”: Multiplication
• “÷” or “/”: Division
• quotient:
  • \( \frac{numerator}{denominator} = \frac{dividend}{divisor} \)

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relational symbols

• ”\( \equiv \)”: “equivalent to” \( x/y \equiv \frac{x}{y}\)
  • a little more than equal to,
  • inclusive of the notation
• ”\(\simeq \)”: “is approximately equal to “ \( \frac{1}{3} \simeq 0.33 \)
  • a good numerical approximation for instance
• ”\( \propto \)”: “is proportional to” \( ax \propto x \)
• ”>”: “greater than”
• ”\( \geq \)”: “greater than or equal to”
• ”<”: “less than”
• ”\( \leq \)”: “less than or equal to”
• ”\( \gg \)”: “much greater than”
• ”\( \ll \)”: “much lesser than”

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greek characters

…	\( \alpha \): a	“alpha”
…	\( \beta \): b	“bay-ta”
\( \Gamma \): G	\( \gamma \): g	“gamma”
\( \Delta \): D	\( \delta \): d	“delta”
…	\( \epsilon \): e	“epsilon”
…	\( \zeta \): z	“zeta”
…	\( \eta \): second type of e (h)	“eta”
…	\( \theta \): th (q)	“theta”
…	\( \kappa \): k	“kappa”
\( \Lambda \): L	\( \lambda \): l	“lambda”
…	\( \mu \): m	“mu”
…	\( \nu \): n	“nu”
\( \Xi \): X	\( \xi \): x	“xi”
\( \Pi \): P	\( \pi \): p	“pi”
…	\( \rho \): r	“rho”
\( \Sigma \): S	\( \sigma \): s	“sigma”
…	\( \tau \): t	“tau”
\( \Phi \): ph	\( \phi \): ph (f)	“phi”
…	\( \chi \): ch	“chi”
\( \Psi \): psi	\( \psi \): psy	“psy”
\( \Omega \): O	\( \omega \): o	“omega”

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algebra notations

multiplication

• \( 2 \times 3 = 6 \)
• \( a \times b = c \)
  • \( ab = c \): implicit multiplication
• parenthesis for grouping
  • \( 2 \times ( 3 \times 4 ) = 2 \times 7 = 14 \)
  • \( 2 \times [ 3 \times 4 ] = 2 \times 7 = 14 \)
  • \( 2 \times { 3 \times 4 } = 2 \times 7 = 14 \)

associative property

• operations are associative if grouping does not matter
• addition:
  • \( (a+b)+c = a+(b+c) \)
• multiplication:
  • \( (a \times b) \times c = a \times (b \times c) \)
• division:
  • \( \frac{(\frac{8}{4})}{2} = \frac{2}{2} = 1 \)
  • \( \frac{8}{(\frac{4}{2})} = \frac{8}{2} = 4 \)

distributive property

• property where removing parenthesis distributes the operation
• multiplication over addition:
  • \( a \times ( b + c ) = a \times b + a \times c \)
• addition is not distributive over multiplication
  • \( 3 + (2 \times 5) = 13 \neq (3 + 2) \times (3 + 5) = 40 \)

commutative property

• property where the order can be switched around
• addition: \( a + b = b + a \)
• multiplicative: \( a \times b = b \times a \)
• subtraction and division are not commutative
  • subtraction: \( 5 - 3 = 2 \neq 3 - 5 = -2 \)

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algebra notation and functions

parenthesis and functions

• a function is something that relates or “maps”
  • one set of values to another
  • takes an argument variable to output a value dependent on that argument
  • \( f(x) = x + \frac{1}{4} \)
• conventionally we say “f of x” when we read \(f(x)\)
  • not “f times x”
• only round brackets ‘()’ are used for arguments,
  • not square ‘[]’ or flower brackets ‘{}’
• sometimes the brackets are simply dropped
  • like trigonometric functions
  • \( sin \theta \equiv sin (\theta) \)

trigonometry

• derived from angles in a right angled triangle

• \( \cos \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)

• \( \cot \theta = \frac{1}{\tan \theta} \)

• inverse sine functions:
  • outputs angle taking in a ratio
  • example: \( \sin^{-1} \theta \neq \frac{1}{\sin \theta}\)
  • also: \( \sin^{-1} \theta = \arcsin \)
• squared trigonometric functions:
  • \( \sin^2 \theta = \sin \theta \times \sin \theta = (\sin \theta)^2 \neq \sin{ (\sin \theta) } \)