[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”
• ”$\ge$”: “greater than or equal to”
• ”<”: “less than”
• ”$\le$”: “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”
…	$ϵ$: 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	$\varphi$: 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:
  • $\frac{\left(\frac{8}{4}\right)}{2}=\frac{2}{2}=1$
  • $\frac{8}{\left(\frac{4}{2}\right)}=\frac{8}{2}=4$

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
  • $3+(2\times 5)=13\ne (3+2)\times (3+5)=40$

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: $5-3=2\ne 3-5=-2$

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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
  • $f\left(x\right)=x+\frac{1}{4}$
• conventionally we say “f of x” when we read $f\left(x\right)$
  • 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
  • $sin\theta \equiv sin\left(\theta \right)$

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: ${\sin }^{-1}\theta \ne \frac{1}{\sin \theta }$
  • also: ${\sin }^{-1}\theta =\arcsin$
• squared trigonometric functions:
  • ${\sin }^2\theta =\sin \theta \times \sin \theta =(\sin \theta {)}^2\ne \sin \left(\sin \theta \right)$