Additive | Multiplicative | |
---|---|---|
Commutativity | $a+b=b+a$ | $a·b=b·a$ |
Associativity | $(a+b)+c=a+(b+c)$ | $(a·b)·c=a·(b·c)$ |
Identity | $a+0=a=0+a$ | $a·1=a=1·a$ |
Inverse | $a+(-a)=0$ | $a·a^{-1}=1$ |
Distributive property | $a·(b+c)=a·b+a·c$ |
Distribution of one negative | $\frac{-1}{\phantom{-}1}=\frac{\phantom{-}1}{-1}=-\frac{1}{1}=-1$ |
Distribution of two negatives | $\frac{-1}{-1}=-\big(-\frac{1}{1}\big)=1$ |
For expressions | $a=\frac{1}{1/a}$ |
For evaluating fractions | $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$ |
Symbol | Meaning | Example | Translation |
---|---|---|---|
$||$ | absolute value | $|x|$ | The positive value or magnitude of $x$ |
$!$ | factorial | $5!=1·2·3·4·5$ | The product of all integers to the specified value |
$\therefore$ | therefore | $x^2=4 \therefore x= \pm 2$ | One is true, therefore the other is true. |
$\bmod$ | modulus | $4 \bmod 3 = 1$ | 4/3 has a remainder of 1. |
$\forall$ | for all | $\forall x≥0$ | for every non-negative value |
$\isin$ | element of | $\forall x \isin \Z$ | $x$ is an integer. |
$\land$ | and | $x>0 \land y>0$ | Both of these statements are true. |
$\lor$ | or | $x>0 \lor y>0$ | One or both of these statements are true. |
Real $$\R$$ | All common numbers and multiples of one |
Integer $$\Z$$ | Real whole numbers that can be expressed without using a fraction $$\lbrace …,-2,-1,0,1,2,… \rbrace$$ |
Natural $$\N$$ | Positive real integers (argued whether or not zero is included) |
Rational | Any number that can be represented as a fraction inclusive of only natural numbers, and always either repeats or terminates as a decimal $$ \begin{array}{cc} 1/2 & 2/1 & 1.2 & 2.\overline{1} \end{array} $$ |
Irrational | Any number that cannot be represented as a fraction, and never repeats or terminates as a decimal $$ \begin{array}{cc} \sqrt{2} & e & \pi & \phi \end{array} $$ |
Imaginary | All common numbers that are multiples of $i$ |
Complex $$\Complex$$ | Numbers comprised of both real and imaginary numbers |
$$.\overline{2}=\frac{2}{9}$$ | $$.\overline{137}=\frac{137}{999}$$ | $$.\overline{142857}=\frac{142857}{999999}=\frac{1}{7}$$ |
$$433.\overline{3}=400+\frac{3}{9}·10^2=\frac{1300}{3}$$ | $$.35\overline{7}=\frac{35}{100}+\frac{7}{9}·10^{-2}=\frac{161}{450}$$ | $$3.8\overline{3}=\frac{1}{2}+\frac{30}{9}=\frac{23}{6}$$ |
$$\frac{0}{0}$$ | $$\frac{∞}{∞}$$ | $$0·∞$$ | $$∞-∞$$ | $$0^0$$ | $$1^∞$$ | $$∞^0$$ |
$17+5=22$ | sum |
$17-5=12$ | difference |
$17·5=22$ | product |
$\frac{17}{5}=3+\frac{2}{5}$ |
$17$: dividend $5$: divisor $17/5$: simplified fraction $3$: quotient $2$: remainder $3+2/5$: proper fraction |
$\frac{x}{x_∘}+\frac{y}{y_∘}=1$ |
$x$: abscissa $x_∘$: horizontal axis intercept $+$: operator $y$: ordinate $y$∘: vertical axis intercept $1$: value |
$y=a·x^2+b·x+c$ |
$y$: dependent variable $a$: leading coefficient $x$: independent variable $2$: order $b$: coefficient $c$: constant |
$(a+b·i)(a-b·i)=a^2+b^2$ |
( expression = expression ) ← equation factored form = expanded form $a$: real component $b·i$: imaginary component (a±b∙i): roots |
$\sqrt[n]{x}$ |
$n$: $n$th root $x$: radicand |
$x/x_∘$ $y/y_∘$ $a·x^2$ $b·x$ $(a±b·i)$ $a^2$ $b^2$ $\sqrt[n]{x}$ |
terms |
Zeros of $x$ only (specific) $$x=\frac{-b \pm \sqrt{b^2-4·a·y_∘}}{2·a}$$ | All values of $x$ (general) $$x=\frac{-b \pm \sqrt{b^2+4·a·(y-y_∘)}}{2·a}$$ |
Even - symmetric about $y$-axis $$f(-x)=f(x)$$ |
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Odd - symmetric about origin $$f(-x)=-f(x)$$ |
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Inverse - reflected about $y=x$ $$f(g(x))=g(f(x))$$ |
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Absolute value - magnitude; non-negative $$ |x| = \begin{cases} a &x ≥ 0 \\ c &x \lt 0 \end{cases} $$ |
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Periodic - repeated segment ($P$) $$f(x+P)=f(x)$$ |
$$x^3=1·x·x·x$$ | $$3^3=27$$ |
$$x^2=1·x·x$$ | $$3^2=9$$ |
$$x=1·x$$ | $$3^1=3$$ |
$$x^0=1$$ | $$3^0=1$$ |
$$x^{-1}=1/x$$ | $$3^{-1}=1/3$$ |
$$x^{-2}=1/(x·x)$$ | $$3^{-2}=1/9$$ |
$$x^{-3}=1/(x·x·x)$$ | $$3^{-3}=1/27$$ |
$$i^{-2}=i^2=i^6=-1$$ | $$i^{-4}=i^0=i^4=1$$ |
$$i^{-3}=i^1=i^5=i$$ | $$i^{-1}=i^3=i^7=-i$$ |
$$z=x+y·i$$ | $$z^*=x-y·i$$ |
1 | ||||||||||||||
1 | 1 | |||||||||||||
1 | 2 | 1 | ||||||||||||
1 | 3 | 3 | 1 | |||||||||||
1 | 4 | 6 | 4 | 1 | ||||||||||
1 | 5 | 10 | 10 | 5 | 1 | |||||||||
1 | 6 | 15 | 20 | 15 | 6 | 1 | ||||||||
1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 |
$$\log_b x = \frac{1}{log_x b}$$ | $$\log_{c^n} x=\frac{\log_c x}{n}$$ | $$a^{\log_b x}=x^{\log_b a}$$ |
Slope of a line | $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$ |
Midpoint formula | $\big(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\big)$ |
Horizontal line | $y=c, \medspace m=0$ |
Vertical line | $x=c, \medspace m=±∞$ |
General equation | $A·x+B·y=C, \medspace x_∘=\frac{C}{A} \land y_∘=\frac{C}{B}$ |
Slope-intercept equation | $y(x)=m·x+y_∘$ |
Point-slope equation | $y-y_1=m·(x-x_1)$ |
Two point equation | $y-y_1=\frac{y_2-y_1}{x_2-x_1}·(x-x_1)$ |
Two-intercept equation | $\frac{x}{x_∘}+\frac{y}{y_∘}=1, \medspace x_∘≠0 \land y_∘≠0$ |
Parallel lines | $m_1=m_2$ |
Intercepting lines | $m_1 \ne m_2$ |
Perpendicular lines | $m_1=-1/m_2$ |
Perimeter | $a+b+c$ |
Area (non-obtuse) | $b·h/2$ |
Sum of Angles | $\pi$ |
Right (all others are oblique) | |
Obtuse | |
Acute | |
Equilateral | |
Isosceles | |
Scalene | |
Congruent triangles |
$$\sin(\theta)=\frac{\text{opp}}{\text{hyp}}$$ | $$\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$$ |
$$\csc(\theta)=\frac{\text{hyp}}{\text{opp}}$$ | $$\sec(\theta)=\frac{\text{hyp}}{\text{adj}}$$ |
$$\tan(\theta)=\frac{\text{opp}}{\text{adj}}$$ | $$\cot(\theta)=\frac{\text{adj}}{\text{opp}}$$ |
Function | $$0$$ | $$\pi/6$$ | $$\pi/4$$ | $$\pi/3$$ | $$\pi/2$$ |
---|---|---|---|---|---|
$\sin(\theta)$ | $$0$$ | $$1/2$$ | $$1/\sqrt{2}$$ | $$\sqrt{3}/2$$ | $$1$$ |
$\cos(\theta)$ | $$1$$ | $$\sqrt{3}/2$$ | $$1/\sqrt{2}$$ | $$1/2$$ | $$0$$ |
$\csc(\theta)$ | $$∞$$ | $$2$$ | $$\sqrt{2}$$ | $$2/\sqrt{3}$$ | $$1$$ |
$\sec(\theta)$ | $$1$$ | $$2/\sqrt{3}$$ | $$\sqrt{2}$$ | $$2$$ | $$∞$$ |
$\tan(\theta)$ | $$0$$ | $$1/\sqrt{3}$$ | $$1$$ | $$\sqrt{3}$$ | $$∞$$ |
$\cot(\theta)$ | $$∞$$ | $$\sqrt{3}$$ | $$1$$ | $$1/\sqrt{3}$$ | $$0$$ |
Perimeter | Sum of side lengths |
Area | Sum of areas of triangles divided within shape |
Sum of Angles | $(\text{number of sides}-2)·\pi$ |
Equilateral | |
Equiangular | |
Regular | |
Irregular | |
Convex & Concave |
Perimeter | $n·s$ |
Area | $n·s·h/2$ |
Sum of angles | $(n-2)·\pi$ |
Angle (h,r) | $\pi/n$ |
Circumradius | $h·\sec \Big(\frac{\pi}{n}\Big) \land \frac{s}{2}·\csc \Big(\frac{\pi}{n}\Big)$ |
Area using apothem | $A=n·h^2·\tan \Big(\frac{\pi}{n}\Big)$ |
Area using side | $A=\frac{n·s^2}{4}·\cot \Big(\frac{\pi}{n}\Big)$ |
Area using circumradius | $A=n·r^2·\sin \Big(\frac{\pi}{n}\Big)·\cos \Big(\frac{\pi}{n}\Big)$ |
$$\sec \Big(\frac{\pi}{n}\Big)=\frac{r}{h}$$ | $$\csc \Big(\frac{\pi}{n}\Big)=\frac{r}{s/2}$$ |
No result | $A⋅x^2+A⋅y^2=-1$ |
Point | $A⋅x^2+A⋅y^2=0$ |
Line | $D⋅x+E⋅y+F=0$ |
Intersecting lines | $(x-a)⋅(y+a)=0$ |
Parallel lines | $(x-a)⋅(x-b)=0$ |
Circumference | $2·\pi·r$ |
Area | $\pi·r^2$ |
Arc length | $\theta·r$ |
Sector area | $\theta·r^2/2$ |
Chord length (k) | $2·r·\sin\Big(\frac{\theta}{2}\Big)$ |
Segment area | $\frac{r^2}{2}·\Big(\theta-2·\sin\Big(\frac{\theta}{2}\Big)·\cos\Big(\frac{\theta}{2}\Big)\Big)$ |
Conic general equation | $A·(x^2+y^2)+D·x+E·y+F=0,A≠0$ |
Standard equation | ${(x-x_∘)}^2+{(y-y_∘)}^2=r^2$ |
Conic-standard conversions | $x_∘=-\frac{D}{2·A}$ $y_∘=-\frac{E}{2·A}$ $r^2=\frac{D^2+E^2-4·A·F}{4·A^2}$ |
Focus Coordinates | $(x_∘,y_∘)$ |
Eccentricity | $0$ |
Directrix | None |
$$k=2·r·\sin\Big(\frac{\theta}{2}\Big)$$ $$h=r·\cos\Big(\frac{\theta}{2}\Big)$$ |
Perimeter | $2·\pi·(R+r)$ |
Area | $\pi·(R^2-r^2)$ |
Sector Area | $\theta·(R^2-r^2)/2$ |
Perimeter | $4·a\lt 2·\pi·a, a\gt b$ |
Area | $\pi·a·b$ |
Conic General Equation | $A·x^2+C·y^2+D·x+E·y+F=0,A·C>0$ |
Standard Equation | ${(x-x_∘)}^2/a^2+{(y-y_∘)}^2/b^2=1$ |
Conic-Standard Conversions | $A=b^2$ $C=a^2$ $D=-2·b^2·x_∘$ $E=-2·a^2·y_∘$ $F=b^2·{x_∘}^2+a^2·{y_∘}^2-a^2·b^2$ $a^2=C$ $b^2=A$ $x_∘=-\frac{D}{2·A}$ $y_∘=-\frac{E}{2·C}$ |
Orientation | horizontal if $C>A \land a>b$ vertical if $A>C \land b>a$ |
Center Coordinates | $(x_∘,y_∘)$ |
Foci Coordinates | $f=\big(x_∘ \pm \sqrt{a^2-b^2}, y_∘\big),a>b$ $f=\left(x_∘,y_∘ \pm \sqrt{b^2-a^2}\right),b>a$ |
Eccentricity | $e=f/a=\sqrt{1-b^2/a^2},a>b$
$e=f/b=\sqrt{1-a^2/b^2},b>a$ |
Directrix | $x=\pm a/e,a>b$
$y=\pm b/e,b>a$ |
Conic General Equations | vertical: $A·x^2+D·x+E·y+F=0$ horizontal: $C·y^2+D·x+E·y+F=0$ |
Standard Equation | $y=a·x^2+b·x+y_∘$ |
Vertex Equation | $y=a·(x-x_∘)^2+y_∘$ |
Intercept Equation | $y=a·(x-x_1)(x-x_2)$ |
Discriminant | $\Delta=b^2-4·a·y_∘$
If $Δ \lt 0$, two $x$-intercepts If $Δ=0$, one $x$-intercept If $Δ>0$, no $x$-intercepts |
Conic-Standard Conversion | $a=-A/E$ $b=-D/E$ $y_∘=-F/E$ |
Vertex Coordinates | $(x_∘,y_∘)=\big(-\frac{b}{2·a},y_∘-\frac{b^2}{4·a}\big)$ |
$x$-Intercepts | $\lbrace (x_1,0),(x_2,0) \rbrace = \big(\frac{-b \pm \sqrt{b^2-4·a·y_∘}}{2·a},0\big)$ |
Fucus Length from Vertex | $f=\frac{1}{4·a}$ |
Focus Coordinates | $\big(-\frac{b}{2·a},y_∘-\frac{b^2+1}{4·a}\big)$ |
Eccentricity | $1$ |
Drectrix | $y=-f$ |
Conic General Equation | $A·x^2+C·y^2+D·x+E·y+F=0, A·C \lt 0$ |
Standard Equation | $(x-x_∘)^2/a^2+(y-y_∘)^2/b^2=1, a^2·b^2 \lt 0$ |
Standard Equation (Real Terms Only) | $\pm (x-x_∘)^2/a^2 \mp (y-y_∘)^2/b^2=1$ |
Conic-Standard Conversion | $A=b^2$ $C=a^2$ $D=-2·b^2·x_∘$ $E=-2·a^2·y_∘$ $F=b^2·{x_∘}^2+a^2·{y_∘}^2-a^2·b^2$ $a^2=C$ $b^2=A$ $x_∘=-\frac{D}{2·A}$ $y_∘=-\frac{E}{2·C}$ |
Orientation | horizontal if $C \lt 0; a \isin ℂ$ vertical if $A \lt 0; b \isin ℂ$ |
Vertices | $(x_∘ \pm a,y_∘) \lor (x_∘,y_∘ \pm b)$ |
Focus Coordinates | $(x_∘ \pm \sqrt{a^2+b^2},y_∘) \lor (x_∘,y_∘ \pm \sqrt{a^2+b^2})$ |
Eccentricity | $e=f/a=\sqrt{1+b^2/a^2}$ |
Directrix | $x=\pm a^2/f \lor y=\pm b^2/f$ |
Asymptotes | $y= \pm b·(x-x_∘)/a+y_∘$ |
$\text{II: $\pi/2\lt 0\le\pi$}$ | $\text{I: $0\le\theta\le\pi/2$}$ |
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$$ \begin{array}{cc} \sin(\theta):+ & \csc(\theta):+ \\ \cos(\theta):- & \sec(\theta):- \\ \tan(\theta):- & \cot(\theta):- \end{array} $$ | $$ \begin{array}{cc} \sin(\theta):+ & \csc(\theta):+ \\ \cos(\theta):+ & \sec(\theta):+ \\ \tan(\theta):+ & \cot(\theta):+ \end{array} $$ |
$\text{III: $\pi/2\lt\theta\lt3⋅\pi/2$}$ | $\text{IV: $3⋅\pi/2\le\theta\lt 2⋅\pi$}$ |
$$ \begin{array}{cc} \sin(\theta):- & \csc(\theta):- \\ \cos(\theta):- & \sec(\theta):- \\ \tan(\theta):+ & \cot(\theta):+ \end{array} $$ | $$ \begin{array}{cc} \sin(\theta):- & \csc(\theta):- \\ \cos(\theta):+ & \sec(\theta):+ \\ \tan(\theta):- & \cot(\theta):- \end{array} $$ |
Function | Period | Domain | Range | Graph |
---|---|---|---|---|
$$\sin(\theta)$$ | $$2·\pi$$ | $(-∞,∞)$ | $[-1,1]$ | |
$$\cos(\theta)$$ | $$2·\pi$$ | $(-∞,∞)$ | $[-1,1]$ | |
$$\tan(\theta)$$ | $$\pi$$ | $(-∞,∞),\medspace x\ne(2·n-1)·\frac{\pi}{2}\medspace \forall n \isin \Z$ | $(-∞,∞)$ | |
$$\cot(\theta)$$ | $$\pi$$ | $(-∞,∞),\medspace x\ne(2·n-1)·\frac{\pi}{2}\medspace \forall n \isin \Z$ | $(-∞,∞)$ | |
$$\sec(\theta)$$ | $$2·\pi$$ | $(-∞,∞),\medspace x\ne(2·n-1)·\frac{\pi}{2}\medspace \forall n \isin \Z$ | $|y|\ge 1$ | |
$$\csc(\theta)$$ | $$2·\pi$$ | $(-∞,∞),\medspace x\ne 2·n·\pi\medspace \forall n \isin \Z$ | $|y|\ge 1$ | |
$$\sin^{-1}(\theta)$$ | $$-$$ | $|x|\le 1$ | $|y|\le \pi/2$ | |
$$\cos^{-1}(\theta)$$ | $$-$$ | $|x|\le 1$ | $0\le y\le\pi$ | |
$$\tan^{-1}(\theta)$$ | $$-$$ | $(-∞,∞)$ | $|y|\le \pi/2$ | |
$$\cot^{-1}(\theta)$$ | $$-$$ | $(-∞,∞)$ | $0\lt y\lt\pi$ | |
$$\sec^{-1}(\theta)$$ | $$-$$ | $|x|\ge 1$ | $0\le y\le \pi, y\ne\pi/2$ | |
$$\csc^{-1}(\theta)$$ | $$-$$ | $|x|\ge 1$ | $|y|\le \pi/2,y\ne 0$ |
$$\sin(2·\theta)=2·\sin(\theta)·\cos(\theta)$$ | $$\cos(2·\theta)=$$ |
$$\csc(2·\theta)=\frac{1}{2}·\sec(\theta)·\csc(\theta)$$ | $$\sec(2·\theta)=$$ |
$$\tan(2·\theta)=$$ | $$\cot(2·\theta)=$$ |
$$\lim\limits_{x→a^+}f(x)$$ | The limit as a function approaches a point from the right |
$$\lim\limits_{x→a^-}f(x)$$ | The limit as a function approaches a point from the left |
$$\lim\limits_{x→a}f(x)$$ | The limit as a function approaches a point from the both sides |
Equation of a line | $\lim_{x→a}f(x)=f(a)=m·a+b$ |
Horizontal asymptote | The line $x=c$ is defined as $\lim_{x→c}f(x)=\pm ∞$ |
Vertical asymptote | The line $y=c$ is defined as $\lim_{x→\pm ∞}f(x)=c$ |
Notation | First | Second |
---|---|---|
Leibniz | $$\frac{dy}{dx} \medspace\text{or}\medspace \frac{d}{dx}f{x}$$ | $$\frac{d^2y}{dx^2} \medspace\text{or}\medspace \frac{d^2}{dx^2}f{x}$$ |
Langrange | $$f'(x)$$ | $$f''(x)$$ |
Newton | $$\dot{x}$$ | $$\ddot{x}$$ |
Euler | $$Df$$ | $$D^2f$$ |