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 | $$ \frac{-1}{1}=\frac{1}{-1}=-\frac{1}{1}=-1 $$ |
Distribution of two negatives | $$ \frac{-1}{-1}=-(-\frac{1}{1})=1 $$ |
For expressions | $$ a=\frac{1}{1/a} $$ |
For evaluating fractions | $$ \frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c} $$ |
Real $$\R$$ | All common numbers and multiples of one. |
Integer $$\Z$$ | Real whole numbers that can be expressed without using a fraction. $$(-2, -1, 0, 1, 2)$$ |
Natural $$\N$$ | Positive real integers. It is argued whether or not zero is included. |
Rational | Any number that can be represented as a fraction inclusive of only natural numbers. Always either repeats or terminates as a decimal. $$1/2, 2/1, 1.2, 2.\overline{1}$$ |
Irrational | Any number that cannot be represented as a fraction. Never repeats or terminates as a decimal. $$ \sqrt{2}, e, \pi, \phi $$ |
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: nth 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) $$x=\frac{-b \pm \sqrt{b^2+4·a·(y-y_∘)}}{2·a}$$ |
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. |
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. |
$$i^{-2}=i^2=i^6=-1$$ | $$i^{-4}=i^0=i^4=1$$ |
$$i^{-3}=i^1=i^5=i$$ | $$i^{-1}=i^3=i^7=-i$$ |
$$z=x+y·i$$ | $$z^*=x-y·i$$ |
$$x^3=1·x·x·x$$ | $$3^3=27$$ |
$$x^2=1·x·x$$ | $$3^2=9$$ |
$$x=1·x$$ | $$3^1=3$$ |
$$x^0=1$$ | $$3^0=1$$ |
$$x^{-1}=1/x$$ | $$3^{-1}=1/3$$ |
$$x^{-2}=1/(x·x)$$ | $$3^{-2}=1/9$$ |
$$x^{-3}=1/(x·x·x)$$ | $$3^{-3}=1/27$$ |
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 |
$$\log_b x = \frac{1}{log_x b}$$ | $$\log_{c^n} x=\frac{\log_c x}{n}$$ | $$a^{\log_b x}=x^{\log_b a}$$ |
Perimeter | $$a+b+c$$ |
Area (non-obtuse) | $$\frac{1}{2}·b·h$$ |
Sum of angles | $$\pi$$ |
$$\sin(\theta)=\frac{\text{opp}}{\text{hyp}}$$ | $$\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$$ |
$$\csc(\theta)=\frac{\text{hyp}}{\text{opp}}$$ | $$\sec(\theta)=\frac{\text{hyp}}{\text{adj}}$$ |
$$\tan(\theta)=\frac{\text{opp}}{\text{adj}}$$ | $$\cot(\theta)=\frac{\text{adj}}{\text{opp}}$$ |
Perimeter | Sum of side lengths |
Area | Sum of areas of triangles divided within shape |
Sum of angles | $$(\text{sides}-2)·\pi$$ |
Perimeter | $$n·s$$ |
Area | $$\frac{n·s·h}{2}$$ |
Sum of angles | $$(n-2)·\pi$$ |
Angle (h,r) | $$\frac{\pi}{n}$$ |
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 | $$\frac{\theta·r^2}{2}$$ |
Chord length (k) | $$2·r·\sin\Big(\frac{\theta}{2}\Big)$$ |
Segment area | $$\frac{r^2}{2}·\Big(\theta-\sin\Big(\frac{\theta}{2}\Big)·\cos\Big(\frac{\theta}{2}\Big)\Big)$$ |
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 | $$\text{None}$$ |
$$b=r·\sin\Big(\frac{\theta}{2}\Big)$$ | $$h=r·\cos\Big(\frac{\theta}{2}\Big)$$ |