Tables and formulas

Calculations with real numbers

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[[$ a+b=b+a,\quad ab=ba $]]	additive commutative property
[[$ a +(b+c) = (a+b)+c, \quad a(bc)=(ab)c \quad $]]	additive associative property
[[$ a(b+c)=ab+ac $]]	multiplicative associative property
[[$ a+(-a)=0 $]]	number a's opposite number –a
[[$ a \cdot \dfrac{1}{a}=1 $]]	number a's reciprocal[[$ \dfrac{1}{a} \quad (a \neq 0) $]]
[[$ \|a\| $]]	absolute value Graphical interpretation: [[$ \|a\| $]] = number a's points distance from zero

Calculations with fractions

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[[$ \dfrac{a}{b} = \dfrac{ka}{kb} \text{, where } k \neq 0 \quad $]]	expanding (→) and reducing (←)
[[$ \dfrac{a}{b} +\dfrac{c}{d} = \dfrac{ad+bc}{bd} $]]	addition (with expanding)
[[$ \dfrac{a}{b} -\dfrac{c}{d} = \dfrac{ad-bc}{bd} $]]	subtraction (with expanding)
[[$ \dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd} $]]	multiplication
[[$ \dfrac{a}{b} : \dfrac{c}{d} = \dfrac{ad}{bc} $]]	division

Powers

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[[$ a^n = a \cdot a \cdot ... \cdot a $]]	n factors, a = base number, n = exponent
[[$ a^0 = 1 $]]	[[$ a \neq 0$]], [[$0^0 $]] not defined
[[$ a^{-p} = \dfrac{1}{a^p} $]]	[[$ a \neq 0 $]]
[[$ \left( \dfrac{a}{b} \right)^{-p} = \left( \dfrac{b}{a}\right)^{p} $]]	[[$ a \neq 0 $]]

Calculation rules

[[$ a^ma^n = a^{m+n} $]]	product of powers with the same base number
[[$ \dfrac{a^m}{a^n} = a^{m-n} $]]	quotient of powers with the same base number
[[$ (ab)^n = a^nb^n $]]	power of products
[[$ \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$]]	power of a quotient
[[$ (a^m)^n = a^{mn} = (a^n)^m $]]	power of a power

Dividing a polynomial into factors

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[[$ ab + ac = a(b + c) $]]	common factor
[[$ ac + ad + bc + bd = a(c + d) + b(c + d) = (a + b)(c + d) $]]	grouping
[[$ a^2+2ab+b^2 = (a+b)^2 $]] [[$ a^2-2ab+b^2 = (a-b)^2 $]] [[$ a^2-b^2 = (a-b)(a+b) $]]	formulas

Square roots

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If [[$ \sqrt{a} = b$]], then [[$ b^2 =a $]] and [[$ b\geq 0$]] (applies both ways).

Calculation rules

[[$ (\sqrt{a})^2 = a$]]

[[$ \sqrt{a^2} = |a|$]]

[[$ \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$]]

[[$ \sqrt{ab} = \sqrt{a}\sqrt{b}$]]

Sequences

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Arithmetic number sequence

[[$d=a_2 -a_1$]]	difference number
[[$a_n = a_1 +(n-1)d$]]	general term

Geometric number sequence

[[$ q = \dfrac{a_2}{a_1}$]]	ratio
[[$ a_n =a_1q^{n-1} $]]	general term

Quadratic equations

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Standard form	[[$ ax^2+bx+c=0, \quad a\neq 1 $]]
Quadratic formula:	[[$ \quad x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} $]]

Parabola opening direction and shape:

If [[$ a > 0 $]], the parabola opens upwards.
If [[$ a < 0 $]], the parabola opens downwards.
If [[$ |a| $]] is small, the parabola is wide.
If [[$ |a| $]] is great, the parabola is narrow.

Incomplete quadratic equations

The number of solutions to equation [[$ ax^2 + c = 0 $]] depends on the constant [[$c$]]:

[[$c < 0$]]: two solutions, solutions of opposite numbers
[[$c= 0$]]: the only solution is [[$x = 0$]]
[[$c > 0$]]: no solution

Solutions of the equation [[$ ax^2 + bx = 0 $]]:

always two solutions, the other is always [[$x = 0$]].

Trigonometry of a right triangle

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[[$ a^2 + b^2 = c^2 $]] (Pythagorean theorem)

[[$ A = \dfrac{1}{2}ab $]]

Trigonometric functions

[[$ \text{sin} \space \alpha = \dfrac{a}{c}, \text{cos}\space\alpha = \dfrac{b}{c}, \text{tan} \space \alpha = \dfrac{a}{b} $]]

Lines

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The slope of a line passing through points [[$x_1, y_1 $]] and [[$x_2, y_2 $]]:

[[$ k = \text{tan} \space \alpha = \dfrac{y_2-y_1}{x_2-x_1} $]]

A line is

ascending if [[$ k > 0 $]]
descending if [[$ k < 0 $]]
parallel to the [[$x$]]- axis if [[$k = 0 $]]
parallel to the [[$y$]]- axis if [[$k$]] cannot be determined.

Consider the lines [[$s_1$]] and [[$s_2$]], whose slopes are [[$k_1$]] and [[$k_2$]].

The lines are parallel, i.e. [[$s_1||s_2$]], if [[$k_1=k_2$]] or lines are parallel to the [[$y$]]- axis.
The lines are perpendicular to each other, i.e. [[$s_1 \perp s_2$]], if [[$k_1 \cdot k_2 = -1$]] or one line is parallel to the [[$x$]]- axis and the other to the [[$y$]]- axi.

Standard form of the equation of a line:
[[$ax + by + c = 0$]]

Solved form of the equation of a line:
[[$y = kx + b$]], where [[$k$]] is the slope an [[$b$]] is the constant term (the [[$y$]] coordinate of the interesection point of the line and the [[$y$]]- axis).

The equation of a line parallel to the [[$x$]]-axis:
[[$y = t$]], where [[$t$]] is the [[$y$]] coordinate of the intersection point of the line and the [[$y$]] axis

The equation of a line parallel to the [[$y$]]-axis:
[[$x = u$]], where [[$u$]] is the [[$x$]] coordinate of the intersection point of the line and the [[$x$]] axis

Two-dimensional figures

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Square

[[$ \qquad A =a^2 $]]
[[$ \qquad d = \sqrt {2}a $]]

Rectangle

[[$ \qquad A =ab $]]
[[$ \qquad d = \sqrt{a^2+b^2} $]]

Rhombus

[[$ \qquad A =ah $]]

Parallelogram

[[$ \qquad A = ah = ab \space \text{sin} \space \alpha $]]

Trapezium

[[$ \qquad A = \dfrac{1}{2}(a+b)h = \dfrac{1}{2}(a+b)s \space \text{sin} \space \alpha $]]

Triangle

[[$ \qquad A= \dfrac{1}{2}ah = \dfrac{1}{2}ab \text{ sin } \alpha $]]

Circle

[[$ \qquad A = \pi r^2 = \dfrac{1}{4} \pi d^2 $]]
[[$ \qquad p = 2 \pi r = \pi d $]]

Sector

[[$ \qquad b = \dfrac{\alpha}{360°} 2\pi r $]] (length of arc)
[[$ \qquad A = \dfrac{\alpha}{360°} \pi r^2 = \dfrac{br}{2} $]]

Three-dimensional objects

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Cube

[[$ \qquad a = s\sqrt{2} $]]
[[$ \qquad d = s\sqrt{3} $]]
[[$ \qquad A=6s^2 $]]
[[$ \qquad V= s^3 $]]

Rectangular prism

[[$ \qquad d = \sqrt{a^2+b^2+c^2} $]]
[[$ \qquad A = 2(ab+ac+bc) $]]
[[$ \qquad V=abc $]]

Straight circular cone

[[$ \qquad A_v = \pi rs $]]
[[$ \qquad V = \dfrac{1}{3}\pi r^2 h $]]

Straight cylinder

[[$ \qquad A_v = 2 \pi rh $]]
[[$ \qquad A_{kok} = A_v + 2\pi r^2 = 2 \pi r(r+h) $]]
[[$ \qquad V = \pi r^2 h $]]

Ball

[[$ \qquad A = 4 \pi r^2 $]]
[[$ \qquad V = \dfrac{4}{3} \pi r^3 $]]

An approximation of [[$ \pi$]] to the first 500 decimal places

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[[$ 3, 14159 \quad 26535 \quad 89793 \quad 23846 \quad 26433 \quad 83279 \quad 50288 \quad 41971 \quad 69399 \quad 37510 \quad 58209 \quad 74944 \quad 59230 \quad 78164 \quad 06286 \quad 20899 \quad 86280 \quad 34825 \quad 34211 \quad 70679 \quad 82148 \quad 08651 \quad 32823 \quad 06647 \quad 09384 \quad 46095 \quad 50582 \quad 23172 \quad 53594 \quad 08128 \quad 48111 \quad 74502 \quad 84102 \quad 70193 \quad 85211 \quad 05559 \quad 64462 \quad 29489 \quad 54930 \quad 38196 \quad 44288 \quad 10975 \quad 66593 \quad 34461 \quad 28475 \quad 64823 \quad 37867 \quad 83165 \quad 27120 \quad 19091 \quad 45648 \quad 56692 \quad 34603 \quad 48610 \quad 45432 \quad 66482 \quad 13393 \quad 60726 \quad 02491 \quad 41273 \quad 72458 \quad 70066 \quad 06315 \quad 58817 \quad 48815 \quad 20920 \quad 96282 \quad 92540 \quad 91715 \quad 36436 \quad 78925 \quad 90360 \quad 01133 \quad 05305 \quad 48820 \quad 46652 \quad 13841 \quad 46951 \quad 94151 \quad 16094 \quad 33057 \quad 27036 \quad 57595 \quad 91953 \quad 09218 \quad 61173 \quad 81932 \quad 61179 \quad 31051 \quad 18548 \quad 07446 \quad 23799 \quad 62749 \quad 56735 \quad 18857 \quad 52724 \quad 89122 \quad 79381 \quad 83011 \quad 94912 $]]

Mathematical statistics

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Means

Mean	[[$ \bar{x} = \dfrac{x_1+x_2+x_3+...+x_n}{n} $]]
Weighted average	[[$ \bar{x} = \dfrac{q_1x_1+q_2x_2+...+q_nx_n}{q_1+q_2+...+q_n} $]], where [[$ q_1, q_2,...,q_n $]] are wheights.

The mode, or type value, means the most common, most common value of a variable.

The median means the mean value (or the average of the two mean values) when the data is arranged in order of magnitude.

Scatter numbers

The standard deviation indicates how far the values of the variable are from the mean on average.

The range indicates how in which interval the observations vary.

The length of the range is the difference between the maximum and minimum values of the variable.

Probability

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Classical probability	[[$ P(A)=\dfrac{\text{number of favorable cases}}{\text{number of all cases}} $]]
Probability of a counter-even	[[$ P(\bar{A}) = P(A \text{ does not occur)} = 1-P(A) $]]

Addition rule

When A and B are separate cases	[[$ P(A \text{ or } B) = P(A) + P(B) $]]
When A and B are not separate	[[$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$]]

Multiplication rule

When A and B are independent	[[$ P(A \text{ and } B) = P(A) \cdot P(B) $]]
When A and B are dependent (general multiplication rule)	[[$P(\text{first } A \text{ and then }B) = P(A) \cdot P(B, \text{when A} \text{ has happened})$]]

The SI system

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Units of measure
Name	Symbol	Coefficient	Name	Symbol	Coefficient
exa	E	[[$ 10^{18}$]]	deci	d	[[$10^{-1}$]]
peta	P	[[$ 10^{15}$]]	centi	c	[[$10^{-2}$]]
tera	T	[[$ 10^{12}$]]	milli	m	[[$10^{-3}$]]
giga	G	[[$10^{9}$]]	micro	μ	[[$10^{-6}$]]
mega	M	[[$10^{6}$]]	nano	n	[[$10^{-9}$]]
kilo	k	[[$10^{3}$]]	pico	p	[[$10^{-12}$]]
hecto	h	[[$10^{2}$]]	femto	f	[[$10^{-15}$]]
deca	da	[[$10^{1}$]]	atto	a	[[$10^{-18}$]]

Additional units
Quantity	Unit	Symbol	Equivalency
time	minute	min	1 min = 60 s
	hour	h	1 h = 60 min
	day	d	1 d = 24 h
	year	a	1 a [[$\approx $]] 365 d
plane angle	degree	°	1° = 60'
	minute	'	1' = 60''
	second	''
volume	litre	l	1 l = 1 dm³
mass	tonne	t	1 t = 1000 kg
	atomic mass unit	u	1 u = 1.6605402 [[$\cdot$]] 10^-27 kg
length	astronomical unit	AU	1 AU = 0.1495979 [[$\cdot$]] 10¹² m
	parsec	pc	1 pc = 30.85678 [[$ \cdot$]] 10¹⁵ m

**Conversion factors**
Length	1 ″ = 1 in = 1 inch = 25.40 mm
	1 ′ = 1 ft = 1 foot = 0.3048 m
	1 yd = 1 yard = 0.9144 m
	1 mi = 1 mile = 1.609344 km
	1 NM = 1 M = 1 nautical mile = 1852 m
	1 light year = 9.46055 [[$ \cdot$]] 10¹⁵ m
	1 AU = astronomical unit = 149.5979 [[$ \cdot $]] 10⁹ m
Mass	1 ct = 1 carat = 0.2 g
	1 u = 1.6605402 [[$\cdot$]] 10^-27 kg
	1 lb = 1 pound = 0.4536 kg
	1 oz = 1 ounce = 28.35 g
Plane angle	1° = 2[[$ \pi / $]]360 rad
Area	1 b = 1 barn = 10^-28 m²
	1 acre = 4.0469 [[$ \cdot $]] 10³ m²
Volume	1 l = 1 dm³ = 0.001 m³
	1 bbl = 1 barrel = 0.1589873 m³
	1 gal = 1 gallon (UK) = 4.546092 l
	1 gal = 1 gallon (US) =3.785412 l
Speed	1 knot = 1 NM/h = 1.852 km/h = 0.5144 m/s

Physical constants
Name	Symbol	Numeric value and unit
acceleration of free fall	[[$g$]]	[[$ \text{9.80665} $]] m/s²
light speed	[[$c$]]	[[$ \text{2.99792458}\cdot 10^{8}$]] m/s
electron mass	[[$m_e$]]	[[$ \text{9.1093897}\cdot 10^{-31}$]] kg
proton mass	[[$m_p$]]	[[$ \text{1.6726231} \cdot 10^{-27}$]] kg
neutron mass	[[$m_n$]]	[[$\text{1.6749286} \cdot 10^{-27}$]] kg