Tables and formulas
Calculations with real numbers
Calculations with fractions
[[$ \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} $]] | division |
Powers
[[$ 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 $]] | power of a power |
Dividing a polynomial into factors
[[$ 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) $]] |
formulas |
Square roots
Calculation rules
[[$ (\sqrt{a})^2 = a$]]
[[$ \sqrt{a^2} = |a|$]]
[[$ \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$]]
[[$ \sqrt{ab} = \sqrt{a}\sqrt{b}$]]
Sequences
Arithmetic number sequence
[[$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} $]] | general term |
Quadratic equations
Standard form | [[$ 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$]]: no solution
- always two solutions, the other is always [[$x = 0$]].
Trigonometry of a right triangle
Lines
[[$ 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$]] cannot be determined.
- The lines are parallel, i.e. [[$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$]]- axi.
[[$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$]] axis
Two-dimensional figures
Square
[[$ \qquad A =a^2 $]]
[[$ \qquad d = \sqrt {2}a $]]
Rectangle
[[$ \qquad A =ab $]]
[[$ \qquad d = \sqrt{a^2+b^2} $]]
Rhombus
Parallelogram
[[$ \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 $]]
Circle
![Circle.](https://peda.net/qis/2022-2023/mathematics/matematiikka-8/taulukko-osio/tl/taulukko_ympyra-png:file/photo/d61fee57919c6723f676bfec54e9dd229f1b479e/taulukko_ympyra.png)
[[$ \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} $]]
Three-dimensional objects
Cube
[[$ \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$]] to the first 500 decimal places
Mathematical statistics
Means
Mean |
[[$ \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 $]] are wheights. |
The mode, or type value, means the most common, most common value of a variable.
The median means the mean value (or the average of the two mean values) when the data is arranged in order of magnitude.
Scatter numbers
The standard deviation indicates how far the values of the variable are from the mean on average.
The range indicates how in which interval the observations vary.
The length of the range is the difference between the maximum and minimum values of the variable.
Probability
Classical probability |
[[$ 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})$]] |
The SI system
Name | Symbol | Coefficient | Name | Symbol | Coefficient |
---|---|---|---|---|---|
exa | E | [[$ 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 | [[$10^{-18}$]] |
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$]] 1015 m |
Length | 1 ″ = 1 in = 1 inch = 25.40 mm |
---|---|
1 ′ = 1 ft = 1 foot = 0.3048 m | |
1 yd = 1 yard = 0.9144 m | |
1 mi = 1 mile = 1.609344 km | |
1 NM = 1 M = 1 nautical mile = 1852 m | |
1 light year = 9.46055 [[$ \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 $]] 103 m2 | |
Volume | 1 l = 1 dm3 = 0.001 m3 |
1 bbl = 1 barrel = 0.1589873 m3 | |
1 gal = 1 gallon (UK) = 4.546092 l | |
1 gal = 1 gallon (US) =3.785412 l | |
Speed | 1 knot = 1 NM/h = 1.852 km/h = 0.5144 m/s |
Name | Symbol | Numeric value and unit |
---|---|---|
acceleration of free fall | [[$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 |