www.wolframalpha.com        Examples:

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3.5 days -> hours  |  10000 sec

minimal coin form 26.85 euros

1 euro  |  image euro

Visualizzare come risolvere un problema di sottrazione con l'aiuto della linea dei numeri: "Il cinema Star ha 216 posti. 148 persone sono già occupati. Quanti posti sono ancora liberi?" Visualization of how to solve a subtraction problem with the help of the number line: "The Star cinema has a total of 216 seats and 148 persons are already sitting there. How many seats are stll free?"

 places values of 6135   |   bar chart 6,1,3,5

6,7,8,9,10,11,12,13 | line (7, 12) | brown polygon (7,0),(12,0),(12,1),(7,1)
(A bar of chocolate ranges from 7cm to 12cm. 7 to 12: 1, 2, ..., 6. Is the bar 6 cm long?)

12-7     |     a=15, b=7, c=2, a-b*c
 912/7 remainder
 simplify 102/360   |  simplify sqrt(x^2) if x < 0

plot (2,1),(2,2),(2,3), (-3,-3),(-2,-2),(-1,-1), (-1,3),(-3,3),(-5,3) color red

line segment (0,4), (2,0), line segment (2,0),(8,2), line segment (8,2),(0,4)
Posso disegnare solo 3 figure con comandi esepliciti; ma con una descrizione algebrica non ho questi limiti: vedi
I can only draw 3 figures with explicit commands; but with an algebraic description I don't have these limits: see

line segment(0,0),(cos 6°,sin 6°), line segment(0,0),(cos 98°,sin 98°), line segment(0,0),(cos(((6+98)/2)°),sin(((6+98)/2)°))
bisettrice / bisector

 L'asse del segmento di estremi (1,3),(-2,-3) The perpendicular bisector of the line segment with endpoints (1,3),(-2,-3) punto medio di [midpoint of]  (a,b), (p,q):  (u,v) u =(a+p)/2, v = (b+q)/2 segment line (a,b),(c,d), line (u+(b-q), v+(p-a)),(u-(b-q), v-(p-a))

a = 1, b = 3, p = -2, q = -3, u = (a+p)/2, v = (b+q)/2

interval (0, 4), (2, 6], [4, 11]

multiplication table 10

from 0 to 90 by 7

floor( 123456/17, 100 )  |  round( 123456/17, 100 )  |  round( 123456/17, 0.01 )

round(5138 / 12746 * 100, 0.01) %

accuracy | interval arithmetic    vedi/see,   vedi/see   or:

A car, traveling at constant speed, covers 240±2 m in 12±0.2 s. What is its speed?    20 ± 0.5 m/s
minmax x/y if( abs(x-240)<2, abs(y-12)<0.2)
I need to perform the following calculation with rounded values:  13.7 · 0.096 / 2.45    [0.531, 0.543] = 0.537 ± 0.006
minmax x*y/z if (abs(x-13.7)<0.05, abs(y-0.096)<0.0005, abs(z-2.45)<0.005)

cuboid surface area  |  complementary 39°   (rectangular cuboid = parallelepipedo rettangolo)

angle 850 °

(0.333...) * 2 + 4/3 + 2.5   |   (sqrt(20)+sqrt(80))/(2*sqrt(15))

8/(6-2)*(2/3)/4-(3/2/(2*5)+6)  |  8/(6-2)*(2/3)/4-(3+2/(2*5)+6)

 Trees to represent thestructure of terms: vedi/see A->B, A->C, A->D, B->E, B->F, D->G, D->H, D->L, F->M, F->P, L->R | directed

(5+?)*8 = 100

f(-2)  if  f(x) = x^3 + 3*x^2 - 2

x^3 + 3*x^2 - 2  if  x = -3, -2, -1, 0, 1, 2, 3

solve x^3 + 3*x^2 - 2 = 0 using bisection method for 0 <= x <= 2 with 1 digits precision

 bisection method x^3 + 3*x^2 - 2 = 0 if 0 < x < 2

sqrt(300)+sqrt(500), 12 digits | sqrt(300)+sqrt(500), 13 digits | round( sqrt(300)+sqrt(500), 0.1)

 0.2999...
 3/8 -> % tape diagram 3/8  diagramma a nastro
 5/6 -1/6

7/8  |  mixed fraction 12.6  |  66.66...%

value of 7 in 351.8705

number line 0, sqrt(2), pi, 5

integer ^ (positive integer)

double of 3/4 liter | ratio of 0.4 and 1.2| -10, 0, 10/3, 2*10/3,10 | rational number | is sqrt(10) rational?   vedi/see

An irrational number is a non-rational number, it is not a number whose digits follow each other without rules. Two examples:
0.10100001000000000100000000000000001...
0.202002000200002000002000000200000002...

for n=1 to 3 sum(1/10^(n*n))  |  for n=1 to oo sum(1/10^(n*n))

for n=1 to oo sum(2/10^(n*(n+1)/2))

relative difference

75/100, 2/3, 3/4, 7/10, 5/8 | sort 75/100, 2/3, 3/4, 7/10, 5/8

sort("maria","luigi","dario","piera","rosa","alfonso","mario")

sort | median | mode | mean 1,5,4,2,3,3,2,4,5,11,4,11,20,1,2,3,1,4,2
 dot plot {1,5,4,2,3,3,2,4,5,11,4,11,20,1,2,3,1,4,2}

{{0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0}, {1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1}}
a block diagram / un diagramma a barre

 Posso usare anche semplici script      I can use simple scripts too                   vedi /see

geometric mean | geometric mean 89/100, 70/100, 51/100

1,1,2,2,2,2, 3 ,3,3,3,4,4, 4 ,4,5,5,6,6,7, 8 ,9,10,11,11,18,23
length | quartiles | interquartile range     For large amounts of data  vedi/see

vedi/see

translate chart from English to Italian | translate grafico from Italian to English

compare italiano, english

Morse code MACOSA

pie chart(1,2,3)   or   pie chart 1,2,3

Bar Chart(1,2,3)   or   bar chart 1,2,3

floor( (1, 2, 3)/sum(1, 2, 3)*100)  |  round( (1, 2, 3)/sum(1, 2, 3)*100, 0.1)

Bar Chart(1/6*100, 2/6*100, 3/6*100)  /  Bar Chart(16.6667, 33.3333, 50)
 1000*(1+20%)*(1+10%)   20% increase + 10% increase = 32% increase

stem and leaf plot 25,35,10,17,29,14,21,31,23,12,28,41,8     or:
staistics 25,35,10,17,29,14,21,31,23,12,28,41,8     (click "more" if necessary)

histogram{156,168,162,150,167,157,170,157,159,164,157,165,163,165,160,163,162,155}
histogram [{156,168,162,150,167,157,170,157,159,164,157,165,163,165,160,163,162,155},1]

1/6+5/9
È scritto anche in notazione egiziana / also in Egyptian notation: 1/2 + 1/5 + 1/45   vedi/see
convert 1,2,3,4,5 to roman numerals | convert ... to babylonian numerals | ... mayan numerals | ... greek numerals

convert 1752 to Babylonian

1563 to roman

(1 - 4/3 + 2/5) / (3/2 - 4/5 + 1/6)

19:32:55

this month  |  now

(A or B) and (not C)

 polyomino vedi/see

{{0,0,1,1,0,0},{0,1,0,0,1,0},{1,0,0,0,0,1},{1,0,0,0,0,1},{1,0,0,0,0,1},{1,1,1,1,1,1}}

acute triangle, obtuse triangle, right triangle

7, 2, 8 triangle    (angles are in rad; if you clik 0.918336 rad you have 52.62°)   vedi/see

triangle side 33, angle 12, angle 90

or     SSS 7 8 9       SAA 33 12deg 90deg

triangle (0,0),(2,3),(8,-1)   or  polygon (0,0),(2,3),(8,-1)

distance from (2,3) to line (0,0),(8,-1)

parallelogram side lengths 3,4 angle 30 degrees

triangle area  |  trapezium area  |  trapezoid 4, 5, 2, 2.5

rectangle (0,2),(4,-2),(7,1),(3,5), triangle (0,0),(0,4),(-3,0), circle (0,4),(0,0),(3,5)   or
polygon (0,2),(4,-2),(7,1),(3,5), polygon (0,0),(0,4),(-3,0), circle (0,4),(0,0),(3,5)
circle (1,3),(1,5),(4,3), rectangle (1,3),(4,5),(4,5),(1,3), rectangle (1,5),(4,3),(1,5),(4,3)

I can draw three figures on the same Cartesian plane
recatangle (a,b),(c,d),(c,d),(a,b) is the segment from (a,b) to (c,d)
 max(abs(x-10),abs(y-5))=7, max(abs(x-10),abs(y-5))=5, max(abs(x-10),abs(y-5))=3

minmax sqrt((x+y-z)(x-y+z)(-x+y+z)(x+y+z))/4*0.03 if (10-0.05<x<10+0.05, 14-0.05<y<14+0.05, 8.2-0.05<z<8.2+0.05)
With a cardboard weighing 300 g per m² I build a triangle with sides of 10.0, 14.0 and 8.2 cm. How much does it weight?   [1.21±0.02 g]

number line |x+5| > 3  |  number line |x+5| >= 3

line, slope=2/3, x-intercept=4  |  atan(2/3)

line (-1, 3), (4, -0.5)

complete the square 6x^2+10x+28

4*K+3 = 9

solve A = (B+b)*h/2 for b

distribute a*x+x*c+c*a

plot y = x^2+a*x+1 for a = -1, 0, 1, 2, 3, -4 < x < 5, -2 < y < 7

plot y = a*x^2+x+1 for a = -1, 0, 1, 2, 3, -4 < x < 5, -2 < y < 7
plot y = x^2+x+a for a = -1, 0, 1, 2, 3, -4 < x < 5, -2 < y < 7

handwritten style (5/2 + 3) - 7 = -1.5  |  handwritten style solve x/3 - 1 = 0.2 for x

plot {3/(1-x), 2/x}, -2 < x < 3   |   solve 3/(1-x) > 2/x  |  plot 3/(1-x) > 2/x

The period of revolution in years T of the planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto) and their average distance from the Sun D, taken as the unit of measure the Sun-Earth distance: the relationship that links these two variables is of the type D^3 = k*T^2.  T = (0.241, 0.615,1.000, 1.881, 11.861,29.457,84.008, 164.784,248.350)  D = (0.387, 0.723, 1.000, 1.523, 5.203, 9.541, 19.190, 30.086, 39.507)
(0.387,0.723,1.000,1.523,5.203,9.541,19.190,30.086,39.507)^3 / (0.241,0.615,1.000,1.881,11.861,29.457,84.008,164.784,248.350)^2
{0.997927, 0.999228, 1, 0.99844, 1.0012, 1.00093, 1.00134, 1.00291, 0.999756}

(10 , 20, 30, 40, 50, 60, 70, 80, 90, 100)^3 | plot V = L^3, V = 500, V = 1000, 0 < L < 10

intersect y = |x| and y = x^2

inverse of y = sqrt(x + 1/x)

subsets of {a,b,c}

100°C -> °F | 0°C -> °F | 99°F -> °C

circle, diameter=10

circle, area 50 cm^2

soccer ball

plot { (0, 41), (26, 63), (35, 50), (49, 84) }
 Puoi aggiungere i comandi sotto al grafico e copiare l'immagine facilmente usando il bottone "Customize" e i "click" opportuni.    You can add commands at the bottom of the graph and copy the image easily using the "Customize" button and the appropriate "clicks".     Puoi cambiare le dimensioni dell'immagine.     You can change the size of the image.

plot {(0, 12), (11, 17), (15, 29), (19, 24),(31,43),(48,109),(56,163),(66,192),(83,325),(101,317),(114,389),(121,397),(132,473),(158,806)}

plot {(8,0);(8,20);(8,12);(10,12.5);(11,14.6);(12,17.2);(12.5,18);(13,19);(13.5,19.3);(14,19);(14.5,18);(15,17);(15.5,15.1);(16.5,12.8);(18,12.4)}

Nei grafici per punti non è sempre facile includere la quota y=0. Un semplice trucco è inseire all'inizio un punto alla quota 0 e uno oltre la quota massima.
Poi, volendo, il segmento verticale aggiunto può esere cancellato.
In point plots it is not always easy to include the y=0 axis. A simple trick is to insert points at the start at the quota 0 and one beyond the maximum quota.
Then, if desired, the added vertical segment can be deleted.

plot {(1,0), (1,200),(1,101.8),(2, 74.0),(3,81.7),(4,88.0),(5,72.4),(6, 58.2),(7,24.2),(8,69.3),(9,136.4),(10,171.3),(11,108.8),(12,93.1)}
plot {(1,0), (1,200),(1,97.5),(2, 109.9),(3,78.2),(4,65.1),(5,36.2),(6, 17.9),(7,6.7),(8,31.8),(9,65.3),(10,105.6),(11,117.5),(12,123.7)}

Questo "trucco" può essere impiegato anche per tracciare grafici per punti con la stessa scala (es: i grafici dei mm di pioggia mensili in due diverse città).
This "trick" can also be used to draw point plots with the same scale (eg: graphs of monthly rainfall in two different cities).

Come costruire istogrammi con intervalli di diversa ampiezza.  /  How to build histograms with intervals of different widths.  Two examples.
In una certa città, nel 1970, gli abitanti con meno di 15 anni sono il 18%, quelli col almeno 15 anni e meno di 25 il 16.5%, quelli con almeno 25 e meno di 55 il 50%, quelli con almeno 55 e meno di 65 il 10%, quelli con almeno 65 il 5.5%.
In a certain city, in 1970, the inhabitants under the age of 15 are 18%, those with at least 15 years and under 25 16.5%, those with at least 25 and under 55 50%, those with at least 55 and less than 65 10%, those with at least 65 5.5%.
 Istogrammi di questo tipo sono facilmente realizzabili con degli script / Histograms of this kind are easily made with scriptsvedi /see

plot (0;0),(0;18/15),(15;18/15),(15;0),(15;16.5/10),(25;16.5/10),(25;0),(25;50/30),(55;50/30),(55;0),(55;10/10),(65;10/10),(65;0),(65;5.5/35),(100;5.5/35),(100;0),(0;0)

 In Italy in 1951, in the age intervals [0,5),[5,10),[10,20),[20,30),[30,40),[40,50),[50,60),[60,75),[75,100) are dead 729,35,77,132,134,285,457,1401,1569 thousand people. The histogram represents the unitary percentage frequencies: the percentage frequencies divided by the amplitude of each interval, so that the area of each vertical rectangle represents the percentage frequency of the outputs that fall in the interval that is the base. The sum of the areas of the rectangles is 100. (see here)
plot (0;0),(0;3.03),(5;3.03),(5;.15),(10;.15),(10;.16),(20;.16),(20;.27),(30;.27),(30;.28),(40;.28),(40;.59), (50;.59),(50;.95), (60;.95),(60;1.94), (75;1.94),(75;1.3), (100;1.3),(100;0),(0;0)

area (1,3.5), (3,5), (4,3), (2.7,2), (3.5,1), (1.6,2) | perimeter (1,3.5), (3,5), (4,3), (2.7,2), (3.5,1), (1.6,2)

plot (5,15),(15,5),(50,5),(70,15),(30,15),(60,30),(25,65),(25,15),(5,15)  | polygon ... ,(25,15) color red
polygon (5,15),(15,5),(50,5),(70,15),(30,15),(60,30),(25,65),(25,15); polygon (0,0),(0,0)

area = 1412.5     perimeter = 5 (29 + 9 sqrt(2) + 5 sqrt(5)) ≈ 264.541

 By adding a second command "polygon P,P" I avoid the appearance of the coordinates of the points and I can make the image contain the point P ↑ By adding another "polygon(150,150),(150,150)" I can enlarge the part of the plane represented and then reduce the drawing →

angle between (0.6,-1.5) and (2,1.5)     or   angle between vectors (0.6,-1.5) and (2,1.5)

angle between (10,30) and (15,15) | angle between (10,30) and (30,10)

0.12323...

As we have seen above, every number with a terminating decimal representation (0.3) is an abbreviation of a repeating decimal representation with a repetend of 0 (0.3000...) or with a repetend of 9 (0.2999...).  0.3000... in base 2 becomes 0.01001100110011... (the repeating sequence is 0011, or 1100 or 0100 or 1001). See the command [base = ].

(2.33...)/(0.71414...)*(41.66...)

100

number name 1264857.03

one million two hundred sixty-four thousand eight hundred fifty-seven point zero three

place values 1234567.123

population of Italy | population of United Arab Emirates

shoe size

car  |  bicycle  |  motorcycle

Roman Empire

 Ukraine  |  Ukrainian  |  Ukraine regions

Genova coordinate

La Spezia - Imperia

caves in Sicilia

sun, May 7, 2012, Genoa, Italy

weather today

temperature Genoa 2021

calendar 1980

7/8/1971 to 7/8/2011

blue+yellow

fontina 100 g

weight 1 cup of flour, 1 cup of milk, 1 cup of tea

biology

brain

puma, lion, cat

growth curve fir tree   (abete)

SI prefix
 libra=? kg  |  foot=? cm  |  can=? l  |  pint=? l  |  Kelvin=? Celsius

chemical elements | chemistry

caffeine
 molecular mass of water

 atomic mass of He

Cu + H2SO4 -> CuSO4 + H2O + SO2

ISBN 9788836548033

QR code "Genova"

F# G G A G C B G G A G D C  |  music theory  |  intervals C3 Eb3 Bb3 D4 G4 C3 Eb3 Bb3 D4 G4

m__n

fr_ct___

words start q end w

words start italia

words contain division

 pronunciation of mean IPA:  vedi/see

words contain division   vedi/see

Emoticon. Introduci:
:)   :-)   :<)   B-)   O:-)   %-)   :S   :\$   :-X   :*   <3   :@
:(   :-(   :'-(   :|   :-||   :-]   :}   :>   :-/   :-o   >:3   >;3

Ada Lovelace | Anne Isabella Byron | Archimedes | Bertrand Russell | Enrico Fermi | Euler | Newton | Galileo Galilei | Godel | Pythagoras

Giotto | artworks of Giotto | ... Leonardo | ... Raffaello | ... Monet | ... Renoir | ... Van Gogh | ... Kandinsky | ... Matisse | ... Klee | ... Picasso | ... Modigliani | artworks of Giorgio De Chirico

 James Bond movies ...

mobile phone | phone

mother's brother's son (il figlio del fratello della madre) | father's mother's brother's son (il figlio del fratello della madre del padre)

didactics

triangle vertices (1,1),(2,4),(3,3)

circle (1,0.5), (2,4), (3,3)   |   incenter (1,0.5), (2,4), (3,3)

circumscribed circle of 13,14,15 triangle  |  inscribed circle ...

circle (1,0.5), (2,4), (3,3) and line (0,0),(2,1)

formula area of a regular n-gon

 diagonals of a regular 6-gon

divisors 20  |  divisors 20,16,28,180  |  gcd(20,16,28,180)  |  lcm(20,16,28,180)

factor 70560    |    factor 2^257-1

primes from 40 to 100   |   primes near 500000

primes of the form 100k+1

Is 5^(1/3) a rational number?

pi

17.03 from base 10 to base 2

growth curve female

3 dice

RandomInteger[(1,6),170]+RandomInteger[(1,6),170]   |   2 dice

 ```Stem | Leaves 0 | 9 1 | 066889 2 | 255688899 3 | 0112234455677788899 4 | 00011122444455556667788889 5 | 0000001112334445556777777888899 6 | 001113333445555566777778888999 7 | 000000111222233455666777778889999999 8 | 001111111122222222223444445555666666777778889999999 9 | 000000001112222333344445555566666777777788899999999 10 | 00000111122222233334445555666677788889999 11 | 0000111222223344555566667799999999 12 | 000001222333444444555556677778888999999 13 | 001111222344444555555566677788888999 14 | 000002222333555556666666778889 15 | 0011123334455667899 16 | 0112233444566799 17 | 0012244566777999 18 | 0111569 19 | 03 stem units: 1 Stem | Leaves 1 | 779 2 | 013356677778899999 3 | 00112222222333333334455555666666666666677777888999999 4 | 01112223333445555666799 5 | 057 stem units: 10```
(RandomReal [{0,10}, 500]+RandomReal [{0,10}, 500])
(RandomReal[{0,10},100]+RandomReal[{0,10},100]+RandomReal[{0,10},100]+ RandomReal[{0,10},100]+ RandomReal[{0,10},100]+ RandomReal[{0,10},100]+ RandomReal[{0,10},100])

 (0.42051, 0.309013, ..., 1.76063, 0.914661) RandomReal[{0,1},3000]+RandomReal[{0,1},3000] Vedi qui come rappresentare questi 3000 valori in un istogramma See here how to represent these 3000 values in a histogram

one die, 200 number tossies  |  one die, 1000 number tossies  |  one die, 10000 number tossies

probability union events

Pascal's triangle

subset(15) | 15! | 1..15 | Tuples[{1,2,3},2} | permutaion (15,4) | permutaion (15,4)/4! | C(15,4)

continued fraction

solve x^2-x=3.2

handwritten style solve cos(x) = sqrt(3)/2

solve x^500 - x + 0.4 = 0   |   plot x^500 - x + 0.4
"solve" does not always provide the solutions. What to do?   See here

plot {x*y=0, y^2+x^2-x=1, y=x^2, y^2=x}  |  x^2+y^2=1, (x-2)^2+(y-1)^2=3

circle (x-3.3)^2+(y-0.3)^2=1.2, circle (1,-1),(2,1),(0,0)  |  circle (1,-1),(2,1),(0,0)
circle (x-3.3)^2+(y-0.3)^2=1.2, circle (x-7/6)^2+(y-1/6)^2=25/18, segment (3.3,0.3),(7/6,1/6)

I know the radii of two circles and the distance of the centers; how to draw the tangent to both

solve x/2.5 = (x-6.3)/1   (10.5)   |  2.5/sqrt(10.5^2-2.5^2)   (0.2451)
plot {x^2+y^2-2.5^2=0, (x-6.3)^2+y^2-1=0,0.2451*(10.5-x) =y, x=0, y=0, x=6.3}, -3 < x < 11, -4 < y < 4

chord, apothem, sagitta  |  chord, radius, angle

intercepts of y^2=(x+1)*(x-1)^2

plot y = |x^3 - 2x^2 - 16x + 6|+x-3, -4 <= x <= 6  |  corners |x^3 - 2x^2 - 16x + 6|+x-3

Come restringere il dominio di una funzione a un intervallo J? Un trucco: aggiungere un termine che vale 1/0 al di fuori di J
How to restrict the domain of a function to an interval J? A trick: add a term that is 1/0 outside of J
Per restringermi a [A,∞) o a (-∞,B] posso aggiungere  /  To restrict to [A,∞) or (-∞,B] I can add
+1/sqrt((sgn( A -x)-1)^2)-1/abs(sgn( A -x)-1)
+1/sqrt((sgn( B -x)-1)^2)-1/abs(sgn( B -x)-1)

I grafici di x → x^2 e di x → 2*x restringendo il dominio della prima funzione a (-2,∞), (1,∞), (-∞,2), (-3,2)
plot y=x^2+1/sqrt((sgn( A -x)-1)^2)-1/abs(sgn( A -x)-1), y=2x, x = -6..4 for A = -2
plot y=x^2+1/sqrt((sgn( A -x)-1)^2)-1/abs(sgn( A -x)-1), y=2x, x = -6..4 for A = 1
plot y=x^2+1/sqrt((sgn( B -x)+1)^2)-1/abs(sgn( B -x)+1), y=2x, x = -6..4 for B = -2
plot y=x^2+1/sqrt((sgn( A -x)+1)^2)-1/abs(sgn( A -x)+1)+1/sqrt((sgn( B -x)-1)^2)-1/abs(sgn( B -x)-1), y=2x, x = -5..5 for A=2, B=-3

Attention!   On the left, the graph of x → x^x ; part G is the graph where the function is continuous. The command "plot" plots only this part.

plot y = x^x, x = -2..2

Attention!   (sqrt(4*x)/sqrt(x) → sqrt(4*x) → 2)
(see below)
y = x-1+sqrt(4*x)/sqrt(x)

Attention!
La composizione di alcune funzioni (definite anche su numeri complessi che non sono reali) può comportarsi come definita anche dove non lo è.
The composition of some functions (also defined on complex numbers that are not real numbers) can behave as defined even where it is not.
An example:   exp( log(x) ) = x also for x ≤ 0, although for x ≤ 0 log(x) does not have a real number as value
Possiamo tracciare il grafico di exp(log(.)) assegnando a x ≤ 0 output che stiano fuori dal grafico.
We can plot the graph of exp(log(.)) by assigning x ≤ 0 outputs that are outside the graph.
plot piecewise[{ {1e10, x <= 0}, { exp(log(x)), x>0} }], x = -2..2, y = -2..2
Alternativa / An alternative  (see):
plot exp(log(x))+1/sqrt((sgn( -x)-1)^2)-1/abs(sgn(-x)-1), x-x, x = -2..2, y = -2..2

See above:     plot piecewise[ { {x-1+sqrt(4*x)/sqrt(x), x > 0}, {1e10, x<=0} }], x=-2..2, y=-1..3
Attention!   Il calcolo di (-3)^0.2 è eseguito correttamente solo se si scrive (-3)^(1/5) altrimenti l'input viene interpretato nell'ambito dei numeri complessi
The calculation of (-3)^0.2 is performed correctly only if (-3)^(1/5) is written, otherwise the input is interpreted in the context of complex numbers.

A destra il risultato di questa calcolatrice / On the right the result of this calculator

Attention!   Se traccio un grafico con un computer non è detto che venga evidenziata la presenza di "buchi"
If I draw a graph with a computer it is not certain that the presence of "holes" will be highlighted
plot piecewise [{ {x+1, x > 1/3}, {x+1, x < 1/3} , {2.5, x=1/3} }], x=-1..1.5
piecewise [{ {x+1, x > 1/3}, {x+1, x < 1/3} , {2.5, x=1/3} }] continuous?
is not continuous on R     x = 1/3 removable discontinuity

(n; 2^n) where n = {0,1,2,3,4,5,6,7,8}  |  plot 2^n where n = {0,1,2,3,4,5,6,7,8}
plot (0;2^0), (1;2^1), (2;2^2), (3;2^3), (4;2^4),(5;2^5),(6;2^6),(7;2^7),(8;2^8)
 plot {(5;5^2),(4;4^2),(3;3^2),(2;2^2),(1;1),(0;0),(1,1),(2;2),(3;3),(4;4),(5;5)}

centroid of (0,0),(2,2),(6,2),(1,6)  |  centroid of polygon (0,0),(2,2),(6,2),(1,6)  |  polygon centroid     vedi/see

centroid plane curve semicircle   |   plot y>0 and x^2+y^2<1 and x^2+(y-0.5)^2>0.01
centroid semicircle with center (0,0), radius 1, rotation angle 0

centroid of a quarter of disk with rotation angle -45 degrees
plot y>0 and x>0 and x^2+y^2<1 and (x-4/(3*PI))^2+(y-4/(3*PI)))^2>0.001

reflect across y=2x  |  vertical shear 20 degrees  vedi/see

reflect across x+0y+z=0

rotate 90° around the y-axis

 symmetry with respect to y = -x+2; translation x → x+4, y → y-4; rotation of -90° around (0,0); a non-monometric scale transformation + a translation vedi/see
Come realizzare figure con più di 3 poligoni / How to make figures with more than 3 polygons  vedi/see

mathworld subject map projections     vedi/see

area enclosed in parametric curve
 parametric plot (cos(t),sin(t)), parametric plot (cos(t),sin(t)+cos(t)-1) the circle transformed by  (x,y) → (x, y + x - 1) parametric plot (cos(t),sin(t)), parametric plot (cos(t)-1,sin(t)+1.5)a translation:   (x,y) → (x - 1, y + 1.5)

 The transformationsx' = x / (y - x - q),  y' = (x + y) / (y - x - q)with  q = √2  and  q = 1/2
parametric plot (cos(t),sin(t)), parametric plot ( cos(t)/(sin(t)-cos(t)-sqrt(2)), (cos(t)+sin(t))/(sin(t)-cos(t)-sqrt(2)) ), t=0..2*PI
parametric plot (cos(t),sin(t)), parametric plot ( cos(t)/(sin(t)-cos(t)-1/2), (cos(t)+sin(t))/(sin(t)-cos(t)-1/2) ), t=0..2*PI

 The transformation x' = x / (x - 1/2),  y' = ( x+ y) / (x - 1/2)of the square  |x| + |y| = 1
solve abs(x)*abs(y)=1 for y         y = 1-abs(x) for abs(x)<1    y = abs(x)-1 for abs(x)<1
parametric plot (t, 1-abs(t)), parametric plot (t, -1+abs(t)), t = -1..1
parametric plot (t/(t-1/2) ,(t+1-abs(t))/(t-1/2) ), parametric plot (t/(t-1/2) ,(t-1+abs(t))/(t-1/2) ), t = -1..1

parametric plot (cos(t),sin(t)), parametric plot (cos(t)^3, sin(t)^3), t=0..2*PI

quadrifolium   |   fish curve   |   cardioid   |   astroid

pack 3 circles in a circle of radius 1   |   pack 49 circles in a square   |   geometric packing in 2D

Penrose triangle   |   Optical illusion
dissection fallacy | Curry triangle | Haberdasher's problem | tangram paradox | impossible figure | paradox
 mathworld subject tessellationM.C. Escher vedi/see   Escher's Theorem

(1+V)^2/(V^2+V)  |  (x-2)/(x^2-x+1)-1/(x+1)+(x^2+x+3)/(x^3+1)

simplify (x-y)*z+(x+y)*z  |  expand {sin(x+y), cos(x+y)}

factor x^5-3*x^2+x+1

gcd(x^4+2*x^3+x^2+2*x, x^5+2*x^4-x-2)

quotient and remainder of (3*x^4-x^3) / (x^2-x+2)

x*3 + y/4 = 15, x - 5*y = 2/7

y >= -x + 2, y >= x/2

x+y+z=1, x-y-3*z=1, x+2*y+z=0

( (1, 1, 1), (1, -1, -3), (1, 2, 1) ) * (x, y, z) = (1,1,0)

1/8*(0...8)

 1 + 3 + 5 + ... + (2n-1)

2 * 4 * 6 * 8 * ... * 30  |  1/2 + 1/4 + 1/8 + 1/16 + ...

 series representation pi

sum 1/(1+n^2), n = -oo to oo  |  coth

f(n+1)=f(n)/2, f(0)=1  |  a(0) = 1, a(n+1) = a(n)+n^2  |  2^-n, n -> oo

Q(1)=1, Q(2)=1, Q(n+2)=Q(n)+Q(n+1)  |  fibonacci sequence

 ← Se il grafico prosegue in modo analogo quali sono le successive ordinate? / If the graph continues in a similar way, what are the subsequent ordinates?  [il grafico/the graph:  plot (0,-1),(1,3),(2,5),(3,-7),...,(14,29)] -1, 3, 5, -7, 9, 11, -13, 15, 17, -19, 21, 23, -25, 27, 29, -31, 33, 35, -37, 39, 41, -43, 45, 47, -49, 51, 53, -55, 57, 59, -61, 63, 65, -67, 69, 71, -73, 75, 77, -79, 81, 83, -85, 87, 89, -91, 93, 95, ...

 range of (1 - 2*x)/(x - 2) for x > 1 or image of (1 - 2*x)/(x - 2) for x > 1

{0.805396, 0.366574, 0.336665, 0.333667, 0.333367, 0.333337, 0.333334, 0.333333}
round( (tan(x)-sin(x)+1-cos(x)) / (x^2+1-cos(x)), 0.000001) if x=1,1e-1,1e-2,1e-3,1e-4,1e-5,1e-6,1e-7

lim (sin(x)-x)/x^3 as x -> 0

trapezoidal rule

integrate x^4-x^2 midpoint method on [0,1.5] with 10 intervals

integrate sin from 0 to PI

definite integrals containing exp  |  integrals containing log |  ... sqrt |  ... cos  |  ...

y=piecewise [{ {120, 0< x <= 200}, { 120+0.6*(x-200), 200<x<400} }], y=piecewise[{ {180, 0<x<400} }]
solve piecewise [{ {120, 0 < x <= 200}, { 120 + 0.6*(x-200), 200 < x < 400} }] = 180
→   x = 300

(a piecewise function is a function that is defined on a sequence of intervals; "if then else"  is not used in WolframAlpha)

plot piecewise[{ {x, x <= 1}, { (x-1)*2 , 1 < x <= 1.5}, { (2-x)*2 , 1.5 < x} }], x = 0..2  | integrate ...

int sin(x)/x dx, x=0..infinity

area between y = 23*x*x - 37*x +1, y = 5*x

plot y<x+1/2 and x^2+y^2<1 | area between y=x+1/2, x^2+y^2=1

area enclosed by  x^2 - 2*x*y + 4*y^2 + x = 4

d/dx cos(x)^2*x  |  d/dx cos(x)^2*x, x=0

where x*(x^2-4) is increasing?

1+0.6+0.6^2+0.6^3+...

product (1-1/k^2), k=2 to oo

plot y=x^1000+2*x+1, -1.5 < x < 1.5, -1 < y < 4

plot y = x^4 - x^2 + k for k = -2, -1, 0, 1, 2, -2 < x < 2

plot x^3+y^3=0^3, x^3+y^3=1^3, x^3+y^3=2^3, x^3+y^3=3^3, x^3+y^3=4^3, -5 < x < 5, -5 < y < 5

 (x^2+y^2-4)(x^2+y^2-9) < 0

plot 1/5 < |x|^5+|y|^5 < 1/2 | plot (|x|^5+|y|^5 -1)*(|x|^5+|y|^5 - 1/2)=0

(x-2)^2+(y+3)^2 >= 9 and x*y=0, -3 < x < 7, -8 < y < 2

solve x^4 - x^2 + k = 0 for x real

 (3n+85)/(n+5) = m A problem with integers:What are the positive integers n for which the number  (3n+85)/(n+5)  is an integer? 65,  30,  9,  5,  2 {{m == -67, n == -6}, {m == -32, n == -7}, {m == -11, n == -10}, {m == -7, n == -12}, {m == -4, n == -15}, {m == -2, n == -19}, {m == 1, n == -40}, {m == 2, n == -75}, {m == 4, n == 65}, {m == 5, n == 30}, {m == 8, n == 9}, {m == 10, n == 5}, {m == 13, n == 2}, {m == 17, n == 0}, {m == 38, n == -3}, {m == 73, n == -4}}

laws of physics

Second law of thermodynamics

length, time, mass, volume, density, acceleration, force, energy, electric current, magnetic moment

weight measurement devices

mechanical work calculator
stopping distance  |  pendulum  |  spring pendulum  |  coupled pendulum  |   ...

visible light

concave mirror | convex mirror

play 800 Hz sine wave | play 640 Hz sawtooth wave | play 500 Hz square wave
music theory  |  F# G G A G C B G G A G D C     vedi/see

elastic collision

 7500 ohm resistor vedi/see

parallel RC circuit, R=100ohm, C=50microF

vector( 6,-4), vector(3,2), vector(6+3,-4+2)   |  vector (0,0,2), vector (0,2,0), vector (2,0,0), vector (2,2,2)

 vector(cos(15°)*20,sin(15°)*20)

cross product     vedi/see

(2,0,4)x(1,3,1)   |   (2,0,4)+(1,3,1)   |   (2,0,4).(1,3,1)   |   || (2,0,4) ||
(-12, 2, 6)               {3, 3, 5}                       6                   2*sqrt(5)

(1,3,-2) in spherical coordinates  |  spherical coordinates

triangle (3,6,4),(1,2,3),(4,0,5)
 triangle {(0, 0, 0), (1, 1, 1/2), (2, 0, 1)} from (1,2,3)

distance point (1,1,3), plane 2x+3y+5z+1=0

plane (0,1,0), (3,2,1), (4,-1,2)

field of view

parabola focus (3,-2) and vertex (-3,4)
ellipse, foci (0,-2) and (0,2), semimajor axis 3
ellipse, semiaxes 2, 5, center (3,0)
hyperbola, foci (0,-2) and (0,2), semimajor axis 1
hyperbola center (10, 20), focus (11, 18)
...

ellipse, foci (2,-1) and (-2,1), semimajor axis 3 and ellipse, foci (4,0) and (-3,0), semimajor axis 5
sqrt((x-2)^2+(y+1)^2)+sqrt((x+2)^2+(y-1)^2)=6, sqrt((x-4)^2+y^2)+sqrt((x+3)^2+y^2)=10

parabola points (-2,0), (3,4), (10,12)  |  parabola ... , rotation angle 90  |  parabola ... , rotation angle 45

2*x^2-8*x+5*y^2+10*y-37 = 0

(x-2*y)^2+3*y-4=0
 conic section

mathworld subject fractals | Koch snowflake     vedi/see

tangent to 2*x^2+2*y^2-8*y+1=0 passing through (2,3)

normal to 3x^2-2xy+y^2-1=0 at x=0  |  plot {3x^2-2xy+y^2-1=0, y=-x-1, y=1-x, x=0, y=0}

maximize x*(1-x)*e^x

local extrema x*(1-x)*e^x

stationary points (x^2-x-1)^3
 inflection points (x^2-x-1)^3

asymptotes (x^3 + 5)/(3*x^2 + x - 4)

center of curvature of y=x^3-x^2-1 at x=1

curvature of r=t^3+2 at t=1/10

arc length of y=sin(x) from x=0 to PI

polar r = t, 0 <= t <= 4*pi  |  polar r = teta^-1, 2*pi <= teta <= 7*pi
 spiral ...

cusps |x-2|^(1/2)-|x+2|^(1/3)

interpolation polynomial

cubic fit (-1,4), (2,1.5), (3, -0.5), (5,-2), (8,2)

In alternativa posso usare "cubic fit [=]" e introdurre { (-1,4), (2,1.5), (3, -0.5), (5,-2), (8,2) } in "{x,y} values"
Alternatively I can use "cubic fit [=]" and introduce { (-1,4), (2,1.5), (3, -0.5), (5, -2), (8,2) } in "{x y} values"

exp fit (1,10), (3,5), (6,1), (10,0.1)

log fit (12,0),(16,3),(17,12.5),(22,23),(57,42),(135,56),(180,63),(410,90)

linear fit   or   linear fit {(47,18.9),(30,11.6), (17,6.9)...}

 See here how to choose a fixedpoint and to take into accountthe uncertainties  (min / maxslope)

statistics 10, 11, 12, 15, 20, 28, 45, 65, 100, 150 | skewness 10, 11, ...       skewness: coefficient of asymmetry

Battiti cardiaci (numero dei battito al minuto) registrati tra le studentesse di un corso universitario
Heartbeats (number of beats per minute) recorded among female students of a university course:
{96,62,78,82,100,68,96,78,88,62,80,84,61,64,94,60,72,58,88,66,84,62,66,80,78,68,72,82,76,87,90,78,68,86,76}

median   mean   variance   quartiles   IQR

 MovingAverage[ {7.6,6.4,8.4,6,7.6,6.5,6.3,5.4,6.3,5.2,4.8,2.9,7.4,6.9,5.5,7.6,9.9,12.0,9.4,5.1,3}, 3]
{7.4667, 6.9333, 7.3333, 6.7, 6.8, 6.0667, 6, 5.6333, 5.4333, 4.3, 5.0333, 5.7333, 6.6, 6.6667, 7.6667, 9.8333, 10.433, 8.8333, 5.8333}
Per approndimenti tecnici e operativi / For technical and operational insights  vedi/see

data:  x1, ..., xN
population sd = the root mean square = sqrt of ( ((x1-mean)^2+...+(xN-mean)^2) / N )
(sample) sd = sqrt of ( ((x1-mean)^2+...+(xN-mean)^2) / (N-1) )
standard deviation of the mean = (sample sd) / sqrt(N)     (or [*])
sd (7.3, 7.1, 7.2, 6.9, 7.2, 7.3, 7.4, 6.8, 7.0, 7.1, 6.9)   →   0.192117
population sd (7.3, 7.1, 7.2, 6.9, 7.2, 7.3, 7.4, 6.8, 7.0, 7.1, 6.9)   →   0.183177
length of (7.3, 7.1, 7.2, 6.9, 7.2, 7.3, 7.4, 6.8, 7.0, 7.1, 6.9)   →   11
0.192117/ sqrt(11)   →   0.0579255...
[*]   more briefly:
standard error of the mean (7.3,7.1,7.2,6.9,7.2,7.3,7.4, 6.8,7.0,7.1,6.9)   →   0.0579256

formula confidence interval

Per più numerosi dati puoi usare questa "calcolatrice" / For larger amounts of data you can use this "calculator"  vedi/see

Per regressione lineare, quadratica e cubica, correlazione e test χ² / For linear, quadratic and cubic regression, correlation and χ² test  vedi/see

Per il coefficiente di correlazione e una misura della sua affidabilità: / For the correlation coefficient and a measure of its accuracy   vedi/see

chi-squared distribution

gaussian distribution | probability gaussian | exponential distribution | Poisson distribution | Poisson distribution cdf

plot y=exp(-((x-m)/s)^2/2)/(sqrt(2*PI)*s), for s=0.03,0.02,0.01,m=0.5
plot y=exp(-((x-m)/s)^2/2)/(sqrt(2*PI)*s), for s=0.03,0.02,0.01,m=0.5 from x=0.44 to 0.56, 0 < y < 45

Note.  There are few phenomena that have a Gaussian trend (for example, as we have seen above, the heights of adults of a certain population have this but their weights do not). If a certain quantity N of measurements of a certain size is repeatedly made, it is not the various groups of measurements that tend to have a Gaussian trend but it is the N average values gradually obtained that tend to have a Gaussian distribution. This is the reason why I can associate the average value of a series of measurements with a precision that can be determined with the "standard error of the mean".
 A very large aggregate of people is made up of 36% of men and 64% of children of the same age. The men's heights have an average value of 174.2 cm and a standard deviation of 7.1 cm.For children the values are 132.4 and 5.6. The trend of both distributions is Gaussian.How is the distribution of their union?
plot exp(-((x-174.2)/7.1)^2/2)/(sqrt(2*PI)*7.1)*0.36+exp(-((x-132.4)/5.6)^2/2)/(sqrt(2*PI)*5.6)*0.64, 100<x<200

random integer form 0 to 1000 | ... | RandomInteger[(1,6),2]       vedi/see  and  vedi/see
pseudorandom number   history

random permutation 12 elements | 12 random integers from 1 to 12

C(12,n) where n = 0,1,2,3,4,5,6,7,8,9,10,11,12

BarChart (C(n,0),C(n,1),C(n,2),C(n,3),C(n,4),C(n,5),C(n,6),C(n,7),C(n,8)) where n=8

full house | 13*C(4,3)*12*C(4,2)/C(52,5)
probabilities poker  |  mathworld subject card games

prob x > 3 for x binomial with n =12 and p = 0.34   or [*]  |  binomial distribution
prob X<16 for X geometric with p=0.1 | geometric distribution | uniform distribution

[*]   for k=4 to 12 sum( binomial(12,k)*0.34^k*(1-0.34)^(12-k) )
0.62580527579389019981824

The multinomial distribution is a generalization of the binomial distribution.
If A, B, C, … have probability pA, pB, pC, … (pA+pB+pC+…=1), the probability that the outcomes are nA, nB, nC, … is
(nA +nB + nC + ...)! / ( nA! * nB! * nC! * ...) * pA^nA * pB^nB * pC^nB * ...
The probability that,  from a box in which 20 000 red sticks, 50 000 blue sticks and 30 000 yellow sticks (all of equal size and weight) are placed,  2 red and 1 yellow sticks are extracted:
(3)! / ( (2)!*(0)!*(1)! ) * (2/10)^2*(5/10)^0*(3/10)^1     0.036   9/250  (easy handmade calculation: 6/2*4/100*1*3/10)
(the number of sticks extracted is much less than the total number of sticks, so we can evaluate the probabilities as if there were the reinsertion of the extracted sticks; the rigorous calculation: 20000/100000*19999/99999*30000/99998*3 → 0.03599927... ≈ 0.036)

linear independence of (2, 3, -4), (-1, -2, 5), (4, k, 6)

inv {{6,-7,10),{0,3,-1},{0,5,-7}}

inverse matrix  |  det matrix  |  matrix

plot x+2y<=40 && 3x+2y<=60 && 0<=x<=18 && 0<=y
maximize 16000x+10000y in x+2y<=40 && 3x+2y<=60 && 0<=x<=18 && 0<=y

maximize 6x+3y+4z in x+y+z=10 && 3x+y+2z<=26 && x+3z<=18 && x>=0 && y>=0 && z>=0

amortization | mortgage amortization | mortgage

taylor expansion

power series of 1/(2+x) around 1

plot piecewise[ { {-1, -pi < x < 0}, {1, 0< x < pi} } ], -pi < x < pi  |  Fourier series calculator

Hessian matrix x*y^2

rot(-y^3, x^3, -z^3) | div (grad (x*y*z+x^3+y^2))

plot z=sin(x)*cos(y), x=-pi..2*pi, y=-pi/2..2*pi+pi/2

plot z=sin(x)*cos(y), x=-pi..2*pi, y=-pi/2..2*pi+pi/2 view from (10,10,-2)
 plot z=sin(x)*cos(x), x=-pi..2*pi,y=-2..2

plot z = 1 / (1+(1-sqrt(x^2+y^2))^2), x=-4..4, y=-4..4 | tangent plane to z=1/(1+(1-sqrt(x^2+y^2))^2) at (x,y)=(1,0)

 plot z = 1 / (1+(1-sqrt(x^2+y^2))^2), x=-4..4, y=-4..4 view from (10,10,30)
 level curves 1 / (1+(1-sqrt(x^2+y^2))^2) level curves 1 / (1+(1-sqrt(x^2+y^2))^2), x=-2..2, y=-2..2

1/(1+(1-sqrt(x^2+y^2))^2)=0.2,1/(1+(1-sqrt(x^2+y^2))^2)=0.4,1/(1+(1-sqrt(x^2+y^2))^2)=0.6,1/(1+(1-sqrt(x^2+y^2))^2)=0.8

rotate z=x^3, -1 <= z <= 1, about the z-axis

 spherical triangle Passeggiando sulle sfere Walking on the spheres vedi/see trigonometry  |  spherical trigonometry

 plot x^2+y^2+z^2 = 25

curvature of ( 2*sqrt(t)-2, 1.9*cos(t), 2.2*sin(t) ) at t = 3.1

parametric plot ( cos(t), sin(2*t), sin(3*t) )  |  length ( cos(t), sin(2*t), sin(3*t) )

parametric space curves

3d parametric plot (u,v,sqrt(1-(u^2+v^2)), u=-1 to 1, v=-1 to 1 | 3d parametric plot (cos u, sin u + cos v, sin v), u=0 to 2pi, v=0 to 2pi

 mathworld subject platonic solidsregular polyhedra

cone, radius=4, height=6  |  cone, volume v, height 7

 truncated cylinder

solid of revolution calculator     (truncated cone with hole)

center (0,0,0), edge length 1, edges parallel to the axes

cube view from (2,1,1)  |  cube view from (2,0,2)  |  cube view from (100,100,100)
pyramid view from (2,2,0)
|  pyramid view from (0,-0.045,0.5)  |  pyramid view from (0,-0.045,0.49)
prism
view from ..  |  cuboid view from ..  |  parallelepiped view from ..
cone view from ..  |  conical frustum view from ..
 cuboid 4,3,5 view from (4,2,7)pyramid 3 view from (4,2,3) pyramid 4 view from (4,2,3)
 vanishing pointpunto all'infinitopunto di fuga

s''(t) = g, s(0) = 8, s'(0) = 5

Q(t)/C + R*dQ(t)/dt = V, Q(0)=0 |  limit C*(1 - e^(-t/(C*R))) V as t -> oo where C=80*10^-6, R=500, V=45

rot(-omega*y, omega*x, 0)   |   jacobian { r * cos(k*t), r * sin(k*t) } with respect to (r, t)     vedi/see

slope field of dy/dx = x-y, for -1 <= x <= 5, -2 <= y <= 4  |  dy/dx = x-y, y(1) = 3

plot x+3*e^(1-x)-1, x+2*e^(1-x)-1, x+e^(1-x)-1, x-1, x-e^(1-x)-1, x-2*e^(1-x)-1, x-3*e^(1-x)-1 for -1<x<5, -3<y<4

plot abs(abs(x-1)-2)-1, -5 < x < 7, -2 < y < 3, color blue       [ F'(x) = | |x - 1| - 2 | - 1, F(0) = 0, F → red graph]
plot piecewise[{ {-(x+2)^2/2+1,x<= -1}, { x^2/2,-1<x<1}, { -(x-2)^2/2+1,1<x<3}, { (x-4)^2/2, x>3} }], x = -5..7, y = -2..3, color red

y'' + y = 0, y(1)=2, y'(1)=3

y' = -x*y^2/(1+x^2*y)
y' = -x*y^2/(1+x^2*y), y(0) = 2  ...  y' = -x*y^2/(1+x^2*y), y(-2) = -1
plot (-1+sqrt(1+4 x^2))/x^2, (-1+sqrt(1+2 x^2))/x^2, (-1+sqrt(1-2 x^2))/x^2, -(1+sqrt(1+2 x^2))/x^2, (-1+sqrt(1-x^2))/x^2 for -3 < x < 3

integrate 1 dx dy, x=-1..1, y=-1..1  |  sqrt(integrate 1/(2*pi)*exp(-(x^2+y^2)/2) dx dy, x=-1..1, y=-1..1)

simplify (x+ iy)*(4+3 i)  |  {(0+0i),(1+0 i),(1+1i),(0+1 i)}*(4+3 i)
polygon (0,0),(1,0),(1,1),(0,1); polygon (0,0),(4,3),(1,7),(-3,4)

The circle cos(t)+i*sin(t) transfromed by z → z²+z+1  (z = x+i*y)
(cos(t)+i*sin(t))^2 + cos(t)+i*sin(t) + 1
I have:   cos(t) + cos(2 t) + i sin(t) (2 cos(t) + 1) + 1

parametric plot (cos(t), sin(t)), parametric plot ( sin(t)-cos(2*t)+1, (2*sin(t)+1)*cos(t) )

(3+2i)*   |   exp(32+27i)

 (x + iy)^2 + (x + iy) (x, y) →(x^2 + x - y^2, 2 x y + y)how to draw with a script
this was an example of / questo era un esempio di

 conformal mapping vedi/see: mathworld.wolfram.comand  here

factor z^9-z^3  |  solve z^9=z^3 for z complex

conchoid of Nicomedes

Klein bottle image  |  batman lamina

Archimedean solids

mathworld subject complexity of algorithms

rule

mathworld subject notation

mathworld subject terminology

Puoi esaminare esempi sui vari temi matematici:

examples mathematics

Puoi esaminare esempi "a caso":

english keyboard, italian keyboard  |  french | german | us international

Puoi ridimensionare le immagini e salvarle in diversi formati
You can resize images and save them in different formats

pie chart(1,2,3)
In "Paint" o in un'altra applicazione puoi modificare l'immagine  /  In "Paint" or in another application you can edit the image

Puoi copiare formule da un documento e incollarle in WolframAlpha - quasi sempre le accetta
You can copy formulas from a document and paste them into WolframAlpha - it almost always accepts them
Es. Ho: √(z²+5) = y+3   Per ottenere quanto segue posso scrivere in WolframAlpha solve for z, poi copiare e incollare in mezzo  √(z²+5) = y+3
Ex. I have: √(z²+5) = y+3   To get the following I can write in WolframAlpha solve for z, then copy and paste in the middle   √(z²+5) = y+3

english ↔ italiano       (other examples    general themes    exercises)