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3.5 days -> hours  |  10000 sec


minimal coin form 26.85 euros


1 euro  |  image euro


120 - 45 - 30     see also/vedi anche     see also/vedi anche


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


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)
I can draw at most three figures on the same Cartesian plane



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


long division     see also/vedi anche     see also/vedi anche


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 where abs(x-240)<2 and 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 where abs(x-13.7)<0.05 and abs(y-0.096)<0.0005 and 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 the
structure 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


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

      0.2999...



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

{{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



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)


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)


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


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 lenghts 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)


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


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);(8,12);(8,20);(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 scripts
vedi /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)


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)


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




population of Italy | population of United Arab Emirates


shoe size


car  |  bicycle  |  motorcycle


Roman Empire


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


Pilates better posture


fontina 100 g


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


biology


brain


growth curve fir tree   (abete)


SI prefix


chemical elements | chemistry

caffeine


ISBN 9788836548033


QR code "Genova"


F# G G A G C B G G A G D C


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



CAPTCHA "G. Marconi"


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


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)


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)


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


factor 70560



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


plot y^2+x^2-x=1, y=x^2, y^2=x, y=0, x=0  |  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

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)

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


(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)     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


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)


a translation:   (x,y) → (x - 1, y + 1.5)

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


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


(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


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(x) dx from x=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


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


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


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


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


triangle (3,6,4),(1,2,3),(4,0,5)


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


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


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


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)


linear fit   or   linear fit {(47,18.9),(30,11.6), (17,6.9)...} 
 
See here how to choose a fixed
point and to take into account
the uncertainties  (min / max
slope)
   


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



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


chi-squared distribution

La correlazione lineare può essere calcolata anche con "correlation [(..,..,..),(..,..,..)]", ma per associarle una precisione conviene ricorrere ad R (vedi).
The linear correlation can also be calculated with "correlation [(..,..,..),(..,..,..)]", but to associate it with a measure of accuracy it is convenient to use R (see).



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
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



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



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 mtrix 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 = 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)


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

  spherical triangle

Passeggiando sulle sfere
Walking on the spheres
vedi/see


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



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


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 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)




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 exaample of / questo era un esempio di

 conformal mapping

vedi/see:
mathworld.wolfram.com
and  here




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



conchoid of Nicomedes


Klein bottle image  |  batman lamina


Archimedean solids


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




english ↔ italiano       (other examples    general themes)