MATLAB: MATLAB is a widely used tool for numerical computation, visualization, and programming. It is particularly useful for engineering, science, and mathematical applications.

Maple: Maple is another popular tool for advanced mathematics and computation. It provides a variety of tools for symbolic and numerical computations, visualization, and programming.

SageMath: SageMath is a free, open-source software tool for advanced mathematics and computation. It provides a variety of tools for symbolic and numerical computations, visualization, and programming.

Maxima: Maxima is a free, open-source computer algebra system that is particularly useful for symbolic computations in mathematics, science, and engineering.

GeoGebra: GeoGebra is a free, open-source dynamic mathematics software tool that combines geometry, algebra, and calculus. It provides an interactive environment for exploring mathematical concepts and visualizing mathematical relationships.

R: <a href="http://makeatea.com/earl-grey-tea-benefits/">R</a> is a popular programming language and software environment for statistical computing and graphics. It provides a wide range of tools for data analysis, modeling, and visualization.

These are just a few examples of the many interesting software tools available for computer math. Depending on your needs and interests, there may be other tools that are better suited to your particular application.

]]>2x³+6xy²-114=z³

Use Mathway to solve it

Thanks me later]]>

Keckman wrote:

There is six number in divide operation and you get seven right numbers of pi. But is there even better ones? I guess there is not same kind rational number under one trillion.

Well, according to my research, there's a very big jump from your 355/113 to the next one (103993/33102): your Pascal program might be quite busy for a while.

I initially thought of trying brute force like you did, but changed my mind after I googled 'oeis pi fractions approximations 22/7 355/113 104348/33215' (as a single-line search using your fraction, imcute's and the 22/7 I know)...which introduced me to a minefield of info about this topic!

I started here: OEIS (sequence of convergents to Pi): A002485 (numerators) & A002486 (denominators).

Those two links (and a StackExchange post) quoted the following Mathematica formula for obtaining a sequence of increasingly accurate fraction approximations: Convergents[Pi, x].

That interested me as I have Mathematica. I chose 10 for x (for an output of a sequence of 10 fractions) and ran the code in my program and also online at WolframAlpha, with the following output in both:

3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913.

The 2nd fraction in the sequence shows the well-known 22/7, the 4th is yours from post #1, and the 6th is imcute's from post #2.

Links to a couple of other sites I looked at:

Wikipedia: Continued fraction expansion of π and its convergents.

Wolfram MathWorld: Pi Continued Fraction.

]]>In one Universe there are 500 billion galaxies, each of which has 200 billion stars, of which every tenth has a planet that can be quite easily modified into a suitable place for humans to live, and the planet can hold an average of 10 billion people. At the moment, there is only one planet called Earth with life in that Universe.

Let's assume that the human population grows on average at the same rate as it has grown since 1700, when we were 640 million 35 thousand 774 about.

In this picture, the red curve describes when the number of people has multiplied by 1.0076 every year. The black curve is obtained based on actual values. That is, because that red curve "clicks" quite well with the real curve, the coefficient 1.0076 is used

Question:

In what year do man populate the Universe, after which that 1.0076 multiplication per year must be adjusted to 1.0000 multiplication? The math problem assumes, of course, that space travel is under control, and we can travel to distant stars and galaxies. This is a mathematical exercise and not a model of the real situation, as we currently have no way to travel to other stars and galaxies.

Solution: I don't know how to use logarithm calculation formulas now, but I solve the problem with a computer program: (maybe someone show me how this is done with logarithm?)

```
rebol[]
g: 5E+11 ;How Many galaxies?
g: (g * 2E+11) / 10 ;How many planets to populate
g: g * 1E+10 ; How many people there can be in Universe?
population: 640035774
year: 1700
until [
print [year " " population]
population: population * 1.0076
year: year + 1
population >= g
]
halt
```

Output:

1700 640035774

1701 644900045.8824

1702 649801286.231106

1703 654739776.006463

1704 659715798.304112

1705 664729638.371223

1706 669781583.622844

1707 674871923.658378

1708 680000950.278182

1709 685168957.500296

.

.

.

8743 9.2194892244472E+31

8744 9.289557342553E+31

8745 9.36015797835641E+31

8746 9.43129517899192E+31

8747 9.50297302235226E+31

8748 9.57519561732213E+31

8749 9.64796710401378E+31

8750 9.72129165400429E+31

8751 9.79517347057472E+31

8752 9.86961678895109E+31

8753 9.94462587654712E+31

>>

The answer is the year 8753. So pretty soon the entire universe will be populated! Right? Yes, exponential population growth is wild!

]]>I don't know what it is.

It happens too when y approaches 4/3. Sequences get very long.

So for y= 1.3333333 and x= 1 to 999, it found a sequence of length 23,671 at x = 981.

There are 9,799 more odd numbers than even numbers in this sequence.

Some values of y do better than others.

Thanks for the reply though, I was just experimenting, hoping to discover something.

Not easy for me to figure it out. But I had some fun.

there are 256 possible 3i1o truth tables

(null,a,b,ab,c,ac,bc,abc)

(insert 0~255 translated into binary,i dont want to paste this long sequence here)

fisrt remove not pairs

00000000means that a b c are irrelavent

00001111means that a,b are irrelavent

01010101means that b,c are irrelavent

00110011means that a,c are irrelavent

if 1357/2468 are identical,a is irrelavent

if 1245/3487 are identical,b is irrelavent

if 1234/5678 are identical,c is irrelavent

if 2/3 or 6/7 swap,they are ab symmetry pairs

if 2/5 or 4/7 swap,they are ac symmetry pairs

if 3/5 or 4/6 swap,they are bc symmetry pairs

the rest are elementary but maybe uninteresting gates

(the majority of them are include gates projected to each of the 3 axis)

00000001,

00010000,00010010,00010011,00010101,00010110,00010111,

00100000,00100001,00101110,

00110000,00110001,00110010,00110101,00110110,00110111,

01100000,01100001,01101000,01101001,01101110,01101111,

01110000,01110001,01110010,01110011,01110110,01111000,01111001,01111010,01111011,01111110,01111111,]]>

```
\\ Jacobi theta functions
theta1(q,z)=theta(q,z)
theta2(q,z)=2*q^(1/4)*suminf(n=0,q^(n*n+n)*cos((2*n+1)*z))
theta3(q,z)=1+2*suminf(n=1,q^n^2*cos(2*n*z))
theta4(q,z)=theta3(-q,z)
\\ Lattice reduction
ellreducez(w,z)=my(t=ellperiods(w),a,b);b=Mat([real(t),imag(t)]~);a=round(Mat([real(w),imag(w)]~)^(-1)*b);b=round(b^(-1)*[real(z),imag(z)]~);[z-t*b,a*b]
ellperiodratio(w)=w=ellperiods(w);w[1]/w[2]
\\ Elliptic nome
ellnome(k)=if(k,exp(-Pi*agm(1,sqrt(1-k^2))/agm(1,k)),(k^2)/2)
invellnome(q)=my(s=4*q^(1/2),t);if(s,t=log(q)/(Pi*I);s*=(eta(t/2)*eta(2*t)^2/eta(t)^3)^4);s
\\ j-invariant
invellj(j)=ellperiodratio(ellinit(ellfromj(j)))
\\ Neville theta functions
thetaS(k,z)=if(k,my(m=k^2,a=agm(1,sqrt(1-m)),q=exp(-Pi*a/agm(1,k)));theta1(q,z*a)*sqrt(a)*m^(-1/4)*(1-m)^(-1/4),sin(z))
thetaC(k,z)=if(k,my(m=k^2,a=agm(1,sqrt(1-m)),q=exp(-Pi*a/agm(1,k)));theta2(q,z*a)*sqrt(a)*m^(-1/4),cos(z))
thetaD(k,z)=if(k,my(m=k^2,a=agm(1,sqrt(1-m)),q=exp(-Pi*a/agm(1,k)));theta3(q,z*a)*sqrt(a),1.)
thetaN(k,z)=if(k,my(m=k^2,a=agm(1,sqrt(1-m)),q=exp(-Pi*a/agm(1,k)));theta4(q,z*a)*sqrt(a)*(1-m)^(-1/4),1.)
\\ Jacobi elliptic functions
jacobiSN(k,z)=thetaS(k,z)/thetaN(k,z)
jacobiCN(k,z)=thetaC(k,z)/thetaN(k,z)
jacobiTN(k,z)=thetaS(k,z)/thetaC(k,z)
jacobiDN(k,z)=thetaD(k,z)/thetaN(k,z)
jacobiZN(k,z)=((z)->thetaN(bitprecision(k,getlocalbitprec()),z))'(z)/thetaN(k,z)
jacobiEpsilon(k,z)=jacobiZN(k,z)+z*ellE(k)/ellK(k)
jacobiAM(k,z)=if(k,my(c=thetaC(k,z),s=thetaS(k,z),a=c+I*s,b=c-I*s,t=norm(a)-norm(b));z=if(t<0,I,-I)*log(if(t<0,b,a)/thetaN(k,z));if(t,z,real(z)),z)
\\ Alternative implementations
jacobiSN(k,z)=my(a,b,w,p,t,m=k^2);if(m==0,return(sin(z)),m==1,return(tanh(z)));a=ellK(k);b=I*ellK(sqrt(1-m));[z,t]=ellreducez([2*a,2*b],z);w=[4*a,2*b];t=(-1)^t[1];p=iferr(ellwp(w,z,1),E,return(t*z),errname(E)=="e_DOMAIN");-t*2*(p[1]-(1+m)/6)/p[2]
jacobiCN(k,z)=my(a,b,w,p,t,m=k^2);if(m==0,return(cos(z)),m==1,return(1/cosh(z)));a=ellK(k);b=I*ellK(sqrt(1-m));[z,t]=ellreducez([2*a,2*b],z);w=[4*a,2*(a+b)];t=(-1)^(t[1]+t[2]);p=iferr(ellwp(w,z),E,return(t*1.),errname(E)=="e_DOMAIN");t*(1-6/(1+4*m+12*p))
jacobiTN(k,z)=my(a,b,w,p,t,m=k^2);if(m==0,return(tan(z)),m==1,return(sinh(z)));a=ellK(k);b=I*ellK(sqrt(1-m));[z,t]=ellreducez([2*a,2*b],z);w=[2*a,4*b];t=(-1)^t[2];p=iferr(ellwp(w,z,1),E,return(t*z),errname(E)=="e_DOMAIN");-t*2*(p[1]-(m-2)/6)/p[2]
jacobiDN(k,z)=my(a,b,w,p,t,m=k^2);if(m==0,return(1.),m==1,return(1/cosh(z)));a=ellK(k);b=I*ellK(sqrt(1-m));[z,t]=ellreducez([2*a,2*b],z);w=[2*a,4*b];t=(-1)^t[2];p=iferr(ellwp(w,z),E,return(t*1.),errname(E)=="e_DOMAIN");t*(1-6*m/(4+m+12*p))
jacobiZN(k,z)=if(k==0,return(0.),k^2==1,return(tanh(z)));my(a=ellK(k),b=I*ellK(sqrt(1-k^2)),w=[2*a,2*b]);ellzeta(w,z+b)-ellzeta(w,b)-ellzeta(w,a)/a*z
jacobiEpsilon(k,z)=if(k==0,return(z),k^2==1,return(tanh(z)));my(a=ellK(k),b=I*ellK(sqrt(1-k^2)),w=[2*a,2*b]);ellzeta(w,z+b)-ellzeta(w,b)+z*(2-k^2)/3
jacobiAM(k,z)=if(k,my(a=agm(1,k),b=agm(1,sqrt(1-k^2)),c=a*z/2,t);if(b,b=a*Pi/(2*b);t=real(c)\/real(b);c-=t*b);2*(atan(tanh(c))+if(b,t*Pi/2+suminf(i=1,atan(tanh(c+i*b))+atan(tanh(c-i*b))),0)),z)
\\ Elliptic exponential and logarithm
ellexpnum(E,z)=my(P=ellztopoint(E,z));if(#P<2,0.,-P[1]/P[2])
elllognum(E,z)=if(!z,return(z));my(R=polroots(E.a6*z^2+(E.a3*z+E.a4*z^2)*x+(-1+E.a1*z+E.a2*z^2)*x^2+z^2*x^3),r,x);foreach(R,t,my(a=abs(t));if(a>r,r=a;x=t));ellreducez(E.omega,ellpointtoz(E,[x,-x/z]))[1]
\\ Incomplete elliptic integrals
incellF(k,phi)=my(m=k^2);if(m==1,2*atanh(tan(phi/2)),m,my(s=sin(phi),c=cos(phi),d=1+sqrt(1-m*s^2),E);if(!c,return(s));c=if(real(c)<0,s^2/(1+c),1-c);E=ellinit(-[(1-m+m^2)/3,(2-3*m-3*m^2+2*m^3)/27]);2*ellpointtoz(E,[d/c-(1+m)/3,s*(m*c-d)/c^2]),phi*1.)
incellE(k,phi)=jacobiEpsilon(k,incellF(k,phi))
incellD(k,phi)=my(u=incellF(k,phi));(u-jacobiEpsilon(k,u))/k^2
jacobiZeta(k,phi)=jacobiZN(k,incellF(k,phi))
invellwp(w,z)=w=ellperiods(w);my(g2=elleisnum(w,4,1),g3=elleisnum(w,6,1),E=ellinit(-[g2,g3]/4),x,y,l);z=Vec(z);x=z[1];l=#z<2;y=if(l,sqrt(4*x^3-x*g2-g3),z[2]);z=ellreducez(w,ellpointtoz(E,[x,-y/2]))[1];if(l&&real(z)>=0,z,-z)
\\ complete elliptic integrals
ellD(k)=(ellK(k)-ellE(k))/k^2
\\ Lemniscatic elliptic functions
sinlemn(z)=my(w=2*ellK(I),q=round(z/w),s=(-1)^(real(q)+imag(q)),p,r);z=1.*(z-q*w);if(z&&-2*(q=exponent(z))<(r=bitprecision(z)),[w,z]=bitprecision([w*[1+I,1-I],z],r-q);iferr(p=ellwp(w,z,1),E,,errname(E)=="e_DOMAIN"));s*if(p,-bitprecision(2*p[1]/p[2],r),z)
coslemn(z)=my(p=2*ellK(I),q=round(z/p),s=(-1)^(real(q)+imag(q)));z-=q*p;iferr(p=ellwp(p*[1+I,1-I],z),E,return(s*1.),errname(E)=="e_DOMAIN");s*(1-2/(2*p+1))
asinlemn(z)=z*hypergeom([1/4,1/2],5/4,z^4)
acoslemn(z)=ellK(I)-asinlemn(z)
\\ Associated Weierstrass sigma functions
{
my(ellsigma_=((w,z,i)->my([t,e]=ellperiods(w,1),a=Mat([real(t),imag(t)]~),b=round(a^(-1)*Mat([real(w),imag(w)]~))*Col(i)%2);if(real(t[1]/t[2])>0,b[2]*=-1);[a,e]=[t*b,e*b]/2;if(real(z*conj(a))<0,z=-z);ellsigma(w,z+a)/ellsigma(w,a)*exp(-z*e)));
ellsigma1=((w,z)->ellsigma_(w,z,[1,0]));
ellsigma2=((w,z)->ellsigma_(w,z,[0,1]));
ellsigma3=((w,z)->ellsigma_(w,z,[1,1]));
}
\\ q-Pochhammer
qpoch(a,q)=my(p=getlocalbitprec(),r=1,t=1);a=precision(-a,p+10);localbitprec(p+10);bitprecision((1+a)*(1+suminf(i=1,r*=q;t*=a*r/(1-r);t)),p)
\\ Dixon elliptic functions
DixonSM(z)=my(a,b);iferr([a,b]=ellwp(ellinit([0,-1/108]),z,1),E,return(.),errname(E)=="e_DOMAIN");-6*a/(3*b-1)
DixonCM(z)=my(a,b);iferr([a,b]=ellwp(ellinit([0,-1/108]),z,1),E,return(1.),errname(E)=="e_DOMAIN");(3*b+1)/(3*b-1)
DixonThetaS(z)=ellsigma(ellinit([0,1/4]),z)
DixonThetaC(z)=my(t=gamma(1/3)^3/(sqrt(3)*(2*Pi)),E=ellinit([0,1/4]));exp(Pi*(2*z/t-1)/sqrt(27))*ellsigma(E,t-z)
DixonThetaM(z)=DixonThetaC(-z)
ArcDixonSM(z)=z*hypergeom([1/3,2/3],4/3,z^3)
ArcDixonCM(z)=gamma(1/3)^3/(sqrt(3)*(2*Pi))-ArcDixonSM(z)
```

]]>For example, see these formulas:

]]>I had similar results. The most I've found was 75 steps for 830612 towards 13682.

This time, I removed 0 when it came in front.

This was simpler to implement. I see that you have stopped when you'd loose a digit. Seems cool too.

I have the same result for 103682. It takes 74 steps.

Not sure you would post it to OEIS? All fine by me.

]]>How programming is different from algorithms?

pi_cubed wrote:

Hi 666 bro,

In programming we say that we "implement algorithims" into programs, which gives us the idea that they are different. An algorithim is simply a thought process. Programming is implementing the algorithim, or writing it in a language that the interpeter can understand. Think of the programmer as an inventer. The idea for the invention is the algorithim. The invention is the code. Hope this clears things up.

*Sorry for some misspellings*

]]>Why do you think eliminating mathematics is a good idea?

]]>