mathai 0.9.8

Mathematics solving Ai tailored to NCERT

Math AI Documentation

Source

Github repository of the code https://github.com/infinity390/mathai4

Philosophy

I think it is a big realization in computer science and programming to realize that computers can solve mathematics.
This understanding should be made mainstream. It can help transform education, mathematical research, and computation of mathematical equations for work.

Societal Implications Of Such A Computer Program And The Author's Comment On Universities Of India

I think mathematics is valued by society because of education. Schools and universities teach them.
So this kind of software, if made mainstream, could bring real change.

The Summary Of How Computer "Solves" Math

Math equations are a tree data structure (TreeNode class).
We can manipulate the math equations using various algorithms (functions provided by the mathai library).
We first parse the math equation strings to get the tree data structure (parse function in mathai).

The Library

Import the library by doing:

from mathai import *

str_form

It is the string representation of a TreeNode math equation.

Example

(cos(x)^2)+(sin(x)^2)

Is represented internally as:

f_add
 f_pow
  f_cos
   v_0
  d_2
 f_pow
  f_sin
   v_0
  d_2

Leaf Nodes

Variables (start with a v_ prefix):

• v_0 -> x
• v_1 -> y
• v_2 -> z
• v_3 -> a

Numbers (start with d_ prefix; only integers):

• d_-1 -> -1
• d_0 -> 0
• d_1 -> 1
• d_2 -> 2

Branch Nodes

• f_add -> addition
• f_mul -> multiplication
• f_pow -> power

parse

Takes a math equation string and outputs a TreeNode object.

from mathai import *

equation = parse("sin(x)^2+cos(x)^2")
print(equation)

Output

(cos(x)^2)+(sin(x)^2)

simplify

It simplifies and cleans up a given math equation.

from mathai import *

equation = simplify(parse("(x+x+x+x-1-1-1-1)*(4*x-4)*sin(sin(x+x+x)*sin(3*x))"))
printeq(equation)

Output

((-4+(4*x))^2)*sin((sin((3*x))^2))

Incomplete Documentation, Will be updated and completed later on

Demonstrations

Example Demonstration 1 (absolute value inequalities)

from mathai import *
question_list_from_lecture = [
    "2*x/(2*x^2 + 5*x + 2) > 1/(x + 1)",
    "(x + 2)*(x + 3)/((x - 2)*(x - 3)) <= 1",
    "(5*x - 1) < (x + 1)^2 & (x + 1)^2 < 7*x - 3",
    "(2*x - 1)/(2*x^3 + 3*x^2 + x) > 0",
    "abs(x + 5)*x + 2*abs(x + 7) - 2 = 0",
    "x*abs(x) - 5*abs(x + 2) + 6 = 0",
    "x^2 - abs(x + 2) + x > 0",
    "abs(abs(x - 2) - 3) <= 2",
    "abs(3*x - 5) + abs(8 - x) = abs(3 + 2*x)",
    "abs(x^2 + 5*x + 9) < abs(x^2 + 2*x + 2) + abs(3*x + 7)"
]

for item in question_list_from_lecture:
  eq = simplify(parse(item))
  eq = dowhile(eq, absolute)
  eq = simplify(factor1(fraction(eq)))
  eq = prepare(eq)
  eq = factor2(eq)
  c = wavycurvy(eq & domain(eq)).fix()
  print(c)

Output

(-2,-1)U(-(2/3),-(1/2))
(-inf,0)U(2,3)U{0}
(2,4)
(-inf,-1)U(-(1/2),0)U(1/2,+inf)
{-4,-3,-(3/2)-(sqrt(57)/2)}
{-1,(5/2)-(sqrt(89)/2),(5/2)+(sqrt(41)/2)}
(-inf,-sqrt(2))U((2*sqrt(2))/2,+inf)
(-3,1)U(3,7)U{1,-3,7,3}
(5/3,8)U{5/3,8}
(-inf,-(7/3))

Example Demonstration 2 (trigonometry)

from mathai import *
def nested_func(eq_node):
    eq_node = fraction(eq_node)
    eq_node = simplify(eq_node)
    eq_node = trig1(eq_node)
    eq_node = trig0(eq_node)
    return eq_node
for item in ["(cosec(x)-cot(x))^2=(1-cos(x))/(1+cos(x))", "cos(x)/(1+sin(x)) + (1+sin(x))/cos(x) = 2*sec(x)",\
             "tan(x)/(1-cot(x)) + cot(x)/(1-tan(x)) = 1 + sec(x)*cosec(x)", "(1+sec(x))/sec(x) = sin(x)^2/(1-cos(x))",\
             "(cos(x)-sin(x)+1)/(cos(x)+sin(x)-1) = cosec(x)+cot(x)"]:
  eq = logic0(dowhile(parse(item), nested_func))
  print(eq)

Output

true
true
true
true
true

Example Demonstration 3 (integration)

from mathai import *

eq = simplify(parse("integrate(2*x/(x^2+1),x)"))
eq = integrate_const(eq)
eq = integrate_fraction(eq)
print(simplify(fraction(simplify(eq))))

eq = simplify(parse("integrate(sin(cos(x))*sin(x),x)"))
eq = integrate_subs(eq)
eq = integrate_const(eq)
eq = integrate_formula(eq)
eq = integrate_clean(eq)
print(simplify(eq))

eq = simplify(parse("integrate(x*sqrt(x+2),x)"))
eq = integrate_subs(eq)
eq = integrate_const(eq)
eq = integrate_formula(eq)
eq = expand(eq)
eq = integrate_const(eq)
eq = integrate_summation(eq)
eq = simplify(eq)
eq = integrate_const(eq)
eq = integrate_formula(eq)
eq = integrate_clean(eq)
print(simplify(fraction(simplify(eq))))

eq = simplify(parse("integrate(x/(e^(x^2)),x)"))
eq = integrate_subs(eq)
eq = integrate_const(eq)
eq = integrate_formula(eq)
eq = simplify(eq)
eq = integrate_formula(eq)
eq = integrate_clean(eq)
print(simplify(eq))

eq = fraction(trig0(trig1(simplify(parse("integrate(sin(x)^4,x)")))))
eq = integrate_const(eq)
eq = integrate_summation(eq)
eq = integrate_formula(eq)
eq = integrate_const(eq)
eq = integrate_formula(eq)
print(factor0(simplify(fraction(simplify(eq)))))

Output

log(abs((1+(x^2))))
cos(cos(x))
((6*((2+x)^(5/2)))-(20*((2+x)^(3/2))))/15
-((e^-(x^2))/2)
-(((8*sin((2*x)))-(12*x)-sin((4*x)))/32)

Example Demonstration 4 (derivation of hydrogen atom's ground state energy in electron volts using the variational principle in quantum physics)

from mathai import *;
def auto_integration(eq):
    for _ in range(3):
        eq=dowhile(integrate_subs(eq),lambda x:integrate_summation(integrate_const(integrate_formula(simplify(expand(x))))));
        out=integrate_clean(copy.deepcopy(eq));
        if "f_integrate" not in str_form(out):return dowhile(out,lambda x:simplify(fraction(x)));
        eq=integrate_byparts(eq);
    return eq;
z,k,m,e1,hbar=map(lambda s:simplify(parse(s)),["1","8987551787","9109383701*10^(-40)","1602176634*10^(-28)","1054571817*10^(-43)"]);
pi,euler,r=tree_form("s_pi"),tree_form("s_e"),parse("r");a0=hbar**2/(k*e1**2*m);psi=((z**3/(pi*a0**3)).fx("sqrt"))*euler**(-(z/a0)*r);
laplace_psi=diff(r**2*diff(psi,r.name),r.name)/r**2;V=-(k*z*e1**2)/r;Hpsi=-hbar**2/(2*m)*laplace_psi+V*psi;
norm=lambda f:simplify(
    limit3(limit2(expand(TreeNode("f_limitpinf",[auto_integration(TreeNode("f_integrate",[f*parse("4")*pi*r**2,r])),r]))))
    -limit1(TreeNode("f_limit",[auto_integration(TreeNode("f_integrate",[f*parse("4")*pi*r**2,r])),r]))
);
print(compute(norm(psi*Hpsi)/(norm(psi**2)*e1)));

Output

-13.605693122882867

Example Demonstration 5 (boolean algebra)

from mathai import *
print(logic_n(simplify(parse("~(p<->q)<->(~p<->q)"))))
print(logic_n(simplify(parse("(p->q)<->(~q->~p)"))))

Output

true
true

Example Demonstration 6 (limits)

from mathai import *
limits = ["(e^(tan(x)) - 1 - tan(x)) / x^2", "sin(x)/x", "(1-cos(x))/x^2", "(sin(x)-x)/sin(x)^3"]
for q in limits:
    q = fraction(simplify(TreeNode("f_limit",[parse(q),parse("x")])))
    q = limit1(q)
    print(q)

Output

1/2
1
1/2
-(1/6)