Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Integral( 0, pi/2, sin^2(x)*cos(x), dx ) = ???
Integral( 0, pi/2, sin(x)sin(x)cos(x), dx ) = 1/3
Notice that there are two easy ways to calculate this integral:
1) F(x) = 1/3 * sin3(x); F'(x) = sin2(x)cos(x)
2) sin2(x)cos(x) = sin2(x)sin'(x); change variable cos(x)=t
Integral( pi/6, pi/2, ctg(x), dx ) = ???
Integral( pi/6, pi/2, ctg(x), dx ) = log(2)
hint: [ log(sin(x)) ]' = ctg(x)
sin(pi/6) = 1/2
Integral( 0, pi/2, sin^2(2x), dx ) = ???
Integral( 0, pi/2, sin^2(2x), dx ) = pi/4
Use: Integral( 0, pi, sin^2(x), dx ) = pi/2
Integral( 0, pi/2, sin^2(x)*cos^2(x), dx ) = ???
Integral( 0, pi/2, sin^2(x)*cos^2(x), dx ) = pi/16
Use: Integral( 0, pi/2, sin^2(2x), dx ) = pi/4
hint: sin(2x)=2cos(x)sin(x)
Integer( 0, 1, x^2 * sqrt(1-x^2), dx ) = ???
Integer( 0, 1, x^2 * sqrt(1-x^2), dx ) = pi/16
Use: Integral( 0, pi/2, sin2(x)cos2(x), dx ) = pi/16
hint: x=sin(t)
Integral( 0, 2, x^2 * sqrt(1 - 1/4*x^2), dx ) = ???
Integral( 0, 2, x^2 * sqrt(1 - 1/4*x^2), dx ) = pi/2
hint: x=2sin(t)
Integral( 0, pi/2, sin^2(x)*cos^2(x), dx ) = pi/16
Integral( 0, pi, sin^2(x), dx ) = pi/2 (page 87 in OLDTIMER)
/\n:-|N lim(x->0) ((1+x)^n - 1)/x = ???
/\n:-|N lim(x->0) ((1+x)^n - 1)/x = n
If n:-|N, let f(x) = (1+x)^n.
Notice that this limit calculates the derivative of f at 0.
a,b :- |C; a!=1, b!=1, b!=a
1 / (1-a)(1-b) = ???
1 / (1-a)(1-b) = (1/(1-a) - 1/(1-b)) / (a-b)
Let x:-|C\{0,1}.
1 / x(x-1) = ???
1 / x(x-1) = 1/(x-1) - 1/x
Let x:-|C\{0,1}.
1/(x-1) - 1/x = ???
1/(x-1) - 1/x = 1 / x(x-1)