# 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.

A,B c X
1) A c X\B <=> ???
2) A c X\B <=> ???
1) A c X\B <=> B c X\A
2) A c X\B <=> B n A = O
A,B c X
1) A n (X\B) = O <=> ???
2) A n (X\B) = O <=> ???
1) A n (X\B) = O <=> A c B
2) A n (X\B) = O <=> X\B c X\A
A,B c X
1) A c B <=> ???
2) A c B <=> ???
3) A c B <=> ???
1) A c B <=> X\B c X\A
2) A c B <=> A n (X\B) = O
3) A c B <=> A u B = B
A,B c X
A \ (X\B) = ???
A \ (X\B) = A n B = B n A = B \ (X\A)
Let X = {x:|N->|C : +(n=1 to oo) |x(n)| < oo}.
If x:-X, let g(x) = +(n=1 to oo) |x(n)|.
Is g a norm on X?
Yes.
Does this series converge?
1/n * (-1)^n * i^( n*(n+1) )
YES.
a[n] = (-1)^n * i^( n*(n+1) )
a[1]=1 a[2]=-1 a[3]=-1 a[4]=1
a[n+4] = a[n]
Hence the series a[n] is bounded.
Let D be an open subset of |R^2.
Let g : D -> |R. Suppose that the partial derivatives gx, gy both exist on D.
Let (a,b) be a point in D.
Suppose that /\(x,y):-D g(x,y) <= g(a,b).
What can we conclude?
gx(a,b) = gy(a,b) = 0
Prove that
/\x:-|R \/k:-|Z [ -PI < x + k*2*PI <= PI ]
hint: k-1 < (x-PI)/(2*PI) <= k
page 62 in OLDTIMER
-PI <= x,y <= PI
cos(x) = cos(y)
What can we conclude?
x = -y or x = y
Let x,y be real numbers such that cos(x)=cos(y).
What can we conclude?
\/k:-|Z x = y + k*2*PI or x = -y + k*2*PI
page 62 in OLDTIMER