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.

Let a,b be complex numbers.

exp(a) = exp(b) <=> ????

exp(a) = exp(b) <=> ????

a = b + k*2*pi*i

for some integer k

for some integer k

Describe the set {z:-|C : exp(z)=1}.

{k*2*pi*i : k is an integer}

A good proof is on page 68 in the palace notebook.

A bad proof is on page 24 in OLDTIMER.

Be careful not to repeat the bad proof.

A good proof is on page 68 in the palace notebook.

A bad proof is on page 24 in OLDTIMER.

Be careful not to repeat the bad proof.

Describe the set {z:-C : cos(z)=0}.

{ pi/2 * (2*k+1) : k is an integer}

cos(z)=0 <=> z is real and cos(z)=0

cos(z)=0 <=> z is real and cos(z)=0

exp(z) = u(z) + i*v(z), u=Re(exp), v=Im(exp).

Find u and v.

Find u and v.

exp(x,y) = exp(x)*cos(y) + i*exp(x)*sin(y)

Describe the set {z:-C : sin(z)=0}.

{k*pi : k is an integer}

sin(z)=0 <=> z is real and sin(z)=0

sin(z)=0 <=> z is real and sin(z)=0

Is it possible that z is not real and cos(z)=0 ?

NO.

cos(z)=0 => z is real

cos(z)=0 => z is real

Is it possible that z is not real and sin(z)=0 ?

NO.

sin(z)=0 ==> z is real

sin(z)=0 ==> z is real

Is it possible that z is not real and cos(z)=1 ?

NO.

cos(z)=1 ==> z is real

cos(z)=1 ==> z is real

Is it possible that z is not real and sin(z)=1 ?

NO. sin(z)=1 ==> z is real

Follow this downward chain of reasoning:

1) sin(z) = 1

2) cos(z) = 0

3) z is real

Follow this downward chain of reasoning:

1) sin(z) = 1

2) cos(z) = 0

3) z is real

Is it possible that z is not real and cos(z) = -1 ?

NO.

cos(z)=-1 ==> z is real

cos(z)=-1 ==> z is real