heureka

$$\Delta{x} =$$ $${\mathtt{98\,200}}{\mathtt{\,-\,}}{\mathtt{94\,700}} = {\mathtt{3\,500}}$$

\Delta{x} =

$$\Delta{y}=$$ $${\mathtt{29\,500}}{\mathtt{\,-\,}}{\mathtt{28\,200}} = {\mathtt{1\,300}}$$

\Delta{y}=

$$r=\sqrt{\Delta{x}^2+\Delta{y}^2}=$$  $${\sqrt{{{\mathtt{3\,500}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{1\,300}}}^{{\mathtt{2}}}}} = {\mathtt{3\,733.630\: \!940\: \!518\: \!893\: \!867\: \!9}}$$

r=\sqrt{\Delta{x}^2+\Delta{y}^2}=

$$\theta= \tan^{-1}{(\frac{\Delta{y}}{\Delta{x}})} =$$  $$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{1\,300}}}{{\mathtt{3\,500}}}}\right)} = {\mathtt{20.376\: \!435\: \!213\: \!836^{\circ}}}$$

\theta= \tan^{-1}{(\frac{\Delta{y}}{\Delta{x}})} =

so

$$r = 3733.63 \mbox{ units}$$

r = 3733.63  \mbox{ units}

and

$$\theta = 20.3764 \ensuremath{^\circ}$$

\theta = 20.3764 \ensuremath{^\circ}

Hi Alan,

$$\\\cos{(\alpha)}=\textcolor[rgb]{1,0,0}{\pm}\frac{3}{5}\\
\text{if }{+\frac{3}{5}} \Rightarrow \alpha = \cos^{-1}(0.6) = 53.1301023542\ensuremath{^\circ}\\
\text{if }{-\frac{3}{5}} \Rightarrow \alpha = \cos^{-1}(-0.6) = 126.869897646\ensuremath{^\circ}\\
\sin{(53.1301023542\ensuremath{^\circ})}=\sin{(126.869897646\ensuremath{^\circ})}=0.8=\frac{4}{5}$$

Latex code:

\text{if }{+\frac{3}{5}} \Rightarrow \alpha = \cos^{-1}(0.6) = 53.1301023542\ensuremath{^\circ}\\
\text{if }{-\frac{3}{5}} \Rightarrow \alpha = \cos^{-1}(-0.6) = 126.869897646\ensuremath{^\circ}\\
\sin{(53.1301023542\ensuremath{^\circ})}=\sin{(126.869897646\ensuremath{^\circ})}=0.8=\frac{4}{5}

Primzahltest mit dem Satz von Wilson:$$(p-1)! \equiv -1 \bmod p\qquad oder \qquad (p-1)! \equiv p-1 \bmod p$$

Achtung die Fakultätsfunktion erzeugt riesige Zahlen! Der Web-Rechner liefert folgende Ergebnisse:

101: (101-1)! mod 101 = 100 ( 101 ist eine Primzahl! )

111: (111-1)! mod 111 = 0

121: (121-1)! mod 121 = 0

131: (131-1)! mod 131 = 130 ( 131 ist eine Primzahl! )

141: (141-1)! mod 141 = 0

151: (151-1)! mod 151 = 150 ( 151 ist eine Primzahl! )

161: (161-1)! mod 161 = 0

171: (171-1)! mod 171 = 0

181: (181-1)! mod 181 = 180 ( 181 ist eine Primzahl! )

191: (191-1)! mod 191 = 190 ( 191 ist eine Primzahl! )

Hi Melody,

i agree! The Calculator is wrong!

If p is a prime Number: $$(p-1)!\equiv -1 \bmod p\quad or \quad (p-1)!\equiv (p-1) \bmod p$$

Example p = 23 (23 is a prime number):

(23-1)! mod 23 must be 22 ( or -1).

The Calculator said: 12

Bye

Hi Melody,

thank you for $$\pm\infty$$.

Now to the limits:

\prod\limits_{i=1}^n x_i

$$\prod\limits_{i=1}^n x_i$$

\sum\limits_{i=1}^n i

$$\sum\limits_{i=1}^n i$$

 \lim\limits_{ n \to \infty }x_n

$$\lim\limits_{ n \to \infty }x_n$$

\int\limits_{x=0}^{x=1}

$$\int\limits_{x=0}^{x=1}$$

 \frac{40}{(1/10)} $$\frac{40}{(1/10)}$$ ?

$$\frac{40}{\big(1/10\big)}$$

\frac{40}{\big(1/10\big)}

Bye

let x=1:   $${\frac{{\mathtt{40}}}{{\mathtt{1}}}} = {\mathtt{40}}$$

let x=$$\frac{1}{10}$$=0.1: $${\frac{{\mathtt{40}}}{\left({\frac{{\mathtt{1}}}{{\mathtt{10}}}}\right)}} = {\mathtt{400}}$$

latex code: \frac{1}{10}

let x=$$\frac{1}{1000}$$=0.001: $${\frac{{\mathtt{40}}}{\left({\frac{{\mathtt{1}}}{{\mathtt{1\,000}}}}\right)}} = {\mathtt{40\,000}}$$

latex code: \frac{1}{1000}

let x=$$\frac{1}{10000000000}$$=0.0000000001: $${\frac{{\mathtt{40}}}{\left({\frac{{\mathtt{1}}}{{\mathtt{10\,000\,000\,000}}}}\right)}} = {\mathtt{400\,000\,000\,000}}$$

latex code: \frac{1}{10000000000}

let x=$$\frac{1}{100000000000000000000}$$=0.00000000000000000001:$${\frac{{\mathtt{40}}}{\left({\frac{{\mathtt{1}}}{{\mathtt{100\,000\,000\,000\,000\,000\,000}}}}\right)}} = {\mathtt{4\,000\,000\,000\,000\,000\,000\,000}}$$

latex code: \frac{1}{100000000000000000000}

...

let x=0=0.0000000000000000...: $$\boxed{\frac{40}{0}=\infty}$$

latex code: \boxed{\frac{40}{0}=\infty}

40/x  if x=0  is infinite!

Hallo Melody,

use

\usepackage{pgfplots}

Bye

heureka

$$\\\left(
\frac
{
150
\;\textcolor[rgb]{1,0,0}{\not}c\textcolor[rgb]{1,0,0}{\not}m
}
{
1\;rod
}
\right)*
\left( \frac{1\;piece
}
{1\frac{1}{4}\;\textcolor[rgb]{1,0,0}{\not}c\textcolor[rgb]{1,0,0}{\not}m} \right)=
\frac{150}{\frac{5}{4}}\;\frac{piece}{rod}=\frac{150*4}{5}\;\frac{piece}{rod}=
30*4\;\frac{piece}{rod}=120\;\frac{piece}{rod}
\\\\
\Rightarrow 120\;\mbox{pieces per one rod}$$

\\\left(
\frac
{
150
\;\textcolor[rgb]{1,0,0}{\not}c\textcolor[rgb]{1,0,0}{\not}m
}
{
1\;rod
}
\right)*
\left( \frac{1\;piece
}
{1\frac{1}{4}\;\textcolor[rgb]{1,0,0}{\not}c\textcolor[rgb]{1,0,0}{\not}m} \right)=
\frac{150}{\frac{5}{4}}\;\frac{piece}{rod}=\frac{150*4}{5}\;\frac{piece}{rod}=
30*4\;\frac{piece}{rod}=120\;\frac{piece}{rod}
\\\\
\Rightarrow  120\;\mbox{pieces per one rod}

$$\\\left(
\frac
{
1\;glass\
}
{
\frac{1}{8}\;\textcolor[rgb]{1,0,0}{\not}l
}
\right)*
\left( \frac{2.5\;
\textcolor[rgb]{1,0,0}{\not}{l}
}
{1\;bottle} \right)=2.5*8\;\frac{glass}{bottle}=20\;\frac{glass}{bottle}\\\\
\Rightarrow 20\;\mbox{glass per one bottle}$$

\\\left(
\frac
{
1\;glass\
}
{
\frac{1}{8}\;\textcolor[rgb]{1,0,0}{\not}l
}
\right)*
\left( \frac{2.5\;
\textcolor[rgb]{1,0,0}{\not}{l}
}
{1\;bottle} \right)=2.5*8\;\frac{glass}{bottle}=20\;\frac{glass}{bottle}\\\\
\Rightarrow  20\;\mbox{glass per one bottle}

$$\\\left(\frac{0.084\;kg}{1\;
\textcolor[rgb]{1,0,0}{\not}{w}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{k}
}\right)* \left( \frac{1\;
\textcolor[rgb]{1,0,0}{\not}{w}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{k}
}{7\;days} \right)=\frac{0.084}{7}\frac{kg}{days}=0.012\;\frac{kg}{days}\\\\
\Rightarrow 0.012\;\mbox{kg per day}$$

\\\left(\frac{0.084\;kg}{1\;
\textcolor[rgb]{1,0,0}{\not}{w}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{k}
}\right)* \left( \frac{1\;
\textcolor[rgb]{1,0,0}{\not}{w}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{e}\textcolor[rgb]{1,0,0}{\not}{k}
}{7\;days} \right)=\frac{0.084}{7}\frac{kg}{days}=0.012\;\frac{kg}{days}\\\\
\Rightarrow  0.012\;\mbox{kg per day}