Contents

• Number Sets & Definitions
• Repeating Values
• Angular Units
• Indeterminate Values

1.1.1 Number Sets & Definitions

Real $$\R$$	All common numbers and multiples of one
Integer $$\Z$$	Real whole numbers that can be expressed without using a fraction $$\lbrace …,-2,-1,0,1,2,… \rbrace$$
Natural $$\N$$	Positive real integers (argued whether or not zero is included)
Rational	Any number that can be represented as a fraction inclusive of only natural numbers, and always either repeats or terminates as a decimal $$ \begin{array}{cc} 1/2 & 2/1 & 1.2 & 2.\overline{1} \end{array} $$
Irrational	Any number that cannot be represented as a fraction, and never repeats or terminates as a decimal $$ \begin{array}{cc} \sqrt{2} & e & \pi & \phi \end{array} $$
Imaginary	All common numbers that are multiples of $i$
Complex $$\Complex$$	Numbers comprised of both real and imaginary numbers

1.1.2 Repeating Values

Series of Nine

Repeating values can always be represented as a fraction with the repeated value over an equal number of digits of 9’s. $$.\overline{2}=\frac{2}{9}\qquad.\overline{137}=\frac{137}{999}$$ $$.\overline{142857}=\frac{142857}{999999}=\frac{1}{7}$$

Orders of Ten

Where a number starts repeating is a matter of multiples of 10, which can be isolated. $$433.\overline{3}=400+\frac{3}{9}·100=\frac{1300}{3}$$ $$.35\overline{7}=\frac{35}{100}+\frac{7}{9}·\frac{1}{100}=\frac{161}{450}$$ $$3.8\overline{3}=.5+3.\overline{33}=\frac{1}{2}+\frac{30}{9}=\frac{23}{6}$$

Proof Nine Repeating Equals One

Let $.\bar{9}$ equal a variable $$x=.\overline{9}$$ Multiply by $10$ $$10·x=9.\overline{9}$$ Separate $9.\bar{9}$ into $9$ and $.\bar{9}$ $$10·x=9+.\overline{9}$$ Substitute $.\bar{9}$ for $x$ $$10·x=9+x$$ Subtract $x$ $$9·x=9$$ Divide by $9$

1.1.3 Angular Units

A full rotation in one angular direction is 360° or two radians. This page will always use radians. $$180°=\pi$$

1.1.4 Indeterminate Values

Undefined forms with no number value. Arithmetic operation is forbidden because they create logical fallacies. $$\frac{0}{0}\qquad\frac{∞}{∞}$$ $$0·∞\quad ∞-∞$$ $$0^0\quad 1^∞\quad ∞^0$$