1.1.1 Order of Operations
Please Excuse My Dear Aunt Sally
Parentheses, Exponents, Multiplication, Division, Addition & Subtraction
1.1.2 Transitive Property
If two quantities are equal with a third, then they are equal with each other.
$$ \text{If } a=c \text{ and } b=c \text{, then } a=b. $$
1.1.3 Fundamental Properties
|
Additive |
Multiplicative |
|---|
| Commutativity |
$a+b=b+a$ |
$a·b=b·a$ |
| Associativity |
$(a+b)+c=a+(b+c)$ |
$(a·b)·c=a·(b·c)$ |
| Identity |
$a+0=a=0+a$ |
$a·1=a=1·a$ |
| Inverse |
$a+(-a)=0$ |
$a·a^{-1}=1$ |
1.1.4 Distribution
| Distributive property |
$a·(b+c)=a·b+a·c$ |
| Distribution of one negative |
$\frac{-1}{\phantom{-}1}=\frac{\phantom{-}1}{-1}=-\frac{1}{1}=-1$ |
| Distribution of two negatives |
$\frac{-1}{-1}=-\big(-\frac{1}{1}\big)=1$ |
1.1.5 Reciprocal Rule of Division
| For expressions |
$a=\frac{1}{1/a}$ |
| For evaluating fractions |
$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$ |
1.1.6 Plus or Minus Notation
If $±$ is used on one side of an equation, then the equation has two solutions
$$x=a±b\medspace→\medspace
\begin{cases}
x=a+b \\
x=a-b
\end{cases}$$
If $±$ is used on two sides of an equation, then when it is one sign on one side, it is the same on the other
$$±x=a±b\medspace→\medspace
\begin{cases}
+x=a+b \\
-x=a-b
\end{cases}$$
If $±$ and $∓$ are used, then when one is positive, the other is negative
$$ ±x=a∓b\medspace→\medspace
\begin{cases}
+x=a-b \\
-x=a+b
\end{cases}$$
If $±$ is used in an exponent, it indicates multiplication and division, and the same rules above apply
$$x·y^{±1}=a±b\medspace→\medspace
\begin{cases}
x·y=a+b \\
x/y=a-b
\end{cases}$$
1.1.7 Mathematical & Logical Operators
| Symbol |
Meaning |
Example |
Translation |
| $|x|$ |
absolute value |
$|-x|=x$ |
The positive value or magnitude of $x$ |
| $\Vert\vec{v}\Vert$ |
magnitude |
$\Vert\vec{v}\Vert=\sqrt{v_x^2+v_y^2}$ |
The length of a line or vector |
| $\parallel$ |
parallel |
$\vec{v}\parallel\vec{w}$ |
$\vec{v}$ is parallel to $\vec{w}$ |
| $\perp$ |
perpendicular |
$\vec{v}\perp\vec{w}$ |
$\vec{v}$ is perpendicular to $\vec{w}$ |
| $!$ |
factorial |
$5!=1·2·3·4·5$ |
The product of all integers to the specified value |
| $\therefore$ |
therefore |
$x^2=4 \therefore x= \pm 2$ |
One is true, therefore the other is true |
| $\bmod$ |
modulus |
$4 \bmod 3 = 1$ |
4/3 has a remainder of 1 |
| $\forall$ |
for all |
$\forall x≥0$ |
for every non-negative value |
| $\isin$ |
element of |
$\forall x \isin \Z$ |
$x$ is an integer |
| $\land$ |
and |
$x>0 \land y>0$ |
Both of these statements are true |
| $\lor$ |
or |
$x>0 \lor y>0$ |
One or both of these statements are true |
1.1.8 No Solution
When a numerical conclusion such as $2=0$ occurs, it either means that no solutions exist to a given scenario or that arithmetical rules were broken to arrive at the conclusion