Pre-Algebra

Arithmetics

Number Types

$\mathbb{N} = \{1,2,3,\dots\} \quad \mathbb{N}_0 = \mathbb{N} \cup \{0\} \quad \mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\}$ $\mathbb{Q} = \left\{\frac{a}{b} \mid a,b \in \mathbb{Z},\ b \ne 0 \right\}$ $\mathbb{I} = \Bigl\{ x \in \mathbb{R} \;\Big|\; \forall a,b \in \mathbb{Z},\ b\neq 0, \ x \neq \frac{a}{b} \Bigr\}$ $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ Each set is contained in the next: $\mathbb{N} \subset \mathbb{N}_0 \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ .

Order of Operations (PEMDAS)

P --- Parentheses
E --- Exponents
M/D --- Multiplication & Division (left to right, equal precedence)
A/S --- Addition & Subtraction (left to right, equal precedence)

Properties of Arithmetic

Property	Addition	Multiplication
Commutative	$a + b = b + a$	$ab = ba$
Associative	$(a+b)+c = a+(b+c)$	$(ab)c = a(bc)$
Identity	$a + 0 = a$	$a \cdot 1 = a$
Inverse	$a + (-a) = 0$	$a \cdot \frac{1}{a} = 1$ ( $a \neq 0$ )

Distributive	$a(b + c) = ab + ac$
Zero Property	$a \cdot 0 = 0$

Primes, Factors, GCF & LCM

$3 \times 4 = 12$ : 3 and 4 are factors of 12; 12 is a multiple of both.

A prime has no factors besides 1 and itself. By the Fundamental Theorem of Arithmetic, every integer $> 1$ has a unique prime factorization:

$60 = 2^2 \times 3 \times 5$

Given prime factorizations, e.g.\ $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$ :

GCF: min power of each shared prime $\to 2^1 \times 3^1 = 6$
LCM: max power of each prime $\to 2^2 \times 3^2 = 36$

Fractions

$\frac{a}{b} = a \div b$ , with $b \neq 0$ .

Equivalent fractions: $\frac{a}{b} = \frac{ak}{bk}$ for $k \neq 0$ . To simplify, divide by the GCF.

Adding/subtracting: requires a common denominator (LCD):

$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Exponents & Roots

Exponents

$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$

All rules follow from the definition: $a^0 = 1$ $a^m \cdot a^n = a^{m+n}$ $a^{-n} = \frac{1}{a^n}$ $(a^m)^n = a^{mn}$ $(ab)^n = a^n b^n$

Roots

The $n$ -th root inverts the $n$ -th power: $\sqrt[n]{a} = b \iff b^n = a,\; b \geq 0$ (for even $n$ ).

Simplify by extracting the largest perfect power: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

Odd roots are defined for negative arguments: $\sqrt[3]{-8} = -2$ .

Scientific Notation

$n \times 10^{k}, \quad 1 \leq n < 10, \quad k \in \mathbb{Z}$

Algebra

Expressions and Simplification

Variables stand for unknown or varying quantities. In $3x^2 + 5x - 7$ : terms are $3x^2$ , $5x$ , $-7$ ; coefficients are 3, 5; constant is $-7$ .

Like terms share the same variable and exponent: $5x + 3x = 8x$ , but $5x + 3x^2$ cannot be combined.

Distributive property: $a(b + c) = ab + ac$ . Expanding and factoring are the same operation in opposite directions.

Solving Equations

Isolate the variable by applying inverse operations to both sides, outermost first.

Three outcomes:

$x = \text{number}$ --- unique solution
Contradiction ( $3 = 7$ ) --- no solution
Tautology ( $5 = 5$ ) --- infinitely many solutions

Inequalities

Same as equations, except multiplying or dividing by a negative reverses the inequality (negation reverses the order on $\mathbb{R}$ ).

Polynomials and the Distributive Property

Polynomial multiplication is repeated distribution: every term in the first factor multiplies every term in the second. For binomials (FOIL):

\begin{align*} (x + 3)(x + 2) &= x^2 + 2x + 3x + 6 = x^2 + 5x + 6 \end{align*}

FOIL is not a separate rule --- it is distribution. Generalizes to polynomials of any degree.

Quadratics

Standard form: $y = ax^2 + bx + c$ , $a \neq 0$ . Graph is a parabola.

$a > 0$ : opens up; $a < 0$ : opens down
Vertex (axis of symmetry) at $x = -\frac{b}{2a}$

Roots of $ax^2 + bx + c = 0$ :

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant $\Delta = b^2 - 4ac$ classifies solutions:

$\Delta > 0$ : two distinct real roots
$\Delta = 0$ : one repeated real root
$\Delta < 0$ : no real roots (two complex conjugate roots)

Graphs & Functions

The Coordinate Plane

Two perpendicular axes assign each point in $\mathbb{R}^2$ a unique ordered pair $(x, y)$ .

Slope

Slope is the rate of change of $y$ with respect to $x$ .

$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

$m > 0$ : increasing. $m < 0$ : decreasing. $m = 0$ : horizontal. $m$ undefined: vertical.

Lines

A line is determined by two pieces of data:

Slope-intercept: $y = mx + b$
Point-slope: $y - y_1 = m(x - x_1)$

$x$ -intercept: set $y = 0$ . $\quad$ $y$ -intercept: set $x = 0$ .

Parallel $\iff$ equal slopes (or both vertical). Perpendicular $\iff m_1 \cdot m_2 = -1$ .

Distance and Midpoint

Distance is the Pythagorean theorem on coordinate differences. Midpoint averages each coordinate.

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$

Functions

A function $f: D \to R$ assigns each input exactly one output. Domain $D$ : set of valid inputs. Range $R$ : set of attained outputs.

Linear ( $f(x) = mx + b$ ): constant rate of change.
Nonlinear (e.g.\ $f(x) = x^2$ ): rate of change varies.

Systems of Equations

Find values satisfying all equations simultaneously. Geometrically: intersection of graphs.

Substitution: express one variable from one equation, substitute into the other.

Outcomes (for two linear equations in two unknowns):

Intersecting lines $\to$ unique solution
Parallel lines $\to$ no solution
Coincident lines $\to$ infinitely many solutions

Geometry

Angles

An angle measures rotation between two rays sharing a vertex.

Acute: $< 90^{\circ}$ $\quad$ Right: $= 90^{\circ}$ $\quad$ Obtuse: $90^{\circ}$ to $180^{\circ}$ $\quad$ Straight: $= 180^{\circ}$

Complementary: sum to $90^{\circ}$ . Supplementary: sum to $180^{\circ}$ . Vertical angles (opposite angles at an intersection) are equal:

Triangles

By sides: scalene / isosceles / equilateral. By angles: acute / right / obtuse. Interior angles sum to $180^{\circ}$ .

Parallel Lines & Transversals

A transversal crossing two parallel lines creates 8 angles with only two distinct measures: $\alpha$ and $180^{\circ} - \alpha$ . All named relationships follow from this.

Corresponding (same position at each crossing): $\angle 1 = \angle 5$ , etc.
Alternate interior (opposite sides, between the lines): $\angle 3 = \angle 6$ , $\angle 4 = \angle 5$
Co-interior (same side, between the lines): $\angle 3 + \angle 5 = 180^{\circ}$

Pythagorean Theorem

For a right triangle with legs $a, b$ and hypotenuse $c$ :

$c^2 = a^2 + b^2$

The distance formula is a direct consequence.

Perimeter & Area

Shape	Area	Perimeter / Circumference
Rectangle	$A = lw$	$P = 2(l + w)$
Triangle	$A = \frac{1}{2}bh$
Parallelogram	$A = bh$
Trapezoid	$A = \frac{1}{2}h(b_1 + b_2)$
Circle	$A = \pi r^2$	$C = 2\pi r$

Volume & Surface Area

$V = (\text{base area}) \times h$ . Shapes tapering to a point get the $\frac{1}{3}$ factor.

Prisms and Cylinders (constant cross-section)

Pointed shapes ( $\frac{1}{3}$ factor)

( $s$ = slant height of cone, $l$ = slant height of pyramid, $B$ = base area, $P$ = base perimeter)

Sphere

$V = \frac{4}{3}\pi r^3, \qquad SA = 4\pi r^2$

Transformations

Rigid motions (isometries) preserve distances: reflection, translation, rotation. Dilation preserves shape but scales distances.

Data & Probability

Statistics

Mean: $\bar{x} = \frac{1}{n}\sum x_i$ . Sensitive to outliers.
Median: middle value of sorted data. Robust to outliers.
Mode: most frequent value.
Range: $\max - \min$ .
IQR: $Q_3 - Q_1$ (spread of the middle 50%).

Probability

For a finite sample space with equally likely outcomes:

$P(A) = \frac{|A|}{|\Omega|}, \qquad 0 \leq P(A) \leq 1$

When outcomes are not equally likely, estimate $P(A)$ empirically via relative frequency.