TSI MATH CRAM SHEET

A Comprehensive review set for the TSI Math exam covering number classifications, formulas, geometry, and algebraic principles.

What is a natural number?

A natural number is the number $1$ or any number obtained by adding $1$ to it one or more times ($1, 2, 3, 4, 5, …$).

How are whole numbers defined?

Whole numbers are positive integers or zero ($0, 1, 2, 3, 4, 5, …$) and do not include fractions or decimal parts.

What is an integer?

An integer is any whole number or its opposite ($-5, -4, -3, -2, -1, 0, 1, 2, 3 …$).

How is a rational number defined?

A number that can be expressed as a ratio or fraction.

What is the definition of a complex number?

A complex number is written in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is a formal square root of $-1$ ($i = \text{√-1}$, $i^2 = -1$).

What is the definition of absolute value?

Absolute value is the distance away from zero a number is on the number line; it is always positive and written as $|x|$.

What do the mathematical symbols $ϕ$, $∞$, and $e$ represent?

$ϕ$ is the Golden Ratio, $∞$ is Infinity, and $e$ is Euler's Number.

What do the mathematical symbols $∀$ and $∈$ represent?

$∀$ is the Universal Quantifier and $∈$ is the Membership Sign.

What are the simplified rules for exponents when multiplying and dividing powers with the same base number?

To multiply, add the exponents ($a^n \times a^m = a^{n+m}$). To divide, subtract the exponents ($a^n ÷ a^m = a^{n-m}$).

What is the result of any number raised to the power of $0$?

Any number raised to the power of $0$ is equal to $1$ ($a^0 = 1$).

What is the distance formula for two points on a coordinate plane?

$$d = \text{√}((x_2 - x_1)^2 + (y_2 - y_1)^2)$$

What is the slope-intercept form of a linear equation?

$y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

What is the point-slope form of a linear equation?

$y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ are point coordinates.

What is the Midpoint Formula?

M = \begin{pmatrix} \frac{x_1 + x_2}{2}, & \frac{y_1 + y_2}{2} \right)

What are the four types of slope shown on a graph?

Positive slope, negative slope, slope of zero, and undefined slope.

What does the FOIL method stand for when multiplying binomials?

First, Outer, Inner, Last.

What is the Quadratic Formula used to solve an equation in the form $ax^2 + bx + c = 0$?

$$x = \frac{-b ± \text{√}(b^2 - 4ac)}{2a}$$

What are the three types of polynomials based on the number of terms?

Monomial (one term), Binomial (two terms), and Trinomial (three terms).

How is a function defined?

A function is a relation between a set of inputs and a set of outputs where each input is related to exactly one output.

What are the formulas for Percent Increase and Percent Decrease?

Percent Increase = $\frac{\text{new value} - \text{original value}}{\text{original value}} \times 100$; Percent Decrease = $\frac{\text{original value} - \text{new value}}{\text{original value}} \times 100$.

What is the rule for dividing fractions?

Keep the first fraction, change division to multiplication, flip the second fraction to its reciprocal, and then multiply ($\frac{A}{B} ÷ \frac{C}{D} = \frac{A \times D}{B \times C}$).

Describe the components of a Box Plot.

A Box Plot shows statistical distribution including the Minimum value, Lower quartile, Median, Upper quartile, and Maximum value.

Define complementary and supplementary angles.

Complementary angles are two angles whose sum is exactly $90^∘$, and supplementary angles are two angles whose sum is exactly $180^∘$.

What are vertical angles?

Angles that are not adjacent because they share a vertex but have no common side.

What is the Sum of Interior Angles theorem for triangles?

The sum of the interior angles of any triangle is always $180^∘$.

What are the side length ratios for a $45-45-90$ special right triangle?

The sides are in the ratio $x : x : x\text{√2}$.

What are the side length ratios for a $30-60-90$ special right triangle?

The sides are in the ratio $x : x\text{√3} : 2x$.

What is the Pythagorean Theorem?

$c^2 = a^2 + b^2$, where $c$ is the hypotenuse and $a$ and $b$ are the legs.

What is the area formula for a Trapezoid?

$$A = \frac{(b_1 + b_2)h}{2}$$

What are the formulas for the volume and surface area of a sphere?

$V = \frac{4}{3}πr^3$ and $SA = 4πr^2$.

What are the formulas for the volume and surface area of a cylinder?

$V = πr^2h$ and $SA = 2πr(r + h)$.

What is the Greatest Common Factor (GCF)?

The largest number that is a factor of two or more whole numbers.

What is the Least Common Multiple (LCM)?

The smallest number that is a multiple of two or more whole numbers.

Define constant and coefficient in an algebraic expression.

A constant is a fixed value that does not change, while a coefficient is a constant factor of a term as distinguished from a variable.

In a system of linear equations, what are the characteristics of an 'Inconsistent' system?

An inconsistent system has parallel lines and no solutions.