ASVAB Mathematics Study Guide

Vocabulary flashcards featuring key definitions, formulas, and concepts from the ASVAB Mathematics review book.

Fraction

A mathematical representation of a part of a whole, written as $\frac{a}{b}$, where $a$ is the number of parts you have and $b$ is the total number of equal parts that make up the whole.

Improper Fraction Conversion

A method to convert an improper fraction $\frac{a}{b}$ into a mixed number $q\frac{r}{b}$, where $q$ is the quotient and $r$ is the remainder obtained from dividing $a$ by $b$.

Percent

A representation of a part of $100$ (e.g., $30\%$ means $30$ out of $100$ parts), calculated using the formula $\text{Percent of a number} = \frac{\text{Percent}}{100} \times \text{Number}$.

Percent Change

The relative change between an initial and final quantity, determined by the formula $\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$.

Product of Powers

An exponent rule stating that when multiplying powers with the same base, the exponents are added: $a^m \times a^n = a^{m+n}$.

Quotient of Powers

An exponent rule stating that when dividing powers with the same base, the exponent of the denominator is subtracted from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$.

Power of a Power

An exponent rule stating that raising a power to another power requires multiplying the exponents: $(a^m)^n = a^{m \times n}$.

Power of a Product

An exponent rule stating that a product raised to a power equals each factor raised to that power: $(a \times b)^n = a^n \times b^n$.

Zero Exponent Rule

An exponent property stating that any non-zero value raised to the power of zero equals one: $a^0 = 1$ (where $a \neq 0$).

Negative Exponent Rule

An exponent property stating that a non-zero base raised to a negative exponent equals the reciprocal raised to the positive exponent: $a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$).

Factorial

A mathematical operation denoted by $n!$ that calculates the product of all positive integers less than or equal to $n$ ($n! = n \times (n-1) \times (n-2) \times \dots \times 1$), with $0! = 1$ by definition.

Permutation

A formula used to determine the number of ordered arrangements of $n$ items taken $r$ at a time: $P(n,r) = \frac{n!}{(n-r)!}$.

Combination

A formula used to calculate the number of subsets of $r$ items chosen from $n$ total items without regard to order: $C(n,r) = \frac{n!}{r!(n-r)!}$.

Basic Logarithm Definition

The inverse of an exponential relationship, stating that $\log_b(x) = y$ if and only if $b^y = x$.

Common Logarithm

A logarithm with base $10$, conventionally written without an explicit base subscript: $\log_{10}(x) = \log(x)$.

Natural Logarithm

A logarithm with base $e$, denoted as $\log_e(x) = \ln(x)$.

Logarithm Product Rule

A logarithmic identity stating that the log of a product equals the sum of the logs of the factors: $\log_b(xy) = \log_b(x) + \log_b(y)$.

Logarithm Quotient Rule

A logarithmic identity stating that the log of a quotient equals the difference between the numerator log and the denominator log: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$.

Logarithm Power Rule

A logarithmic property stating that the log of a base raised to an exponent equals the exponent multiplied by the log: $\log_b(x^n) = n \cdot \log_b(x)$.

Change of Base Formula

A formula enabling conversion of a logarithm from base $b$ to base $c$: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$.

Mean (Average)

A measure of central tendency found by adding all data values and dividing by the total number of values: $\text{Mean} = \frac{\sum x_i}{n}$.

Median

A measure of central tendency that represents the middle value in an ordered dataset.

Mode

A measure of central tendency defined as the value that occurs most frequently in a dataset.

Range (Statistics)

The difference between the highest and lowest data values in a dataset, given by $\text{Range} = \text{Maximum value} - \text{Minimum value}$.

Variance

A statistical measure of data dispersion calculated as the average of the squared deviations from the mean: $\text{Variance}(\sigma^2) = \frac{\sum (x_i - \mu)^2}{n}$.

Standard Deviation

A statistical measure of data dispersion calculated as the square root of the variance: $\sigma = \sqrt{\text{Variance}}$.

Slope-Intercept Form

The linear equation $y = mx + c$, where $m$ is the slope indicating line steepness and $c$ is the y-intercept where the line crosses the y-axis.

Quadratic Formula

The formula used to solve any quadratic equation in the form $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Difference of Squares

An algebraic identity used for factoring that equates the subtraction of two squares to the product of their sum and difference: $a^2 - b^2 = (a - b)(a + b)$.

Pythagorean Theorem

A fundamental geometric relation in a right triangle stating that the sum of the squares of the legs equals the square of the hypotenuse: $a^2 + b^2 = c^2$.

Circumference of a Circle

The total distance around the edge of a circle, calculated using the formula $C = 2\pi r$, where $r$ is the radius.

Volume of a Rectangular Prism

The amount of space occupied by a rectangular prism, calculated using the formula $V = \text{length} \times \text{width} \times \text{height}$.

Volume of a Cylinder

The volume enclosed by a cylinder, determined by the formula $V = \pi r^2 h$, where $r$ is the radius of the circular base and $h$ is the height.

Surface Area of a Sphere

The total area of the outer surface of a sphere, calculated using the formula $A = 4\pi r^2$, where $r$ is the radius.