Math Reviewer

Real Number System

A classification of numbers that includes positive, negative, and zero values, which can be rational or irrational.

Rational Numbers

Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

Irrational Numbers

Numbers that cannot be expressed as a simple fraction, having non-repeating, non-terminating decimal expansions.

Closure Property

The principle stating that performing an operation on two numbers in a set results in a number that is also in the same set.

Commutative Property

A property indicating that the order of numbers does not affect the result of an operation, applicable to both addition and multiplication.

Associative Property

A property that states the way numbers are grouped in an operation does not change their result.

Distributive Property

A property that relates multiplication to addition, stating that multiplying a number by a sum is the same as multiplying each addend separately and then adding the results.

Identity Element

A number that, when used in an operation with another number, does not change the value of that number.

Inverse Element

A number that, when combined with another number in an operation, results in the identity element.

Absolute Value

The distance of a number from zero on the number line, always expressed as a non-negative value.

Opposite Numbers

Two numbers that are equidistant from zero on the number line but have opposite signs.

Double-Negative Property

A mathematical rule stating that the negation of a negative number results in a positive number.

Linear Equation

An equation that represents a straight line when graphed, typically in the form ax + b = 0.

Quadratic Equation

An equation that can be expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Inequalities

Mathematical expressions that show the relationship between two values that are not necessarily equal.

Mean

The average value of a set of numbers, calculated by dividing the sum of the numbers by the count of the numbers.

Median

The middle value in a list of numbers, which separates the higher half from the lower half.

Mode

The value that appears most frequently in a data set.

Permutations

Different arrangements of a set of items, where the order matters.

Combinations

Selections of items from a larger set, where the order does not matter.

Probability

A measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

Data Interpretation

The process of analyzing and making sense of data presented in various forms, such as graphs or tables.

Polynomials

Mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

Rational Expressions

Fractions that contain polynomials in the numerator and denominator.

Complex Numbers

Numbers that have a real part and an imaginary part, typically expressed in the form a + bi, where i is the imaginary unit.

Trigonometric Functions

Functions that relate the angles of a triangle to the lengths of its sides, commonly used in geometry and physics.

Pythagorean Theorem

A fundamental relation in Euclidean geometry among the three sides of a right triangle, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Word Problems

Mathematical problems presented in a narrative form, requiring translation into mathematical expressions or equations to solve.

Solution to x + 5 = -3

The value of x that satisfies the equation is found by isolating x, resulting in x = -8.

Additive Identity

A property stating that adding zero to any number does not change the value of that number.

Reciprocals

Two numbers whose product equals one; for example, 4 and 0.25 are reciprocals.

Negative Exponents

A mathematical notation indicating that a number is to be taken as the reciprocal raised to the positive exponent.

Law of Signed Numbers

Rules governing the addition, subtraction, multiplication, and division of real numbers based on their signs.

PEMDAS

An acronym representing the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Operations with Decimals

The process of performing arithmetic operations such as addition, subtraction, multiplication, and division with decimal numbers.

Operations with Fractions

Rules for adding, subtracting, multiplying, and dividing fractions, including finding a common denominator.

Changing Percent to Decimals

The process of converting a percentage into a decimal by dividing by 100.

Changing Decimals to Percent

The process of converting a decimal into a percentage by multiplying by 100 and adding the percent sign.

Percentage

A fraction expressed as a part of 100, often denoted by the symbol '%', used to compare relative sizes.

Terminating Decimal

A decimal that has a finite number of digits after the decimal point.

Repeating Decimal

A decimal in which a digit or group of digits repeats infinitely.

Scientific Notation

A way to express numbers as a product of a number between 1 and 10 and a power of ten.

Ratio

A relationship between two numbers indicating how many times the first number contains the second.

Proportion

An equation that states that two ratios are equal.

Direct Variation

A relationship where one variable is a constant multiple of another, expressed as y = kx.

Inverse Variation

A relationship where one variable increases as the other decreases, expressed as y = k/x.

Joint Variation

A relationship where a variable depends on two or more other variables, expressed as y = kxz.

Fraction

A numerical quantity that is not a whole number, represented as a ratio of two integers.

Long Division

A method for dividing large numbers by breaking down the division process into simpler steps.

Decimal

A number that includes a decimal point, representing a fraction of a whole.

Cross Multiplication

A method used to solve proportions by multiplying the numerator of one fraction by the denominator of the other.

Constant of Proportionality

A constant value that relates two variables in a proportional relationship.

Exponent

A mathematical notation indicating the number of times a number is multiplied by itself.

Permutation

An arrangement of objects in a specific order, where the order of selection is important.

Combination

A selection of items from a larger set where the order does not matter.

Factorial

The product of all positive integers up to a specified number, denoted by that number followed by an exclamation mark.

Circular Permutation

The arrangement of objects in a circle, where the arrangement is considered the same if it can be rotated into another arrangement.

Repetition Allowed

A scenario in permutations where the same item can be chosen more than once.

Repetition Not Allowed

A scenario in permutations where each item can only be chosen once.

Distinct Objects

Items that are different from one another, allowing for unique arrangements.

Non-Distinct Objects

Items that are identical in some way, affecting the total number of unique arrangements.

Sample Space

The set of all possible outcomes of a probability experiment.

Event

A specific outcome or a set of outcomes of a probability experiment.

Combinatorial Formula

Mathematical expressions used to calculate combinations and permutations.

nPr

The notation representing the number of permutations of n items taken r at a time.

nCr

The notation representing the number of combinations of n items taken r at a time.

Binomial Coefficient

A coefficient that represents the number of ways to choose a subset of items from a larger set, often denoted as nCr.

Probability Distribution

A function that describes the likelihood of obtaining the possible values that a random variable can take.

Random Variable

A variable whose possible values are numerical outcomes of a random phenomenon.

Independent Events

Two events that do not affect each other's occurrence.

Dependent Events

Two events where the occurrence of one event affects the probability of the other.

Mutually Exclusive Events

Events that cannot occur at the same time.

Complementary Events

Two events where the occurrence of one event means the other cannot occur.

Arithmetic Progression

A sequence of numbers in which the difference between consecutive terms is constant, known as the common difference.

Geometric Progression

A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Common Difference

The fixed amount added to each term in an arithmetic progression to get the next term.

Common Ratio

The fixed factor by which each term in a geometric progression is multiplied to obtain the next term.

Nth Term

The term in a sequence that is located at position n, often expressed as a formula involving n.

Sum of an Arithmetic Series

The total of the terms in an arithmetic progression, calculated using the formula Sn = n/2 * (a1 + a_n), where n is the number of terms.

Outcome

A possible result of a random experiment, such as tossing a coin or rolling a die.

Favorable Outcomes

The outcomes of an event that are considered successful or desired.

Total Outcomes

The complete set of possible outcomes from a random experiment.

Number Series

A sequence of numbers that follows a specific pattern or rule.

Pattern Recognition

The ability to identify regularities and trends in sequences or data.

Fixed Number

A constant value that does not change in a given context.

First Term

The initial term in a geometric progression, often denoted as a₁.

Sum of the First n Terms

The total of the first n terms of a geometric progression, calculated using the formula Sₙ = a₁(1 - rⁿ) / (1 - r) for r ≠ 1.

Exponential Growth

A pattern of data that shows greater increases over time, often modeled by a geometric progression.

Term

An individual element or number in a sequence or series.

Sequence

An ordered list of numbers that follow a specific pattern or rule.

Population Growth

The increase in the number of individuals in a population, often modeled using geometric progression when growth is proportional to the current population.

Example of Geometric Progression

A sequence such as 2, 6, 18, 54, … where each term is multiplied by 3, making the common ratio 3.

Shelter, Clothing and Socials

A category of expenses that includes housing costs, clothing purchases, and social activities.

Education, Transport, Health

A category of expenses dedicated to educational costs, transportation needs, and healthcare services.

Monthly Income

The total amount of money earned by a family in one month.

Budget

A plan that outlines expected income and expenses over a specific period.

Difference

The result of subtracting one quantity from another.

Appropriation

The allocation of funds for a specific purpose or category of expenses.

Savings

Money that is set aside for future use or emergencies.

Insurance

A financial arrangement that provides protection against potential future losses.