Algebra and Rational Numbers Quiz

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Key Concepts:
- Variables: Symbols (e.g., x, y) that represent numbers.
- Expressions: Combinations of variables and constants (e.g., 3x + 2).
- Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
- Functions: Relations that assign exactly one output for each input (e.g., f(x) = x²).
Operations:
- Addition, subtraction, multiplication, and division of algebraic expressions.
- Distributive property: a(b + c) = ab + ac.
- Factoring: breaking down expressions into products of simpler expressions (e.g., x² - 5x = x(x - 5)).
Solving Equations:
- Isolate the variable using inverse operations.
- Check solutions by substituting back into the original equation.

Definition: Numbers that can be expressed as the quotient of two integers (a/b where b ≠ 0).
Key Properties:
- Include integers, fractions, and finite or repeating decimals.
- Closure under addition, subtraction, multiplication, and division (except by zero).
Examples:
- 1/2, -3, 0.75 (3/4), 0.333... (1/3).

Definition: Numbers that cannot be expressed as a simple fraction; non-repeating, non-terminating decimals.
Key Properties:
- Cannot be written as a ratio of integers.
- The square root of a non-perfect square is irrational (e.g., √2, √3).
Examples:
- π (pi), e (Euler's number), √2, √3.
Relation to Rational Numbers:
- Together, rational and irrational numbers make up the real numbers.
Number Line: Irrational numbers are dense on the number line, meaning between any two rational numbers, there exists an irrational number.

A branch of mathematics focused on symbols and the manipulation of these symbols.
Variables represent unknown values, typically denoted by letters such as x and y.
Expressions consist of variables and constants combined through operations, e.g., 3x + 2.
Equations declare the equality of two expressions, e.g., 2x + 3 = 7.
Functions define a relationship where each input corresponds to exactly one output, such as f(x) = x².
Basic operations involve addition, subtraction, multiplication, and division applied to algebraic expressions.
The distributive property states that a(b + c) = ab + ac, facilitating simplification of expressions.
Factoring involves decomposing expressions into simpler products, exemplified by x² - 5x = x(x - 5).
To solve equations, isolate the variable through the use of inverse operations and verify solutions by substitution.

Rational numbers are expressible as a fraction of two integers (a/b) where b is not zero.
These numbers encompass integers, fractions, and decimals that are either finite or repeating.
Rational numbers exhibit closure under the operations of addition, subtraction, multiplication, and division (excluding division by zero).
Examples include 1/2, -3, 0.75 (equivalent to 3/4), and 0.333… (which represents 1/3).

Irrational numbers cannot be represented as a simple fraction; they are non-repeating and non-terminating decimals.
They cannot be expressed as a ratio of integers.
The square root of any non-perfect square yields an irrational number, e.g., √2 and √3.
Notable examples include π (pi), e (Euler's number), √2, and √3.
Irrational and rational numbers together constitute the set of real numbers.
On a number line, irrational numbers are densely packed; between any two rational numbers, there exists at least one irrational number.

A mathematical discipline focused on symbols and the manipulation of those symbols.
Variables serve as placeholders for unknown values, commonly represented by letters.
Expressions consist of numbers, variables, and operations, such as 3x + 2.
Equations are statements that declare equality, exemplified by equations like 2x + 3 = 7.
Functions illustrate relationships between values, typically denoted as f(x).
Fundamental operations include addition, subtraction, multiplication, and division applied to algebraic expressions.
Factoring involves decomposing expressions into simpler multiplicative components.
The Distributive Property allows for distributing multiplication over addition, represented as a(b + c) = ab + ac.
Types of Equations:
- Linear equations follow the form y = mx + b, with m representing the slope and b as the y-intercept.
- Quadratic equations take the form ax² + bx + c = 0, solvable through methods such as factoring, completing the square, or using the quadratic formula.
In graphing, the coordinate plane consists of a horizontal x-axis and a vertical y-axis, where points (x, y) can be plotted to illustrate relationships.

Defined as numbers that can be expressed as fractions p/q, with p and q being integers and q ≠ 0.
Incorporates integers, fractions, and both finite and repeating decimals.
Examples of rational numbers include 1/2, -3, 0.75, and 0.333...
Operations with rational numbers:
- Addition/Subtraction may require a common denominator.
- Multiplication is achieved by multiplying the numerators and denominators, illustrated by p/q * r/s = pr/qs.
- Division is executed by multiplying by the reciprocal, represented as p/q ÷ r/s = p/q * s/r.

Numbers that cannot be expressed as a simple fraction p/q, where p and q are integers.
Characterized by their non-repeating and non-terminating decimal nature.
Common examples include √2, π (pi), and e (Euler's number).
Properties of irrational numbers highlight that they cannot be accurately expressed as fractions.
They frequently emerge in the solutions to equations involving square roots or greater roots.
In relation to rational numbers, irrational numbers cannot be pinpointed on the number line; they fill in the gaps, contributing to a continuous real number line.