Arithmetic Foundations: Integers, Number Line, and Basic Operations

Vocabulary flashcards covering key concepts from the arithmetic foundations: integers, number line, and basic operations.

Integers

All counting numbers and whole numbers, plus their negative counterparts (e.g., -3, -2, -1, 0, 1, 2, 3).

Natural numbers

Counting numbers: 1, 2, 3, … to infinity; numbers you can count on your fingers.

Whole numbers

Natural numbers including zero: 0, 1, 2, 3, …

Zero (0)

The additive identity; located at the center of the number line; included in the whole numbers.

Number line

A visual representation of integers with zero at the center; numbers increase to the right and decrease to the left (can be horizontal or vertical).

Positive infinity

Right end of the number line representing unbounded growth; not a number or integer.

Negative infinity

Left end of the number line representing unbounded negative growth; not a number or integer.

Infinity

The idea of unbounded quantity; not an integer.

Addition

Operation of combining numbers to form a sum; e.g., 10 + 4 = 14.

Subtraction

Operation of taking away one number from another to form a difference; e.g., 15 - 7 = 8.

Multiplication

Repeated addition; a × b means adding a to itself b times (or vice versa); e.g., 4 × 5 = 20.

Division

Inverse of multiplication; splitting a number into equal groups; e.g., 10 ÷ 2 = 5; 2 × 5 = 10.

Exponents

Shorthand for repeated multiplication; a^b means multiply a by itself b times; e.g., 3^3 = 27; 2^2 = 4.

Factorials

Notation n! means multiply n down to 1: n! = n × (n−1) × … × 1; e.g., 6! = 720.

Non-integers

Numbers that are not integers (e.g., 1/2, -1.7, pi); they have fractional or irrational parts.

Arbitrary orientation of the number line

The convention that moving right increases value is a human choice; other orientations (e.g., vertical, or different reading directions) are possible in different contexts.

Rational Numbers

Numbers that can be expressed as a fraction \frac{a}{b} where a and b are integers and b \neq 0; includes integers, terminating decimals, and repeating decimals (e.g., \frac{1}{2}, -3, 0.25, 0.333…).

Irrational Numbers

Numbers that cannot be expressed as a simple fraction \frac{a}{b} ; their decimal representations are non-terminating and non-repeating (e.g., \pi, \sqrt{2}, e).

Real Numbers

All rational and irrational numbers; encompasses all numbers that can be represented on a number line.

Absolute Value

The distance of a number from zero on the number line, always non-negative; denoted by vertical bars (e.g., |{-3}| = 3, |3| = 3).

Order of Operations (PEMDAS

Integers

All counting numbers and whole numbers, plus their negative counterparts (e.g., -3, -2, -1, 0, 1, 2, 3).

Natural numbers

Counting numbers: 1, 2, 3, … to infinity; numbers you can count on your fingers.

Whole numbers

Natural numbers including zero: 0, 1, 2, 3, …

Zero (0)

The additive identity; located at the center of the number line; included in the whole numbers.

Number line

A visual representation of integers with zero at the center; numbers increase to the right and decrease to the left (can be horizontal or vertical).

Positive infinity

Right end of the number line representing unbounded growth; not a number or integer.

Negative infinity

Left end of the number line representing unbounded negative growth; not a number or integer.

Infinity

The idea of unbounded quantity; not an integer.

Addition

Operation of combining numbers to form a sum; e.g., 10 + 4 = 14.

Subtraction

Operation of taking away one number from another to form a difference; e.g., 15 - 7 = 8.

Multiplication

Repeated addition; a × b means adding a to itself b times (or vice versa); e.g., 4 × 5 = 20.

Division

Inverse of multiplication; splitting a number into equal groups; e.g., 10 ÷ 2 = 5; 2 × 5 = 10.

Exponents

Shorthand for repeated multiplication; a^b means multiply a by itself b times; e.g., 3^3 = 27; 2^2 = 4.

Factorials

Notation n! means multiply n down to 1: n! = n × (n−1) × … × 1; e.g., 6! = 720.

Non-integers

Numbers that are not integers (e.g., 1/2, -1.7, pi); they have fractional or irrational parts.

Arbitrary orientation of the number line

The convention that moving right increases value is a human choice; other orientations (e.g., vertical, or different reading directions) are possible in different contexts.

Rational Numbers

Numbers that can be expressed as a fraction \frac{a}{b} where a and b are integers and b \neq 0; includes integers, terminating decimals, and repeating decimals (e.g., \frac{1}{2}, -3, 0.25, 0.333…).

Irrational Numbers

Numbers that cannot be expressed as a simple fraction \frac{a}{b} ; their decimal representations are non-terminating and non-repeating (e.g., \pi, \sqrt{2}, e).

Real Numbers

All rational and irrational numbers; encompasses all numbers that can be represented on a number line.

Absolute Value

The distance of a number from zero on the number line, always non-negative; denoted by vertical bars (e.g., \left|-3\right| = 3, \left|3\right| = 3).

Order of Operations (PEMDAS)

A set of rules for evaluating mathematical expressions: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Commutative Property of Addition

Changing the order of addends does not change the sum (e.g., a + b = b + a; 2 + 3 = 3 + 2).

Commutative Property of Multiplication

Changing the order of factors does not change the product (e.g., a \times b = b \times a; 2 \times 3 = 3 \times 2).

Associative Property of Addition

Changing the grouping of addends does not change the sum (e.g., (a + b) + c = a + (b + c); (1 + 2) + 3 = 1 + (2 + 3)).

Associative Property of Multiplication

Changing the grouping of factors does not change the product (e.g., (a \times b) \times c = a \times (b \times c); $$(1 \times 2) \times

Associative Property of Multiplication

Changing the grouping of factors does not change the product (e.g., (a \times b) \times c = a \times (b \times c); (1 \times 2) \times 3 = 1 \times (2 \times 3)).

Distributive Property

Allows multiplication to be distributed over addition or subtraction (e.g., a \times (b + c) = (a \times b) + (a \times c); 2 \times (3 + 4) = (2 \times 3) + (2 \times 4)).

Identity Property of Addition

The sum of any number and zero is that number itself (e.g., a + 0 = a; 5 + 0 = 5).

Identity Property of Multiplication

The product of any number and one is that number itself (e.g., a \times 1 = a; 5 \times 1 = 5).

Inverse Property of Addition

The sum of a number and its opposite (additive inverse) is zero (e.g., a + (-a) = 0; 5 + (-5) = 0).

Inverse Property of Multiplication

The product of a non-zero number and its reciprocal (multiplicative inverse) is one (e.g., a \times \frac{1}{a} = 1 where a \neq 0; 5 \times \frac{1}{5} = 1).

Prime Number

A natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7).

Composite Number

A natural number greater than 1 that is not prime; it has at least one divisor other than 1 and itself (e.g., 4, 6, 8, 9).

Even Number

An integer that is divisible by 2 (e.g., -4, 0, 2, 8).

Odd Number

An integer that is not divisible by 2 (e.g., -3, 1, 5, 9).

Undefined (Division by Zero)

The result of dividing any non-zero number by zero, which is not a defined mathematical operation (e.g., \frac{5}{0} is undefined).

Expression

A mathematical phrase that can contain numbers, variables, and operation symbols, but does not include an equality or inequality sign (e.g., 3x + 5).

Equation

A mathematical statement that asserts the equality of two expressions, typically separated by an equals sign (e.g., 2x + 1 = 7).

Integers

All counting numbers and whole numbers, plus their negative counterparts (e.g., -3, -2, -1, 0, 1, 2, 3).

Natural numbers

Counting numbers: 1, 2, 3, … to infinity; numbers you can count on your fingers.

Whole numbers

Natural numbers including zero: 0, 1, 2, 3, …

Zero (0)

The additive identity; located at the center of the number line; included in the whole numbers.

Number line

A visual representation of integers with zero at the center; numbers increase to the right and decrease to the left (can be horizontal or vertical).

Positive infinity

Right end of the number line representing unbounded growth; not a number or integer.

Negative infinity

Left end of the number line representing unbounded negative growth; not a number or integer.

Infinity

The idea of unbounded quantity; not an integer.

Addition

Operation of combining numbers to form a sum; e.g., 10 + 4 = 14.

Subtraction

Operation of taking away one number from another to form a difference; e.g., 15 - 7 = 8.

Multiplication

Repeated addition; a \times b means adding a to itself b times (or vice versa); e.g., 4 \times 5 = 20.

Division

Inverse of multiplication; splitting a number into equal groups; e.g., 10 \div 2 = 5; 2 \times 5 = 10.

Exponents

Shorthand for repeated multiplication; a^b means multiply a by itself b times; e.g., 3^3 = 27; 2^2 = 4.

Factorials

Notation n! means multiply n down to 1: n! = n \times (n-1) \times \ldots \times 1; e.g., 6! = 720.

Non-integers

Numbers that are not integers (e.g., 1/2, -1.7, pi); they have fractional or irrational parts.

Arbitrary orientation of the number line

The convention that moving right increases value is a human choice; other orientations (e.g., vertical, or different reading directions) are possible in different contexts.

Rational Numbers

Numbers that can be expressed as a fraction \frac{a}{b} where a and b are integers and b \neq 0; includes integers, terminating decimals, and repeating decimals (e.g., \frac{1}{2}, -3, 0.25, 0.333\ldots).

Irrational Numbers

Numbers that cannot be expressed as a simple fraction \frac{a}{b} ; their decimal representations are non-terminating and non-repeating (e.g., \pi, \sqrt{2}, e).

Real Numbers

All rational and irrational numbers; encompasses all numbers that can be represented on a number line.

Absolute Value

The distance of a number from zero on the number line, always non-negative; denoted by vertical bars (e.g., \left|-3\right| = 3, \left|3\right| = 3).

Order of Operations (PEMDAS)

A set of rules for evaluating mathematical expressions: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Commutative Property of Addition

Changing the order of addends does not change the sum (e.g., a + b = b + a; 2 + 3 = 3 + 2).

Commutative Property of Multiplication

Changing the order of factors does not change the product (e.g., a \times b = b \times a; 2 \times 3 = 3 \times 2).

Associative Property of Addition

Changing the grouping of addends does not change the sum (e.g., (a + b) + c = a + (b + c) ; (1 + 2) + 3 = 1 + (2 + 3)).

Associative Property of Multiplication

Changing the grouping of factors does not change the product (e.g., (a \times b) \times c = a \times (b \times c) ; (1 \times 2) \times 3 = 1 \times (2 \times 3)).

Distributive Property

Allows multiplication to be distributed over addition or subtraction (e.g., a \times (b + c) = (a \times b) + (a \times c) ; 2 \times (3 + 4) = (2 \times 3) + (2 \times 4)).

Identity Property of Addition

The sum of any number and zero is that number itself (e.g., a + 0 = a; 5 + 0 = 5).

Identity Property of Multiplication

The product of any number and one is that number itself (e.g., a \times 1 = a; 5 \times 1 = 5).

Inverse Property of Addition

The sum of a number and its opposite (additive inverse) is zero (e.g., a + (-a) = 0; 5 + (-5) = 0).

Inverse Property of Multiplication

The product of a non-zero number and its reciprocal (multiplicative inverse) is one (e.g., a \times \frac{1}{a} = 1 where a \neq 0; 5 \times \frac{1}{5} = 1).

Prime Number

A natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7).

Composite Number

A natural number greater than 1 that is not prime; it has at least one divisor other than 1 and itself (e.g., 4, 6, 8, 9).

Even Number

An integer that is divisible by 2 (e.g., -4, 0, 2, 8).

Odd Number

An integer that is not divisible by 2 (e.g., -3, 1, 5, 9).

Undefined (Division by Zero)

The result of dividing any non-zero number by zero, which is not a defined mathematical operation (e.g., \frac{5}{0} is undefined).

Expression

A mathematical phrase that can contain numbers, variables, and operation symbols, but does not include an equality or inequality sign (e.g., 3x + 5).

Equation

A mathematical statement that asserts the equality of two expressions, typically separated by an equals sign (e.g., 2x + 1 = 7).

Variable

A symbol (usually a letter) that represents a quantity that can change or take on different values in an expression or equation (e.g., x in 2x+7).

Constant

A value that remains unchanged in an expression or equation (e.g., 7 in 2x+7).

Coefficient

A numerical factor that multiplies a variable in an algebraic term (e.g., 2 in 2x).

Term (algebraic)

A single number, a variable, or a product/quotient of numbers and variables within an expression, separated by addition or subtraction (e.g., in 3x^2 + 2x - 5, the terms are 3x^2, 2x, and -5).