Chapter 1 Prerequisites: Real Numbers and Basic Operations (Vocabulary)

A vocabulary set covering key real numbers, operations, sets, and interval concepts from the notes.

Real Numbers

The set of all rational and irrational numbers; the symbol for this set is R.

Rational Numbers

Numbers that can be written as a fraction m/n where m and n are integers and n ≠ 0.

Irrational Numbers

Numbers that cannot be written as a ratio of integers; examples include √2 and π.

Natural Numbers

The counting numbers: 1, 2, 3, 4, …

Prime Numbers

Natural numbers greater than 1 with exactly two distinct positive divisors: 1 and itself.

Composite Numbers

Natural numbers greater than 1 that are not prime; they have more than two positive divisors.

Integers

All positive and negative whole numbers plus zero: …, -2, -1, 0, 1, 2, …

Even Numbers

Integers divisible by 2 (no remainder).

Odd Numbers

Integers not divisible by 2 (remainder 1).

Whole Numbers

Nonnegative integers: 0, 1, 2, 3, …

Fraction

A number written as m/n with m and n integers and n ≠ 0; another name for a rational number in fractional form.

Numerator

The top part of a fraction; the quantity being divided.

Denominator

The bottom part of a fraction; the divisor in a division.

Reciprocal (Multiplicative Inverse)

The number 1/a for a ≠ 0; a · (1/a) = 1.

Sum

The result of addition; example: a + b.

Additive Identity

Zero; adding zero to any number leaves it unchanged: a + 0 = a.

Additive Inverse

The opposite of a number; a + (-a) = 0.

Product

The result of multiplication; example: a · b.

Multiplicative Identity

One; multiplying by one leaves a number unchanged: a · 1 = a.

Reciprocal (in fractions)

The fraction obtained by inverting the numerator and denominator (a/b)⁻¹ = b/a.

Quotient

The result of division; a/b is the quotient of a and b.

Division by Zero

Undefined; division by 0 is not allowed.

Exponent

The power in a^n; n is the exponent and a is the base.

Base

The number that is repeated by multiplication in an exponent: a in a^n.

Exponent (power)

The exponent n indicates how many times the base is multiplied by itself.

Nth Root

The number b such that b^n = a; the principal root is denoted with a bar over the radical.

Square Root

The special case of the nth root with n = 2; √a is the nonnegative number b with b^2 = a.

Radical

A root expression using the radical symbol (√ or the nth root symbol).

Principal Root

For even roots, the nonnegative root; for odd roots, the real root.

Prime Factorization

Expressing a number as a product of prime numbers.

LCD (Least Common Denominator)

The smallest common denominator that two or more fractions share.

Cancel Common Factors

Dividing numerator and denominator by their common factors to simplify a fraction.

Decimal Fractions

Decimals are fractions where the denominator is a power of 10.

Repeating Decimal

A decimal with a block of digits that repeats forever (bar notation).

Nonrepeating Decimal

A decimal expansion without a repeating block (often irrational).

Decimal Place Value

Digits to the right of the decimal point correspond to tenths, hundredths, thousandths, etc.

Fraction to Decimal Conversion

Any fraction can be written as a decimal by performing the division numerator ÷ denominator.

Absolute Value

The distance of a number from 0 on the real line; |a| = a if a ≥ 0, and |a| = -a if a < 0.

Distance on Real Line

The distance between a and b is d(a,b) = |b − a|; always nonnegative.

Set

A collection of distinct objects called elements.

Element

An object that belongs to a set; a ∈ S means a is in S.

Set-builder Notation

A = {x | condition} reads as 'A is the set of all x such that condition is true.'

Union

The set of elements in S or T (or both): S ∪ T.

Intersection

The set of elements common to S and T: S ∩ T.

Empty Set

A set with no elements; denoted Ø or {}.

Subset

A set A is a subset of B if every element of A is also in B; A ⊆ B.

Interval

A set of real numbers described by inequalities; examples: (a,b), [a,b], [a,b) or (a,b].

Open Interval

(a,b): all x with a < x < b.

Closed Interval

[a,b]: all x with a ≤ x ≤ b.

Half-Open Interval

[a,b) or (a,b], include one endpoint but not the other.

Real Line

The number line representing all real numbers; notation R for the set of all real numbers.

Order Symbols

A ⊆ B Notation

A is a subset of B; every element of A is also an element of B.

PEMDAS (Order of Operations)

Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Distributive Property

a(b + c) = ab + ac; distribution of multiplication over addition.

Commutative Property

Order of addition or multiplication does not affect the result: a + b = b + a and ab = ba.

Associative Property

Grouping does not affect the result: (a + b) + c = a + (b + c) and (ab)c = a(bc).

Notation for Sets and Intervals

Use symbols like ∪, ∩, {x|…}, [a,b], (a,b) to describe sets and intervals.

Cross-multiplication

Method used to compare fractions or solve equations by equating the products of two cross terms.

Simplifying Fractions

Reduce by canceling common factors in numerator and denominator.

Fraction Bar

The line that separates the numerator from the denominator (the division symbol in a fraction).

Irrational vs Rational Decimal Expansion

Rational decimals either terminate or repeat; irrational decimals do not terminate or repeat.

Distance Between Points (Example)

On the real line, distance d(a,b) = |b − a|.

Set Equality

Two sets are equal if they contain exactly the same elements.

Notation for Empty Set

Denoted Ø or {} and means “contains no elements.”

Inequality Notation for Intervals

Intervals can be described by inequalities such as a < x < b or a ≤ x ≤ b.