Whole Numbers, Integers, and Common Fractions

• Whole numbers, integers, and common fractions are fundamental concepts in mathematics, each representing different sets of numbers with specific properties and uses.

Whole Numbers

• Whole numbers are the basic counting numbers and include zero.
• Examples of whole numbers: 0, 1, 2, 3, 4, and so on.
• Whole numbers do not include negative numbers, fractions, or decimals.
• The set of whole numbers is usually denoted by the symbol W.
• W = {0, 1, 2, 3, ...}
• Whole numbers are used for counting and basic arithmetic operations such as addition, subtraction, multiplication, and division.
• When dividing, whole numbers may not always result in a whole number quotient.
• The number line representation of whole numbers starts at 0 and extends infinitely to the right, with each number equally spaced.
• Whole numbers are fundamental for understanding basic mathematical principles and are the building blocks for more complex number systems.
• Properties like commutativity and associativity apply to whole number addition and multiplication.
• The identity element for addition is 0, and for multiplication is 1 (for whole numbers).
• Whole numbers are used in everyday life for counting objects, measuring quantities, and representing numerical data.
• They form the basis for understanding more advanced numerical concepts like integers, rational numbers, and real numbers.
• In set theory, the set of whole numbers is an infinite set, meaning it contains an unlimited number of elements.
• The successor of any whole number n is n + 1.
• Zero is the smallest whole number.

Integers

• Integers are whole numbers along with their negative counterparts.
• Examples of integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Integers include all positive whole numbers, zero, and all negative whole numbers.
• The set of integers is usually denoted by the symbol Z.
• Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Integers are used to represent quantities that can be both positive and negative, such as temperature above or below zero, profit or loss, and elevation above or below sea level.
• The number line representation of integers extends infinitely in both directions, with zero at the center.
• Each integer has an additive inverse, such that the sum of the integer and its inverse is zero (e.g., the additive inverse of 5 is -5, since 5 + (-5) = 0).
• Integers are closed under addition, subtraction, and multiplication, meaning that performing these operations on integers will always result in another integer.
• Division of integers may not always result in an integer quotient.
• The absolute value of an integer is its distance from zero: |x| = x if x ≥ 0, and |x| = -x if x < 0.
• Properties like commutativity, associativity, and distributivity apply to integer addition and multiplication.
• The identity element for integer addition is 0, and for multiplication is 1.
• Integers are used in computer science for representing data, indexing arrays, and performing calculations.
• They are also used in various mathematical fields, including algebra, number theory, and calculus.
• The set of integers is an infinite and countable set.
• Integers can be even (divisible by 2) or odd (not divisible by 2).
• Understanding integers is crucial for solving equations and inequalities involving negative numbers.

Common Fractions

• Common fractions are numbers that represent a part of a whole.
• A common fraction is written in the form a/b, where a is the numerator and b is the denominator.
• The numerator (a) represents the number of parts taken.
• The denominator (b) represents the total number of equal parts the whole is divided into.
• Examples of common fractions: 1/2, 3/4, 2/5, 7/8.
• Common fractions can be proper or improper.
• Proper fractions have a numerator smaller than the denominator (e.g., 2/3).
• Improper fractions have a numerator greater than or equal to the denominator (e.g., 5/3).
• Improper fractions can be converted into mixed numbers, which consist of a whole number and a proper fraction (e.g., 5/3 = 1 2/3).
• Fractions can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• Example: 4/6 can be simplified to 2/3 by dividing both by 2.
• Equivalent fractions represent the same value but have different numerators and denominators (e.g., 1/2 = 2/4 = 3/6).
• To add or subtract fractions, they must have a common denominator. If they don't, find the least common multiple (LCM) of the denominators and convert the fractions accordingly.
• Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
• To multiply fractions, multiply the numerators together and the denominators together: (a/b) * (c/d) = (ac)/(bd).
• Example: 1/2 * 2/3 = 2/6 = 1/3.
• To divide fractions, multiply the first fraction by the reciprocal of the second fraction: (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc).
• Example: 1/2 / 2/3 = 1/2 * 3/2 = 3/4.
• Fractions are used in various real-world applications, such as cooking, measuring, and calculating proportions.
• Converting fractions to decimals can be done by dividing the numerator by the denominator (e.g., 1/4 = 0.25).
• Understanding fractions is essential for mastering more advanced mathematical concepts, such as algebra and calculus.
• The number line representation of fractions shows the position of the fraction between integers.
• Fractions are a part of the set of rational numbers, which also includes integers and terminating or repeating decimals.