Rational and Whole Numbers explained

• Math is the study of topics such as quantity, structure, space, and change
• Other views can be found, and mathematicians seek out patterns and formulate new conjectures

Rational Numbers

• A rational number is any number that can be expressed as the quotient or fraction p/q of two integers
• p is a numerator and q is a non-zero denominator
• Since q may be equal to 1, every integer is a rational number
• A rational number can be expressed as a terminating decimal or a repeating infinite decimal
• 0 is a rational number
• It can be written as 0/1
• 5 is a rational number
• It can be written as 5/1
• 1.5 is a rational number
• It can be written as 3/2
• 0.333... is a rational number
• It can be written as 1/3
• All integers are rational
• All fractions are rational

Whole Numbers

• Whole numbers are the basic counting numbers, starting from zero
• The set of whole numbers includes 0, 1, 2, 3, and so on
• Whole numbers are non-negative integers
• They do not include fractions, decimals, or negative numbers
• Every whole number is a rational number
• Whole numbers are used for counting objects
• The set of whole numbers is usually denoted by the symbol W
• W = {0, 1, 2, 3, ...}
• The successor of a whole number is the next whole number
• The successor of 5 is 6
• There is no largest whole number

Integers

• Integers are whole numbers, but also include negative numbers
• The word integer comes from the Latin word "integer" meaning "whole"
• The set of integers includes all whole numbers (0, 1, 2, 3, ...) and their negatives (...-3, -2, -1, 0, 1, 2, 3, ...)
• Integers do not include fractions or decimals
• Every integer is a rational number
• Examples of integers: -5, 0, 7, 100
• Examples of non-integers: 2.5, -1/3, π
• The set of integers is usually denoted by the symbol Z
• Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Integers can be positive, negative, or zero
• 0 is an integer, and it is neither positive nor negative
• The additive inverse of an integer n is -n
• The additive inverse of 5 is -5
• The further a negative number is from zero, the smaller it is
• -10 < -5
• The absolute value of an integer is its distance from zero
• | -5 | = 5
• | 5 | = 5
• Integers are used in everyday life
• Bank balances can be expressed as integers
• Thermometers use integers to measure temperature

Relationships

• Every whole number is an integer
• Every integer is a rational number
• Not every rational number is an integer
• 1/2 is a rational number, but not an integer
• 3/4 is a rational number, but not an integer
• Not every rational number is a whole number
• 1/2 is a rational number, but not a whole number
• There are numbers that are not rational
• They are known as irrational numbers
• The decimal representation of an irrational number is non-terminating and non-repeating
• π is an irrational number
• √2 is an irrational number

Properties of Integers

• Closure under addition and multiplication
• The sum of two integers is an integer
• The product of two integers is an integer
• Commutativity of addition and multiplication
• a + b = b + a for any integers a and b
• a * b = b * a for any integers a and b
• Associativity of addition and multiplication
• (a + b) + c = a + (b + c) for any integers a, b, and c
• (a * b) * c = a + (b * c) for any integers a, b, and c
• Distributive property
• a * (b + c) = a * b + a * c for any integers a, b, and c
• Identity element of addition
• a + 0 = a for any integer a
• Identity element of multiplication
• a * 1 = a for any integer a
• Additive inverse
• For any integer a, there exists an integer -a such that a + (-a) = 0

Applications

• Integers are used to represent bank balances
• You can have a positive or negative balance
• Integers are used to measure temperature
• Temperatures can be above or below zero
• Integers are used to represent altitude
• Altitude can be above or below sea level
• Rational numbers are used in cooking and baking
• They are used for measurements like 1/2 cup of flour
• Rational numbers are used in construction
• They are used in measurements like 3/4 inch of wood
• Rational numbers are used in finance for interest rates
• Percentages, like 5.25%, can be written as a rational number

Examples of Problem Solving with Rational Numbers

• Adding fractions: 1/2 + 1/4 = 3/4
• Multiplying fractions: 1/2 * 1/4 = 1/8
• Dividing fractions: 1/2 ÷ 1/4 = 2
• Converting fractions to decimals: 1/2 = 0.5
• Converting decimals to fractions: 0.75 = 3/4
• Solving equations with fractions: x + 1/2 = 1
• x = 1/2

Examples of Problem Solving with Integers

• Adding integers: -3 + 5 = 2
• Subtracting integers: 7 - 10 = -3
• Multiplying integers: -4 * 2 = -8
• Dividing integers: 12 ÷ -3 = -4
• Solving equations with integers: x + 5 = 2
• x = -3
• Understanding temperature changes: If the temperature started at -5°C and rose by 10°C, what is the new temperature?
• Answer: 5°C

Explanation of what constitutes rational and whole numbers. Rational numbers can be expressed as a fraction p/q, where p and q are integers. Whole numbers are non-negative integers starting from zero.