1.1 – Types of Numbers

Section 1 (Apex Pg 1 – 8)

Objectives

• Organize collections of numbers into mathematical sets.
Keywords
• Set – a collection of objects.
• Subset – a set that is part of a larger set.
• Element – an element is an individual item of a set.
• Counting Numbers / Natural Numbers – the numbers used to count: {1, 2, 3, 4, 5, …}.
• They go on forever; they are infinite, positive integers.
• Whole Numbers – the counting numbers (positive integers) and also zero:  {0, 1, 2, 3, 4, 5, …}.
• They go on forever.
• WhOle numbers have a zerO in them!
• Integers – whole numbers and their opposites:  {…, –3, –2, –1, 0, 1, 2, 3, …}.
• Positive Integers – these are the counting numbers / natural numbers.
• They are greater than zero:  {1, 2, 3, 4, 5,…}.
• Negative Integers – the integers that are less than zero.
• They are the opposites of the positive integers  {…, –3, –2, –1}.
• Opposites – two numbers that are equally far from zero in opposite directions along the number line.
• Adding a minus (or negative) sign to a positive number creates its opposite.
• Removing the minus (or negative) sign from a negative number creates its opposite.
Examples
1. Months
• Set – all of the months of the year
• Subset – winter months { December, January, February ]
• Element – December
2. Numbers
• Set – all numbers
• Subset – positive, even integers: {2, 4, 6, 8, 10,…}
• Element – 4

Notes

• A set is often shown by listing its elements within curly braces, sometimes called brackets { }.
• The ellipses (…) show that these numbers go on forever.

Section 2 (Apex Pg 9 – 19)

Objectives

• Convert fractions to decimals to percent.
Keywords
• Fraction – a part, or parts, of the whole.
• Numerator – the number of parts being considered.
• Denominator – the total number of parts in the whole item.
• Decimal – you can write every fraction as a decimal by dividing the numerator by the denominator.
• Terminating Decimal – a number that ends
• Repeating Decimal – a number that keeps on going forever
• Percent – “per one hundred”
• The symbol that represents percent is %.
• Convert a decimal to a percentage by moving the decimal point two places to the right and putting % after it.
Notes
• The symbol ≠ means “is not equal to.”

Section 3 (Apex Pg 20 – 32)

Objectives

• Define rational and irrational numbers, and explore their properties.
• Characterize numbers as fractions, exponents, and square roots.
• Show the relationships between numbers with a number line and inequality symbols.

Key Words

• Real Numbers – a set made up of all the numbers on the number line.
• Any real number is either a rational or irrational number.
• Rational Numbers – the set of all numbers that can be written as a fraction (which is a ratio of two integers, where the denominator is not zero).
• The rational numbers are a subset of the real numbers.
• Irrational Numbers – numbers that cannot be written in the form , where a and b are integers.
• Irrational numbers cannot be written as terminating decimals or repeating decimals.
• Proof by Contradiction – a method of proving that a proposition is true by showing that an assumption is false.
• Proposition – a sentence that is either true or false.
• Assumption – a statement that is accepted as true at the beginning of a proof.
• Exponent – a small raised number that tells you how many times to multiply the base number by itself.
• Base – a number in exponential form. It appears beneath the exponent.
Notes
• Squares and Square Roots are inverse (opposite) operations
• Rational Numbers
• Terminating (ending) numbers such as 4.5, 3.75, 2, 1/4, 2
• or Repeating numbers such as 0.222…, 3.876587658765…, 0.7777…
• Irrational Numbers
• They go on forever
• and they do not have any pattern
• Examples
• π
• √2, √3, √5, √7, etc.
• e
• Ratios make up Rational Numbers
• Examples
• 3/5 = 3:5 = 3 to 5
• 2/7 = 2:7 = 2 to 7

Section 4 (Apex Pg 33 – 39)

Objectives

• Show the relationships between numbers with a number line and inequality symbols.

Key Words

• Number Line – a line that uses equally spaced marks to represent numbers.
• Numbers to the right of zero are positive, and numbers to the left of zero are negative.
• The greater a number is, the farther to the right it is placed on the number line.
• Comparison Symbols – Symbols used to compare two numbers or expressions.
• Comparison symbols include:

> (greater than)
< (less than)
≥ (greater than or equal to, at least)
≤ (less than or equal to, at most)
= (equal to)
(not equal to)

• at least : a number or more
• at most: a number or smaller