This document provides an overview of real number properties and operations. It defines natural numbers, integers, rational numbers, and irrational numbers. It discusses order of operations and using properties like commutative, associative, distributive, identity, and inverse. Examples are provided of simplifying algebraic expressions and evaluating expressions using order of operations. Formulas are defined as relationships between quantities that can be used to find variable values. The document concludes with examples of simplifying algebraic expressions using properties of real numbers.

1. 1.1 Real Numbers Properties
Chapter 1 Prerequisites

2. Concepts & Objectives
⚫ Objectives for this section:
⚫ Classify a real number as a natural, whole, integer,
rational, or irrational number.
⚫ Perform calculations using order of operations.
⚫ Use the following properties of real numbers:
commutative, associative, distributive, inverse, and
identity.
⚫ Evaluate algebraic expressions.
⚫ Simplify algebraic expressions.

3. Number Systems
⚫ What we currently know as the set of real numbers was
only formulated around 1879. We usually present this
as sets of numbers.

4. Number Systems
⚫ The set of natural numbers () and the set of integers
() have been around since ancient times, probably
prompted by the need to maintain trade accounts.
Ancient civilizations, such as the Babylonians, also used
ratios to compare quantities.
⚫ One of the greatest mathematical advances was the
introduction of the number 0.

5. Rational Numbers
⚫ A number is a rational number () if and only if it can be
expressed as the ratio (or Quotient) of two integers.
⚫ Rational numbers include decimals as well as fractions.
The definition does not require that a rational number
must be written as a quotient of two integers, only that it
can be.

6. Examples
⚫ Example: Prove that the following numbers are
rational numbers by expressing them as ratios of
integers.
1. 2-4 4.
2. 64-½ 5.
3. 6. –5.4322986


4
20.3
0.9
6.3

7. Examples
⚫ Example: Prove that the following numbers are
rational numbers by expressing them as ratios of
integers.
1. 2-4 4.
2. 64-½ 5.
3. 6. –5.4322986


4
20.3
0.9
6.3
1
16
1
8
4
1
7
=
1 61
20
3 3
−
54322986
10000000

8. Order of Operations
⚫ Parentheses (or other grouping symbols, such as square
brackets or fraction bars) – start with the innermost set,
following the sequence below, and work outward.
⚫ Exponents
⚫ Multiplication
⚫ Division
⚫ Addition
⚫ Subtraction
working from left to right
working from left to right

9. Order of Operations
⚫ Use order of operations to explain why
( )
−  −
2 2
3 3

10. Order of Operations
⚫ Use order of operations to explain why
⚫ We can think of –3 as being –1  3. Therefore we have
It should be easier now to see that on the left side we
multiply first and then apply the exponent, and on the
right side, we apply the exponent and then multiply.
( )
−  −
2 2
3 3
( )
−  −
2 2
1 3 1 3

11. Order of Operations
Work the following examples without using your calculator.
1.
2.
3.
− + 
2 5 12 3
( ) ( )( )
− − + −
3
4 9 8 7 2
( )( )
( )
8 4 6 12
4 3
− + − − 
− −

12. Order of Operations
Work the following examples without using your calculator.
1.
2.
3.
− + 
2 5 12 3
( ) ( )( )
− − + −
3
4 9 8 7 2
( )( )
( )
− + − − 
− −
8 4 6 12
4 3
1. –6
2. –60
−
6
3.
7

13. Properties of Real Numbers
⚫ Closure Property
⚫ a + b  
⚫ ab  
⚫ Commutative Property
⚫ a + b = b + a
⚫ ab = ba
⚫ Associative Property
⚫ (a + b) + c = a + (b + c)
⚫ (ab)c = a(bc)
⚫ Identity Property
⚫ a + 0 = a
⚫ a  1 = a
⚫ Inverse Property
⚫ a + (–a) = 0
⚫
⚫ Distributive Property
⚫ a(b ± c) = ab ± ac
For all real numbers a, b, and c:
1
=1
a
a

14. Properties of Real Numbers
⚫ The properties are also called axioms.
⚫ 0 is called the additive identity and 1 is called the
multiplicative identity.
⚫ Notice the relationships between the identities and the
inverses (called the additive inverse and the
multiplicative inverse).
⚫ Saying that a set is “closed” under an operation (such as
multiplication) means that performing that operation on
numbers in the set will always produce an answer that is
also in the set – there are no answers outside the set.

15. Properties of Real Numbers
⚫ Examples
⚫ The set of natural numbers () is not closed under
the operation of subtraction. Why?
You can end up with a result that is not a natural
number. For example, 5 – 7 = –2, which is not in .
⚫ –20  5 = –4. Does this show that the set of integers
is closed under division?

16. Properties of Real Numbers
⚫ Examples
⚫ The set of natural numbers () is not closed under
the operation of subtraction. Why?
⚫ You can end up with a result that is not a natural
number. For example, 5 – 7 = –2, which is not in .
⚫ –20  5 = –4. Does this show that the set of integers
is closed under division?
⚫ No. Any division that has a remainder is not in .

17. Formulas
⚫ An equation is a mathematical statement indicating that
two expressions are equal. The values that make the
equation true, the solutions, are found using the
properties of real numbers and other results.
⚫ A formula is an equation expressing a relationship
between constant and variable quantities. Very often,
the equation is a means of finding the value of one
quantity (often a single variable) in terms of another or
other quantities.

18. Simplifying Algebraic Expressions
⚫ Sometimes we can simplify an algebraic expression to
make it easier to evaluate or to use in some other way.
⚫ To do so, we use the properties of real numbers and
combine like terms.
⚫ Like terms are terms that are either constants or
variables having the same exponent.
⚫ We can use the same properties in formulas because
they contain algebraic expressions.

19. Simplifying Algebraic Expressions
Example: Simplify each expression:
⚫ 3x ‒ 2y + x ‒ 3y ‒ 7
⚫ 2mn ‒ 5m + 3mn + n

20. Simplifying Algebraic Expressions
Example: Simplify each expression:
⚫ 3x ‒ 2y + x ‒ 3y ‒ 7
= 3x + x ‒ 2y ‒ 3y ‒ 7
= 4x ‒ 5y ‒ 7
⚫ 2mn ‒ 5m + 3mn + n
= 2mn + 3mn ‒ 5m + n
= 5mn ‒ 5m + n

21. Classwork
College Algebra 2e (OpenStax.org)
⚫ 1.1: 4-26 (even)
⚫ 1.1 Classwork Check (in Canvas)
⚫ The classwork check will pick 5 problems at random
from this assignment.
⚫ You do not need to turn in your work.
⚫ You may retake the classwork check as many times as
you like until the end of the six weeks. I will always
take the higher grade.