This document discusses properties of real numbers. It defines rational numbers as numbers that can be expressed as ratios of integers. It also covers order of operations and properties of real numbers such as closure, commutativity, associativity, identities, inverses, and distribution. Examples are provided to illustrate rational numbers and properties like closure. The document contains classwork assignments on real number concepts.

1. 0.2 Real Numbers Properties
Chapter 0 Review of Basic Concepts

2. Concepts & Objectives
⚫ Real numbers and their properties
⚫ Sets of numbers
⚫ Exponents
⚫ Order of operations

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. 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

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

7. 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

8. 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
− + − − 
− −

9. 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

10. 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.

11. 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

12. 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

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
=1a
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. Classwork
⚫ College Algebra & Trigonometry
⚫ Assignment 0.2
⚫ Page 18: 4-36 (×4), page 6: 32-64 (×4)
⚫ 0.2 Classwork Check
⚫ Quiz 0.1