1. 1.1: Properties of Real Numbers
Page 13 5) 81
31)
–5p2 + 21 7) 49
33) D 9)
–32
35)
10n + 24; 44
11)
–10,000
37) 26
13)
–64
39)
15) 64
41)
1/9
17) –5
43)
–7d + 11c
19)
–100
45)
2m2 + n2 – 8m
21) 75
47)
13m2 – 5
23) 6
49)
–8s + 8t
25)
5x + 5
52)
÷ (3 – 1) = 15
27)
13z2 – 2z + 10
53)
(4 + 3) • (5 – 2) = 21
29)
11m – 1
54)
( – 6) ÷ 3 = 9
55)
(3 • 4)2 – (23 + 3)2 = 23
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1.1: Properties of Real Numbers

2. 1.1: Properties of Real Numbers
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1.1: Properties of Real Numbers

3. 1.1: Properties of Real Numbers
Chapter 1: Equations and Inequalities
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1.1: Properties of Real Numbers

4. 1.1: Properties of Real Numbers
What is a number?
What is a number?
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1.1: Properties of Real Numbers

5. 1.1: Properties of Real Numbers
What is a number?
According to M–W.com (Merriam–Webster Dictionary), it means “a word, symbol, letter, or combination of symbols represented”
Etymology: Middle English nombre, from Anglo–French, from Latin numerus
What is a number?
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1.1: Properties of Real Numbers

6. 1.1: Properties of Real Numbers
Definitions
Counting numbers are numbers can be expressed using set notation. They are also known as natural numbers KINDERGARDEN; NATURALLY COUNT NUMBERS
If we include a ZERO we have the set of whole numbers. 1ST GRADE; HOLE = ZERO
Include the OPPOSITES of the whole numbers and you have the set of integers. PRE–ALGEBRA
Rational numbers can be expressed as a QUOTIENT (or ratio) of two integers, where the denominator is not zero. The decimal form of a rational number either terminates or repeats. RATIO. CAN I WRITE THIS AS A FRACTION?
Irrational numbers cannot be expressed as a quotient of two integers, and their decimal forms do not terminate or repeat. OPPOSITE OF A RATIONAL NUMBER
{1, 2, 3, 4, }
{0, 1, 2, 3, 4, }
{ …–3, –2, –1, 0, 1, 2, 3, }
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1.1: Properties of Real Numbers

7. 1.1: Properties of Real Numbers
rational numbers
integers
whole numbers
counting/ natural #s
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1.1: Properties of Real Numbers

8. 1.1: Properties of Real Numbers
There are numbers that cannot be expressed as the quotient of two integers. These are called irrational numbers.
rational numbers
integers
irrational
numbers
whole numbers
Counting/natural numbers
REAL NUMBERS
The rational numbers combined with the irrational numbers make up the set of real numbers.
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1.1: Properties of Real Numbers

9. 1.1: Properties of Real Numbers
Breakdown
REAL NUMBERS
RATIONAL NUMBERS
IRRATIONAL NUMBERS
Integers
Counting
Whole
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1.1: Properties of Real Numbers

10. Example 1 Consider the numbers
Order the numbers from least to greatest and classify each number by the subset of the real number to which it belongs to.
Write each number as a decimal to make it easier to compare them.
Use a decimal approximation for .
 ≈ 3.14
Use a decimal approximation for .
Rewrite in decimal form.
–5.5 < 2.23 < 2.3 < < 3.14
Use < to compare the numbers.
The numbers in order from greatest to least are
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1.1: Properties of Real Numbers

11. 1.1: Properties of Real Numbers
Example 1
Consider the numbers
Order the numbers from least to greatest and classify each number by the subset of the real number to which it belongs to.
Numbers
Real
Rational
Integer
Whole
Natural
Irrational
2.3 
2.7652          
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1.1: Properties of Real Numbers

12. 1.1: Properties of Real Numbers
Example 2
Consider the numbers,
Order the numbers from least to greatest and classify each number by the subset of the real number to which it belongs to.
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1.1: Properties of Real Numbers

13. 1.1: Properties of Real Numbers
Your Turn
Consider the numbers,
Order the numbers from least to greatest and classify each number by the subset of the real number to which it belongs to.
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1.1: Properties of Real Numbers

14. 1.1: Properties of Real Numbers
Properties of Addition and Multiplication
Property
Addition
Multiplication
CLOSURE
a + b is a real number
ab is a real number
COMMUTATIVE
a + b = b + a
ab = ba
ASSOCIATIVE
(a + b) + c = a + (b + c)
(ab)c = a(bc)
IDENTITY
a + 0 = a
a • 1 = a
INVERSE
a + (–a) = 0
a • (1/a) = 1
DISTRIBUTIVE
a (b + c) = ab + ac
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1.1: Properties of Real Numbers

15. 1.1: Properties of Real Numbers
Example 3
Identify the property that illustrates, = 4 + 7
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1.1: Properties of Real Numbers

16. 1.1: Properties of Real Numbers
Example 4
Find the additive and multiplicative inverse of 12
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1.1: Properties of Real Numbers

17. 1.1: Properties of Real Numbers
Example 5
Write the additive and multiplicative inverse property of –9/4
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1.1: Properties of Real Numbers

18. 1.1: Properties of Real Numbers
Your Turn
Write the additive and multiplicative inverse property of 12/11
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1.1: Properties of Real Numbers

19. 1.1: Properties of Real Numbers
Example 6
Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers.
a  b = a, where b = 3
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1.1: Properties of Real Numbers

20. 1.1: Properties of Real Numbers
Example 7
Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers.
(a ÷ b) ÷ c = a ÷ (b ÷ c)
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1.1: Properties of Real Numbers

21. 1.1: Properties of Real Numbers
Your Turn
Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers.
a – (b + c) = (a – b) + (a – c)
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1.1: Properties of Real Numbers

22. 1.1: Properties of Real Numbers
Assignment
Page 6
3-15 odd (3-7 odd also classify the numbers as rational, irrational, integer, whole, and natural), odd
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1.1: Properties of Real Numbers