2.1 integers & rational numbers

Chapter 2
Properties of Real Numbers

• In this chapter, you will learn to work with
the REALS – a set of numbers that
include both positive and negative
numbers, decimals, fractions, and more.
• Learn to identify SETS of Numbers
• We’ll look at all four operations and learn
the number properties for each.
• Find square roots of given numbers

Using Integers and
Rational Numbers
Section 2.1
P. 64 - 70

• Natural or Counting Numbers
{ 1, 2, 3, 4, 5, . . .}
• Whole numbers {0, 1, 2, 3, 4, 5, . . .}

• Integers { . . . -3, -2, -1, 0, 1, 2, 3, . . .}

• Rationals: a number a/b, where a & b are
integers and b is not zero. Includes all
terminating and repeating decimals.

learn classify rational numbers into different sets;
alsoTSW be able to compare rational numbers
(including absolute value

Natural

• Two points that are the same distance
from the origin but on opposite sides (of
the origin) are opposites.

-4 -3 -2 -1 0 1 2 3 4

• Name some opposites on this #-line

• The expression “ -3” can be stated as
“negative three” or “the opposite of three”

• How should you read “-a ” ? Why?

• Does zero have an opposite?

• - (-4) = _____ - [ -(-5)] = _____

EXAMPLE 2 Classify numbers

Tell whether each of the following numbers is a whole
number, an integer, or a rational number: 5, 0.6,
–2 2 and – 24.
3
Number Whole Integer? Rational
number? number?
5 Yes Yes Yes
0.6 No No Yes
2 No No Yes
–2
3
–24 No Yes Yes

GUIDED PRACTICE for Examples 2 and 3

5. 4.5, – 3 , – 2.1, 0.5
4
number? number?
4.5 No No Yes
– 3 No No Yes
4
–2 .1 No No Yes

0.5 No No Yes

ANSWER

– 2.1, – 3 ,0.5 ,– 2.1.(Order the numbers from least to
4
greatest).

GUIDED PRACTICE for Examples 2 and 3

Tell whether each numbers in the list is a whole number, an
integer, or a rational number.Then order the numbers from
least list to greatest.

4. 3, –1.2, –2,0

number? number?
3 Yes Yes Yes
–1.2 No No Yes
–2 No Yes Yes
0 Yes Yes Yes

EXAMPLE 1 Graph and compare integers

Graph – 3 and – 4 on a number line. Then tell which
number is greater.

ANSWER

On the number line, – 3 is to the right of – 4. So, –3 > – 4.

Also TSW be able to compare rational numbers

GUIDED PRACTICE for Example 1

Graph the numbers on a number line. Then tell which
number is greater.
1. 4 and 0

0 4

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

ANSWER

On the number line, 4 is to the right of 0. So, 4 > 0.


2. 2 and –5

–5 2
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

ANSWER

On the number line, 2 is to the right of –4. So, 2 > –5.
alsoTSW be able to compare rational numbers (including
absolute value


3. –6 and –1

–6 –1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

ANSWER

On the number line, –1 is to the right of –6. So, –1 > –6.


EXAMPLE 3 Order rational numbers

ASTRONOMY
A star’s color index is a measure of the temperature of
the star. The greater the color index, the cooler the
star. Order the stars in the table from hottest to
coolest.

Star Rigel Arneb Denebola Shaula
Color index –0.03 0.21 0.09 – 0.22

SOLUTION

Begin by graphing the numbers on a number line.

EXAMPLE 3 alsoTSW be able to compare rational numbers (including
absolute value

Read the numbers from left to right: – 0.22, – 0.03,
0.09, 0.21.

ANSWER

From hottest to coolest, the stars are Shaula, Rigel,
Denebola, and Arneb.

Absolute Value
• The absolute value of a real number is the
distance between the origin and the point
representing the number. The symbol
| a | represents the absolute value of a.

• The absolute value of a number is never
negative.

alsoTSW be able to compare rational numbers (including
absolute value

• If a is a positive number, then | a| = a
• If a is zero, then |a | = 0
• If a is a negative #, then | a | = -a
• Examples:
| 6 | = _______
| 0 | = _______
| -5 | = _______

• Simplify: - | -8 | = _____

• - | 5 | = ______

• - ( -5) = ______

• - ( 0 ) = _____
• learn classify rational numbers into different sets;
TSW be able to compare rational numbers (including absolute
value

EXAMPLE 4 Find opposites of numbers

a. If a = – 2.5, then – a = –(– 2.5) =

b. If a = 3 , then – a = – 3 .
4 4

alsoTSW be able to compare rational numbers (including absolute
value

EXAMPLE 5 Find absolute values of numbers

a. If a = – 2 , then |a | = | 2 | = – ( 2 ) = 2
3 3 3 3
b. If a = 3.2, then |a| = |3.2| = 3.2.

TSW be able to compare rational numbers (including absolute value

GUIDED PRACTICE for Example 4, 5 and 6

For the given value of a, find –a and |a|.

8. a = 5.3

SOLUTION

If a = 5.3, then –a = – (5.3) =

|a| = |5.3| =

GUIDED PRACTICE for Example 4, 5 and 6

9. a = – 7

SOLUTION
If a = – 7, then –a = – (– 7) =

|a| = | – 7| =

10. a = – 4
9
SOLUTION
4
If a = – 4 , then –a = – ( – 4 ) = 9
9 9
|a| = | – 4 | = – ( – 4 ) = 4
9 9 9


Be ready to discuss / define these words:
• Real Numbers *
• Rational Numbers*
• Integers
• Irrational Numbers*
• Whole Numbers
• Absolute Value*
* critical vocabulary

• Assignment: :
• P. 67 (#1 - 3,10,11,13 -number
lines,16, 20, 23-25, 42-44)

2.1 integers & rational numbers

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