2.1 integers & rational numbers

Chapter 2Properties 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 andRational Numbers

Section 2.1P. 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.

• Name some opposites on this #-line

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

• 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)] = _____

Classify numbers

EXAMPLE 2

Tell whether each of the following numbers is a wholenumber, an integer, or a rational number: 5, 0.6,–2 and – 24.2

3

YesYesNo–24

YesNoNo

YesNoNo0.6

YesYesYes5

Rational number?

Integer?Whole number?

Number

23–2

YesYesNo–24

YesNoNo

YesNoNo0.6

YesYesYes5

Rational number?


Number

23–2

GUIDED PRACTICE for Examples 2 and 3

5. 4.5, – , – 2.1, 0.5 34

YesNoNo0.5

YesNoNo –2 .1

YesNoNo

YesNoNo4.5

Rational number?


Number

34

–

ANSWER

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

34

YesNoNo0.5

YesNoNo –2 .1

YesNoNo

YesNoNo4.5

Rational number?


Number

34

–

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.

Number Whole number?

Integer? Rational number?

3 Yes Yes Yes–1.2 No No Yes

–2 No Yes Yes

0 Yes Yes Yes

4. 3, –1.2, –2,0

Graph and compare integersEXAMPLE 1

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

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

ANSWER

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

GUIDED PRACTICE for Example 1

Graph the numbers on a number line. Then tell which number is greater.

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

ANSWER

1. 4 and 0

– 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6

0 4


On the number line, 2 is to the right of –4. So, 2 > –5.

ANSWER

2. 2 and –5

– 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6

2–5

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


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

ANSWER

3. –6 and –1

– 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6

–1–6


Order rational numbers

EXAMPLE 3

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 ShaulaColor index –0.03 0.21 0.09 – 0.22

SOLUTION

Begin by graphing the numbers on a number line.

ASTRONOMY

EXAMPLE 3

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.

• 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

Find opposites of numbersEXAMPLE 4

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

b. If a = , then – a = – . 34

34


Find absolute values of numbersEXAMPLE 5

23

23a. If a = – , then |a | = | | = – ( ) = 2

323

b. If a = 3.2, then |a| = |3.2| = 3.2.

learn classify rational numbers into different sets; 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| =

SOLUTION

10. a = 49–

–49–If a = , then –a = – ( ) = 4

94949

49| – | ( – )4

9|a| = = – =

learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers

(including absolute value

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)

Technology

2.1 integers & rational numbers