2.1 integers & rational numbers
DESCRIPTION
TRANSCRIPT
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?
Integer?Whole 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?
Integer?Whole 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?
Integer?Whole 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
GUIDED PRACTICE for Example 1
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
GUIDED PRACTICE for Example 1
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
learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value
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.
learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value
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 | = _______
learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value
• 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
learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value
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)