1. (BY MISS.MANITA MODPHAI)
REAL NUMBERS
(BY MISS.MANITA MODPHAI)

2. Objective TSW identify the parts of the Real Number System
TSW define rational and irrational numbers
TSW classify numbers as rational or irrational

3. Real Numbers Real Numbers are every number.
Therefore, any number that you can find on the number line.
Real Numbers have two categories.

4. What does it Mean? The number line goes on forever.
Every point on the line is a REAL number.
There are no gaps on the number line.
Between the whole numbers and the fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever.

5. Real Numbers REAL NUMBERS 154,769,852,354 1.333
-5, … -8 61 π
49%

6. Two Kinds of Real Numbers
Rational Numbers
Irrational Numbers

7. Rational Numbers
A rational number is a real number that can be written as a fraction.
A rational number written in decimal form is terminating or repeating.

8. Examples of Rational Numbers16
1/2
3.56 -8
1.3333…
- 3/4

9. One of the subsets of rational numbers
Integers
One of the subsets of rational numbers

10. What are integers? Integers are the whole numbers and their opposites.
Examples of integers are 6
-12
186
-934

11. Integers are rational numbers because they can be written as fraction with 1 as the denominator.

12. Types of Integers
Natural Numbers(N): Natural Numbers are counting numbers from 1,2,3,4,5, N = {1,2,3,4,5, }
Whole Numbers (W): Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5, W = {0,1,2,3,4,5, } W = 0 + N

13. REAL NUMBERS
NATURAL
Numbers
WHOLE
Numbers
IRRATIONAL
Numbers
INTEGERS
RATIONAL
Numbers

14. Irrational Numbers
An irrational number is a number that cannot be written as a fraction of two integers.
Irrational numbers written as decimals are non-terminating and non-repeating.

15. Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number.
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

16. Examples of Irrational NumbersPi

17. Try this! a) Irrational b) Irrational c) Rational d) Rational
e) Irrational

18. Additional Example 1: Classifying Real Numbers
Write all classifications that apply to each number. A. 5
5 is a whole number that is not a perfect square.
irrational, real B.
–12.75
–12.75 is a terminating decimal.
rational, real
16 2
= = 2
4 2
16 2 C.
whole, integer, rational, real

19. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

20. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number. A. 21
irrational
0 3
0 3
= 0 B.
rational

21. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number.
4 0 C.
not a real number

22. Objective TSW compare rational and irrational numbers
TSW order rational and irrational numbers on a number line

23. Comparing Rational and Irrational Numbers
When comparing different forms of rational and irrational numbers, convert the numbers to the same form.
Compare and
(convert -3 to …
… > 3 7 3 7

24. Practice

25. Ordering Rational and Irrational Numbers
To order rational and irrational numbers, convert all of the numbers to the same form.
You can also find the approximate locations of rational and irrational numbers on a number line.

26. Example Order these numbers from least to greatest.
¹/₄, 75%, .04, 10%, ⁹/₇
¹/₄ becomes 0.25
75% becomes 0.75
0.04 stays 0.04
10% becomes 0.10
⁹/₇ becomes …
Answer: , 10%, ¹/₄, 75%, ⁹/₇

27. Practice
Order these from least to greatest:

28. Objectives TSW identify the rules associated computing with integers.
TSW compute with integers

29. Examples: Use the number line if necessary.
1) (-4) + 8 = 4
2) (-1) + (-3) = -4
3) 5 + (-7) = -2

30. SUBTRACT and use the sign of the larger number.
Addition Rule
1) When the signs are the same,
ADD and keep the sign.
(-2) + (-4) = -6
2) When the signs are different,
SUBTRACT and use the sign of the larger number.
(-2) + 4 = 2
2 + (-4) = -2

31. Karaoke Time!
Addition Rule: Sung to the tune of “Row, row, row, your boat”
Same signs add and keep, different signs subtract, keep the sign of the higher number, then it will be exact!
Can your class do different rounds?

32. = ? -4 -2 2 4
Answer Now

33. -6 + (-3) = ? -9 -3 3 9
Answer Now

34. The additive inverses (or opposites) of two numbers add to equal zero.
Example: The additive inverse of 3 is -3
Proof: 3 + (-3) = 0
We will use the additive inverses for subtraction problems.

35. What’s the difference between 7 - 3 and 7 + (-3) ?
The only difference is that is a subtraction problem and 7 + (-3) is an addition problem.
“SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.”
(Keep-change-change)

36. When subtracting, change the subtraction to adding the opposite (keep-change-change) and then follow your addition rule.
Example #1: (-7)
- 4 + (+7)
Diff. Signs --> Subtract and use larger sign. 3
Example #2:
- 3 + (-7)
Same Signs --> Add and keep the sign.
-10

37. Which is equivalent to -12 – (-3)?
12 + 3
12 - 3
Answer Now

38. 7 – (-2) = ? -9 -5 5 9
Answer Now

39. Review
1) If the problem is addition, follow your addition rule. 2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule.

40. State the rule for multiplying and dividing integers….
If the signs are the same,
If the signs are different,
the answer will be negative.
the answer will be positive.

41. 1. -8 * 3 4. 6 ÷ (-3) -24 -2 2. -2 * -61 5. - (20/-5) 122 - (-4) 4
Different
Signs
Negative
Answer
What’s
The
Rule?
* 3
4. 6 ÷ (-3)
-24 -2
Start inside ( ) first
* -61
5. - (20/-5)
122
- (-4)
Same
Signs
Positive
Answer 4
3. (-3)(6)(1)
(-18)(1) 6.
-18
Just take
Two at a time 68

42. 7. At midnight the temperature is 8°C
7. At midnight the temperature is 8°C. If the temperature rises 4°C per hour, what is the temperature at 6 am?
How long
Is it from
Midnight
to 6 am?
How much
does the
temperature
rise each
hour?
6 hours
+4 degrees
(6 hours)(4 degrees per hour)
Add this to
the original temp.
= 24 degrees
8° + 24° = 32°C

43. (5 steps) (11 feet) (55 feet) 5 * 11 = 55
8. A deep-sea diver must move up or down in the water in short steps in order to avoid getting a physical condition called the bends. Suppose a diver moves up to the surface in five steps of 11 feet. Represent her total movements as a product of integers, and find the product.
Multiply
What
does
This
mean?
(5 steps)
(11 feet)
(55 feet)
5 * 11 = 55