Real numbers system

The Real Number System
W N Q Z
IR
Presented By: Prince

Real
Number
Real
Number
Real
Number

Real Numbers
REAL NUMBERS
-8 -5,632.1010101256849765…
61
49%
π
549.23789
154,769,852,354
1.333

The Real Number Line
Any real number corresponds to a point on the real
number line.
Order Property for Real Numbers
Given any two real numbers a and b,
- if a is to the left of b on the number line, then a < b.
- if a is to the right of b on the number line, then a > b.

Real Number System Tree Diagram
Real Numbers
Integers
Terminating
Decimals
Repeating
Decimals
Whole
Numbers
Rational
Numbers
Irrational
Numbers
Negative #’s
Natural #’s Zero
Non-Terminating
And
Non-Repeating
Decimals

Examples of Rational
Numbers
•16
•1/2
•3.56
•-8
•1.3333…
•- 3/4

Integers
One Of the subsets Of
ratiOnal numbers

What are integers?
• Integers are the whole numbers and their
opposites.
• Examples of integers are
6
-12
0
186
-934

What are integers?.......
• Integers are rational numbers because they can be
written as fraction with 1 as the denominator.

WHOLE
Numbers
REAL NUMBERS
IRRATIONAL
Numbers
NATURAL
Numbers
INTEGERS
RATIONAL
Numbers

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.
Caution!
A repeating decimal may not appear to
repeat on a calculator, because
calculators show a finite number of digits.

Examples of Irrational
Numbers
• Pi

Try this!
• a) Irrational
• b) Irrational
• c) Rational
• d) Rational
a) 2
b) 12
c) 25
d)
11
5
e) 66 • e) Irrational

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

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

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

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

Comparing Rational and
Irrational Numbers
• When comparing different forms of
rational and irrational numbers,
convert the numbers to the same
form.
37
Compare -3 and -3.571
(convert -3 to -3.428571…
-3.428571… > -3.571
37

Ordering Rational and
Irrational Numbers

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 1.2857142…
Answer: 0.04, 10%, ¹/₄, 75%, ⁹/₇

Practice
Order these from least to greatest:

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

Examples: Use the number line
if necessary.
-5 0 5
4
2) (-1) + (-3) =
-4
3) 5 + (-7) =
-2
1) (-4) + 8 =

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

-1 + 3 = ?
1. -4
2. -2
3. 2
4. 4
Answer Now

-6 + (-3) = ?
1. -9
2. -3
3. 3
4. 9
Answer Now

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.

What’s the difference
between
7 - 3 and 7 + (-3) ?
7 - 3 = 4 and 7 + (-3) = 4
The only difference is that 7 - 3 is a
subtraction problem and 7 + (-3) is an
addition problem.
“SUBTRACTING IS THE SAME AS
ADDING THE OPPOSITE.”
(Keep-change-change)

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

Which is equivalent to
-12 – (-3)?
1. 12 + 3
2. -12 + 3
3. -12 - 3
4. 12 - 3
Answer Now

7 – (-2) = ?
1. -9
2. -5
3. 5
4. 9
Answer Now

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.

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

1. -8 * 3 What’s
Different
The
Signs
Negative
Rule?
Answer
-24
2. -2 * -61
Same
Signs
Positive
Answer
122
3. (-3)(6)(1)
(-18)(1)
-18
Just take
Two at a time
4. 6 ÷ (-3)
-2
Start inside ( ) first
5. - (20/-5)
- (-4)
4
6.
-
-
68
408
6

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
How much
Is it from
does the
Midnight
temperature
to 6 am?
rise each
hour?
6
hours
+4
degrees
(6 hours)(4 degrees per hour)
= 24 degrees
8° + 24° = 32°C
Add this to
the original temp.

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
(5 steps) (11 feet)
mean?
This
(55 feet)
5 * 11 = 55

Summary
• What did you learn in this lesson?
• What are some important facts to
remember about the real number
system?
• Is there something within the lesson
that you need help on?

Real numbers system

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