REAL NUMBERS
(as opposed to fake numbers?)
Real Numbers
• Real Numbers are every number.
• Therefore, any number that you can
find on the number line.
• Real Numbers have two categories.
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.
Real Numbers
REAL NUMBERS
154,769,852,354
1.333
-8
-5,632.1010101256849765…
61
π
549.23789
49%
Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers
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.
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
Integers
• Integers are rational numbers
because they can be written as
fraction with 1 as the denominator.
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
Number Systems
Real
Rational (fraction)
Integer
Irrational
Whole
Natural
– counting numbers beginning with 1, 2, 3, …
– counting numbers beginning with 0, 1, 2, 3, …
– All the whole numbers and their opposites
{…, –3, –2, –1, 0, 1, 2, 3, …}
– all numbers that can be written as a ratio (fraction)
– a decimal that never ends and never repeats
– all numbers that can be put on a number line
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.
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)
2
b)
12
• a) Irrational
• b) Irrational
• c) Rational
c)
d)
e)
25
• d) Rational
5
11
66
• e) Irrational
Additional Example 1: Classifying Real
Numbers
Write all classifications that apply to each
number.
A.
5 is a whole number that is
not a perfect square.
irrational, real
5
B. –12.75 –12.75 is a terminating decimal.
rational, real
16
16
4
=
=2
C.
2
2
2
whole, integer, rational, real
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.
A.
21
irrational
B.
0
3
0
=0
3
rational
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational,
irrational, or not a real number.
C.
4
0
not a real number
Exit Ticket
• Draw the Real Number System.
• Explain the difference between a
natural number and whole number.
Objective
• TSW compare rational and irrational
numbers
• TSW order rational and irrational
numbers on a number line
Comparing Rational and
Irrational Numbers
• When comparing different forms of
rational and irrational numbers,
convert the numbers to the same
form.
Compare -3 37 and -3.571
(convert -3 37 to -3.428571…
-3.428571… > -3.571
Practice
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.
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
1) (-4) + 8 =
0
4
2) (-1) + (-3) =
-4
3) 5 + (-7) =
-2
5
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
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?
-1 + 3 = ?
1.
2.
3.
4.
-4
-2
2
4
Answer Now
-6 + (-3) = ?
1.
2.
3.
4.
-9
-3
3
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 (keepchange-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.
2.
3.
4.
12 + 3
-12 + 3
-12 - 3
12 - 3
Answer Now
7 – (-2) = ?
1.
2.
3.
4.
-9
-5
5
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,
the
answer
will be
positive.
If the
signs are
different,

the
answer
will be
negative.
1. -8 * 3
-24
Different
What’s
Signs
The
Negative
Rule?
Answer
2. -2 * -61
122
3. (-3)(6)(1)
(-18)(1)
-18
4. 6 ÷ (-3)
-2
Start inside ( ) first
5. - (20/-5)
- (-4)
4
Same
Signs
Positive
Answer
6.  4 0 8
6
68
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?
6
hours
+4
degrees
How much
does the
temperature
rise each
hour?
(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
(5 steps) (11 feet)
(55 feet)
5 * 11 = 55
Exit Ticket
• What is operation is called the
additive inverse?
• What are the rules for subtracting
numbers?