1. Sets of Numbers Click “Slide Show”. Then click “From Beginning”.
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Ms. Andrea Davis Morgan

2. Natural Numbers Natural Numbers are the counting numbers:
1, 2, 3, 4, 5, . . .
There is no largest Natural Number because the numbers keep going infinitely (forever).
The Natural Numbers do not include
zero, fractions, decimals, or negative numbers.
The smallest
Natural Number
is 1.
The three dots after the five . . .
mean the numbers keep going.
(We say “et cetera”)
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3. Natural Numbers
The Natural Numbers are positive numbers. They can be written with or without a positive sign.
1, 2, 3, 4, 5, . . . or
+1, +2, +3, +4, +5, . . .
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4. but, also include the number zero.
Whole Numbers
Zero has no sign. The positive numbers can be
written with or without a positive sign.
0, 1, 2, 3, 4, 5, . . .
Zero has no sign. The positive numbers can be
written with or without a positive sign.
0, 1, 2, 3, 4, 5, . . . or
0, +1, +2, +3, +4, +5, . . .
The Whole Numbers are positive numbers, except the number zero, which is neither positive nor negative.
0, 1, 2, 3, 4, 5, . . .
Whole Numbers include the counting numbers,
0, 1, 2, 3, 4, 5, . . .
but, also include the number zero.
Whole Numbers include the counting numbers,
1, 2, 3, 4, 5, . . .
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5. Integers Integers consist of the Whole Numbers.
0, 1, 2, 3, 4, 5, . . .
Integers consist of the Whole Numbers
, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .
and the opposites of those Whole Numbers
(the negatives).
Opposites always have a sum of zero.
= 0
Zero, which is neither positive nor negative,
is its own opposite.
, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .
= 0 or
= 0
Again, zero has no sign, and the positive numbers can be written with or without a positive sign.
, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . . or
, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, . . .
Each Integer, except zero, has an opposite which is the same number, but with the opposite sign.
, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .
Negative three and positive three are
opposites of each other.
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6. There are three types of numbers involving fractions.
Proper fractions
Improper fractions
Mixed numbers
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7. In proper fractions, the numerator (top number) is smaller than the denominator (bottom number).2 7
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8. In improper fractions, the numerator is larger
than the denominator.. 9 7

9. 4 In mixed numbers, there is a whole number
in front of a proper fraction. 2 7 4

10. There are three types of decimal numbers.
Terminating decimal numbers
Repeating decimal numbers
Non-terminating, Non-repeating decimal numbers
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11. Terminating decimal numbers have a definite end, a last digit.
In the number 5.38, the last digit, the 8, is in
the hundredths place.
In the number , the last digit, the 2, is in
the ten thousandths place.
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12. In Repeating decimal numbers, there is no last number because the numbers behind the decimal keep going and going and never stop. There is a number or numbers that repeat in a pattern over and over.
In the number , the number 8 keeps repeating on and on infinitely (forever). The 3 dots indicate that there are more 8’s following the others.
Another way to write this number is with a repeating bar over the number that keeps going.
= 7.8
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13. The repeating bar never goes over the number in front of the decimal.
In the number , the numbers 4 and 3 repeat in a pattern over and over without end. When you write this number with a repeating bar, you only put the bar over the digits that repeat. The bar does not go over the 6 and 7, only the 4 and 3. =
The repeating bar never goes over the number in front of the decimal.
= 6.6
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14. In Non-terminating, Non-repeating decimal numbers, the numbers behind the decimal keep going and never stop, but but there is no pattern to the numbers.
Examples:
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15. Rational Numbers
The Rational Numbers are numbers that can be written as a ratio of two Integers. That means, they can be written as a proper or improper fraction where the numerator and denominator are Integers.
Example:
In this Rational Number, the numerator, -4, is an integer, and the denominator, 5, is an integer. -4 5
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16. Rational Numbers
Numbers are Rational Numbers if they can be written in fraction form where a and b are Integers. Since the denominator of a fraction cannot be zero, the denominator “b” cannot be zero (b ≠ 0)
Example:
In this Rational Number, the numerator “a” is -7 and the denominator “b” is 2. a b -7 2
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17. Rational Numbers a b
Proper and Improper fractions are already in the form
and Mixed numbers can be put into that form by converting the Mixed number to an improper fraction.
Example: =
Therefore, all positive and negative fractions and mixed numbers are Rational Numbers. -3 4 5
-19
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18. Rational Numbers
All Natural Numbers, Whole Numbers, and Integers can be written as a ratio of two integers simply by putting the number over one, so all Natural Numbers, Whole Numbers, and Integers are included in the Rational Numbers.
3 = = = All of these fractions are the ratios
of two Integers and are equivalent
to the number 3.
The number 3 is a Rational Number because it can be written as a ratio of two Integers. 3 1 6 2
-15 -5
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19. Rational Numbers
All Terminating decimal numbers can be written as a fraction
Example: = = =
All Repeating decimal numbers can be written as a fraction
Example: =
So Terminating decimal numbers and Repeating decimal numbers are Rational numbers. a b 45
100 -3 9 20 -3
-69 20
-3.45
(Reduce and change to an improper fraction.) a b
271
110
(We will go over how to convert these in a different lesson.)
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20. All of the following are Rational Numbers:
Proper fractions
Improper fractions
Mixed numbers
Whole numbers
Terminating decimals
Repeating decimals
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21. Irrational Numbers
That leaves the Non-terminating, Non-repeating decimal numbers. Those are the only decimal numbers that cannot be written in fraction form. Therefore, they are in their own separate set called the Irrational Numbers
Example:
(The prefix “ir-“ means “not”, so irrational means “not rational”).
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22. Irrational Numbers
There are some symbols that we use to represent specific irrational numbers, such as 𝝅, 𝚽, and e.
𝝅 =
𝚽 =
e =
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23. Irrational Numbers
A square root is a number that will multiply times itself to equal the number under the square root sign.
For example = 3 because 3 x 3 = 9.
9 is called a perfect square because its square root is a rational number.
When you take the square root of a number that is not a perfect square, the result is an irrational number. =
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24. Real Numbers
The Natural Numbers are the counting numbers.
Rational Numbers 2 7 -8 3 4 5 -2
8.65
-7.3
, -3, -2, -1, 0,
1, 2, 3, 4, 5,
-0.756
- 7 15
13.2
. . .
The Rational Numbers and the Irrational Numbers together make up the set of Real Numbers.
The Natural Numbers are included in the Whole Numbers.
Integers
, -3, -2, -1
1, 2, 3, 4, 5,
Whole Numbers
1, 2, 3, 4, 5,
The Irrational Numbers are a separate set of numbers.
The Whole Numbers are included in the Integers.
1, 2, 3, 4, 5,
Natural Numbers
Irrational Numbers 𝝅
The Integers are included in the Rational Numbers.
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25. Real Numbers 𝝅
1, 2, 3, 4, 5,
, -3, -2, -1 -2 4 5 2 7 -8 3
8.65
-7.3
- 7 15
-0.756
13.2
. . .
Real Numbers
Rational Numbers 2 7 -8 3 4 5 -2
8.65
-7.3
, -3, -2, -1, 0,
1, 2, 3, 4, 5,
-0.756
- 7 15
13.2
. . .
Integers
, -3, -2, -1
1, 2, 3, 4, 5,
Whole Numbers
1, 2, 3, 4, 5,
Irrational Numbers 𝝅
1, 2, 3, 4, 5,
Natural Numbers
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