Real NumbersReal NumbersTypes of Real NumbersTypes of Real Numbers

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Real Numbers✦ Two major groups -

✦ Rational

& Irrational

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Rational Numbers✦ What does it mean to act rational?

Hmmm....

✦ Rational is something that makes sense.

✦ Rational numbers make sense because they are predictable.

✦ Rational numbers fall into 1 of 2 categories.

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Rational Numbers - Category 1

✦ What do these numbers have in common?

✦ Notice they all stop or terminate.

✦ What about the fractions? Change them to decimals.

✦ Do they terminate now?

✦ Rational numbers are like the terminator because they terminate or end.

✦ The first category Rational Numbers can fall in is terminating numbers.

3

−27

−2.34 1

2

−8

4

123,456

Rational Numbers - Category 2

✦ What do these numbers have in common?

✦ Can you guess what the next number will be for each of these numbers?

✦ What about the fractions?

✦ Change them to decimals. Can you predict the next number?

✦ Notice they these numbers follow a predictable pattern.

✦ The second category Rational Number may fall in is following a predictable pattern.

5

3

9.232323...

−11

3

−0.1111...

7.89789

Irrational Numbers✦ What does it mean when a person

is irrational? Hmm....

✦ What do you notice about these numbers?

✦ Can you predict the next number?

✦ When do they end? (Put the square roots in a calculator to analyze their decimal equivalent.)

✦ Irrational numbers are unpredictable numbers just like an irrational person has unpredictable behavior.

✦ Irrational Numbers never end and the next number cannot be guessed because no pattern exists.

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π =3.14159...

e=2.718281828459...

2.1239257...

2

3

5

Types of Rational Numbers

✦ Natural Numbers (also called the Counting Numbers)

✦ What number do you naturally start with when you count?

✦ 1.

✦ The Natural Numbers are the numbers we count with beginning with 1.

✦ 1, 2, 3, 4, ...

✦ Wh0le Numbers - The natural numbers plus zero

✦ 0, 1, 2, 3, 4, ...

✦ Integers - All whole numbers and their opposites. In-other-words, all positive and negative numbers without decimals or fractions.

✦ ... -3, -2, -1, 0, 1, 2, 3, ...

✦ Notice some numbers fall into more than one category.

Real Numbers

5

RationalIrrational

Integers

Whole

Natural0

1

−7

−42

3

−52

0.6767...

π

e

2 =1.4142135...

5

41

3

Be sure you can do the following...

✦ Describe the difference between an irrational and rational number.

✦ Describe the characteristics of a natural number.

✦ Describe the characteristics of a whole number.

✦ Describe the characteristics of a integer.

✦ Give a example of

✦ an irrational number.

✦ a rational number and one that is not an integer.

✦ an integer and one that is not a whole number.

✦ a whole number and one that is not a natural number.

✦ a natural number.