Real Numbers (Part 2)

[Pages:9]Real Numbers (Part 2)

Learning Objective

At the end of this lesson, you should be able to: differentiate among terminating and non-terminating decimals; recurring or repeating decimals; and real numbers.

Terminating Decimals

Terminating decimals are decimals having a finite number of digits after the decimal point. Examples of terminating decimals are: 0.3 , 0.75 , 0.001 , 2.5625 , ...

Non-Terminating Decimals

Activity 1

Express 2 as a decimal. 17

Since, the denominator cannot be expressed as a power of 10, we need to use long division to express 2 as a decimal.

17

1

This decimal number is called a non-terminating decimal. It has an infinite number of digits after the decimal point.

This division will continue indefinitely (i.e. it will never end).

Recurring or Repeating Decimals

Activity 2

Express 1 as a decimal. 3

Since, the denominator cannot be expressed as a power of 10, we need to use long division to express 1 as a decimal.

3 This non-terminating decimal contains an endless pattern of the digit 3. This type of decimal is known as a recurring or repeating decimal.

2

The recurring or repeating decimal may be expressed using the dot notation.

For example:

(i)

4 9

=

0.444444...

=

. 0.4

A dot is placed over the recurring digit 4.

(ii)

8

.. = 0.72727272... = 0.7 2

11

A dot is placed over each of the two recurring digits.

(iii)

5 12

=

0.41666666...

=

. 0.416

A dot is placed over the recurring digit 6.

(iv)

17 111

=

0.153153153...

=

.. 0.153

A dot is placed over the first and the last recurring digits 1 and 3.

(v)

3

=

0.428571428571...

=

. 0.4

. 28571

7

A dot is placed over the first and the last recurring digits 4 and 1.

Ordinary Notation

0.444444... 0.72727272... 0.41666666... 0.153153153... 0.428571428571...

Dot Notation

. 0.4

.. 0.7 2

. 0.416

.. 0.15 3

.. 0.4 28571

3

Example 1

Express the following rational numbers in decimal form.

(a) 2 5

(b) 2 1

4

(c)

5 -

8

Solution

(a)

2 5

=

2? 5?

2 2

=

4 10

=

0.4

(b)

1 1? 25

=

=

25

= 0.25

4 4? 25 100

2 1 = 2 + 0.25 = 2.25 4

(c) 5 = 5?125 = 625 = 0.625 8 8?125 1000 - 5 = -0.625 8

Example 2

Express the following recurring decimals using dot notation. (a) 0.666666... (b) 0.45454545... (c) 0.0324324324...

4

Solution

. (a) 0.666666... = 0.6

Only 1 recurring digit 1 dot.

.. (b) 0.45454545... = 0.45

2 recurring digits 2 dots.

.. (c) 0.0324324324... = 0.032 4

3 recurring digits 1 dot on 1st recurring digit and 1 dot on last recurring digit.

Example 3

.. Express the recurring decimal 0.27 in ordinary notation.

Solution

.. 0.2 7 = 0.27272727...

Irrational Numbers

Irrational numbers are numbers which cannot be expressed in the form p q

where p and q are integers and q 0 including non-repeating and nonterminating decimals.

5

Examples of irrational numbers are: 5 = 2.2360..., = 3.1415..., 1 = 0.7071..., etc. 2

The set of irrational numbers is an infinite set.

Example 4

Which of the following are irrational numbers? Explain why.

(a) 0.75

(b) 3

(c) 36

. (d) 0.5

Solution

(a) Rational since 0.75 = 75 . 100

(b) Irrational since 3 cannot be expressed as a fraction of integers.

(c) Rational since 36 = 6 .

(d)

Rational

since

. 0.5 =

5.

9

6

Real Numbers

The set of rational numbers and the set of irrational numbers together form the set of real numbers. The set of real numbers is denoted by ? . The set of real numbers is an infinite set.

Example 5

For each of the following, fill in the boxes with one of the symbols < , > or = .

(a) -5

0

..

(b) 0.63

0.63

(c) 4.5

4 1

2

(d)

1 2 3

-2 3 5

Solution

(a) -5 < 0

..

(b) 0.63 < 0.63

(c) 4.5

=

4 1 2

0 is greater than any negative number. ..

0.63 = 0.636363...

7

(d) 12 3

> -2 3

5

Any positive number is greater than any negative number.

Exercises

1. Express the following rational numbers in decimal form.

(a) 3 10

(b)

7 8

(c)

-2 4 5

2. Express the following recurring decimals using dot notation.

(a) 0.111111...

(b) 0.454545...

(c) 0.603603...

3. Which of the following are irrational numbers? Explain why. (a) 81 (b) - 1 3 (c)

. 4. Express the recurring decimal 0.3 in ordinary notation.

5. For each of the following, fill in the boxes with one of the symbols < , > or =.

(a) 3.9

- 7.4

(b) 3

.. 0. 2 7

11

(c) 2.5

2 1

5

8

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