 1.01 The real number system

Worksheet
The real number system
1

Consider the folllowing sets of numbers:

a

Identify the set of all integers.

b

Identify the set of whole numbers.

c

Identify the smallest whole number.

• \left\{0, 1, 2, 3, \ldots\right\}

• \left\{\ldots, - 3 , - 2 , - 1 , 0, 1, 2, 3, \ldots\right\}

• \left\{1, 2, 3, \ldots\right\}

• \left\{\ldots, - 3 , - 2 , - 1 , 0\right\}

2

Classify the set of numbers containing \left\{0, 1, 2, \ldots\right\}

3

Consider the following diagram:

a

Determine whether each of the following statements is true or false:

i

\sqrt{13} is a rational number.

ii

When written in decimal form, irrational numbers have a repeating pattern of decimal digits.

iii

A real number is either a rational or irrational number.

b

Classify the following numbers:

i
38
ii
- 42
iii
\sqrt{6}
iv
3.058
v
\sqrt{49}
vi
\dfrac{\pi}{3}
vii
0.5777 \ldots
viii
\dfrac{26}{45}
4

5

Every whole number is also a rational number but not every rational number is a whole number. Explain why this statement is true.

6

Christa decides to express her favorite rational number as a decimal. Could the decimal be repeating, terminating or either?

7

Akin states that 24.737337333... is a rational number because it has a repeating pattern.

8

State whether each of the following numbers are rational or irrational:

a

-\dfrac{2}{7}

b

\sqrt{2}

c

8

d

20.57

e

3\sqrt{5}

f

1-\sqrt{7}

g

-0.\overline{3}

h

\pi

9

Simplify each of the following numbers and state whether they are rational or irrational:

a

\sqrt{16}

b

\dfrac{2\pi}{8}

c

-\dfrac{4\pi}{5\pi}

d

\dfrac{0.4}{5}

10

State the set of numbers most appropriate to describe the following:

a

The populations of towns.

b

The times of the runners (in seconds) in a 100\text{ m} sprint.

c

Distance between Jupiter and Venus.

d

The position of a submarine relative to sea level. (Note that height above sea level is expressed as a positive quantity.)

e

The goal difference of a hockey team at the end of a season. (Note that the goal difference is the number of goals the team scored in the season minus the number of goals that were scored against them.)

f

The cost (in dollars) of sending a text message.

11

The circumference of a circle is given by 2 \pi r, where r is the radius.

State whether each of the following statements is true:

a

The circumference is an irrational number if the radius is rational.

b

The circumference is a rational number if the radius is equal to \pi.

c

The circumference is a rational number if the radius is irrational.

d

The circumference is a rational number if the radius is equal to \dfrac{a}{\pi} where a is an integer.