This document defines key concepts in the real number system including rational numbers, irrational numbers, integers, and subsets. It explains that real numbers contain both rational numbers (which can be expressed as fractions) and irrational numbers (which cannot). Various properties are discussed such as comparing and ordering real numbers. Examples are provided to demonstrate plotting numbers on a number line, determining if a number is rational or irrational, and arranging sets of numbers in ascending and descending order.

1. THE REAL NUMBER SYSTEM

2. What is a Real Number?
Is any element of R which
contains Rational ( Q)
numbers and Irrational (Q’)
numbers

3. How would you plot the following
real numbers in the number line?
a) A= - 𝟐
b) B= 5
c) C= 𝞹
d) D= 0.5
e) E= ¾

4. Describe the relationship of the sets shown
in the Venn Diagram
RATIONAL
NUMBERS(Q) IRRATIONAL
NUMBERS(Q’)
U
0 1 -2
-3 ¾ 15%
0.5
e −𝟐
𝞹 √𝟑

5. How do we define a Irrational
number?
A irrational number is a real
number that can not be expressed
as
𝑚
𝑛
where m and n is an integer (Z)
an that n ≠ 0 .

6. How do we define a Rational
number?
A rational number is a real number
that can be expressed as
𝑚
𝑛
where m
and n is an integer (Z) an that n ≠ 0
.

7. What are the different subsets of
the rational numbers?
Non – integer
Integers
a) negative integers
b) whole number
- zero and natural numbers

8. REAL NUMBER (R)
RATIONAL NUMBER, (Q) IRRATIONAL NUMBER, (Q’)
NON-INTEGER INTEGERS, (Z)
NEGATIVE INTEGERS,(𝒁−) WHOLE NUMBERS,(W)
ZERO
IRRATIONAL NUMBER, (Q’)
POSITIVE INTEGERS,(𝒁+
)
(N)

9. Perform the indicated operations and write the
answer in roster form and set builder form.
𝑍+ U 𝑍−
R – Q ‘
Q ⋂ Q ‘
(Q ‘)’

10. Consider the given numbers which are on the
left most column of the table. Check as many
appropriate cells as applicable:
numbers +Z -Z Whole
number
Natural
number
Rational
number
Irrational
number
Real
number
Non-
negative
Integer
Non-
integer
25
11
6.30
4
23
0
-
𝞹
2
2e

11. TRUE OR FALSE: Write True if the statement is
always True and false if otherwise.
All integers are real numbers
Every whole number is greater than zero
All integers are signed number
All whole numbers are integers
Every integer is a real number

12. TRUE OR FALSE: Write True if the statement is
always True and false if otherwise.
All real numbers are rational
All rational numbers are integers
All non - integers are rational
Every integer is a whole number
Irrational number are real numbers.

13. RATIONAL OR IRRATIONAL?

14. DETERMINE whether the following is
rational or not.
 -20
¼
0.75
7/9
 3
RATIONAL
RATIONAL
RATIONAL
RATIONAL
IRRATIONAL

15. “ TRICHOTOMY LAW”
COMPARING REAL NUMBERS

16. “ TRICHOTOMY LAW”
If a and b are real numbers, then one
and only one of the following is true:
i) a < b ii) a = b iii) a > b

17. How do we compare numbers?
A= 5 C= 0 E=
𝟑
𝟐𝟕 G = 0.8
B = -5 D =
𝟕
𝟏𝟎
F = 3 H =
𝟒
𝟓

18. −7, −
19
11
, 0 , 𝑒, 7 ,
18
5
, 4
ASCENDING ORDER
How do we write
{4, -7, 0, 18/5, -19/11, 7 , e } into
ascending order ?

19. DESCENDING ORDER
How do we write
{4, -7, 0, 18/5, -19/11, 7 , e } into
descending order ?
4,
18
5
, 7,e, 0,
−19
11
, −7

20. ARRANGE THE Following numbers into
ascending order:
a) 𝞹, - 𝞹 , 0 , 2, -5, 5
b) 3. 5,
7
2
, 0.6 , 0.5, 0.05 , 50%

21. ARRANGE THE Following into descending
order:
a) 𝞹, - 𝞹 , 0 , 2, -5, 5
b) 3. 5,
7
2
, 0.6 , 0.5, 0.05 , 50%
c) -3, -2, -1, 0, 1, 2, 3,