MICROMATHS
GRADE 11
REVISION OF THE NUMBER SYSTEM
Lesson 1: Revision of The Number System
In this lesson we will cover:
• Natural Numbers, Counting Numbers, Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers, Non Real Numbers
• When will a number be undefined
REAL NUMBERS (R)
IRRATIONAL NUMBERS (Q’)
RATIONAL NUMBERS (Q)
• NATURAL NUMBERS (N)
• COUNTING NUMBERS NO
• INTEGERS (Z)
N = {1; 2; 3; 4; 5; 6; ….}
• SURDS e.g.
7
𝑁𝑜 = {0; 1; 2; 3; 4; 5; … . }
Z = {….; -3; -2; -1; 0; 1; 2; 3; 4; ….}
• 𝜋
• FRACTIONS
o COMMON FRACTIONS
e.g.
3
4
;
4
; 𝑒𝑡𝑐.
9
o DECIMAL FRACTIONS
NON-REAL NUMBERS:
UNDEFINED:
 TERMINATING DECIMALS
e.g. 0.75
 RECURRING DECIMALS
e.g. 0, 4
7
0
−4
The Number System
1. Natural Numbers
N = {1; 2; 3; 4; ……} or N = {x: x  N}
2. Whole Numbers (Counting Numbers)
N0 = {0; 1; 2; 3; 4; …..} or No = {x: x  No}
3. Integers
Z = {…….; -2; -1; 0; 1; 2; ……} or Z = {x : x  Z}
4. Rational Numbers
Q = {x: x =
𝑎
𝑏
; a, b  Z , b  0}
Rational numbers can be expressed as
– common fractions e.g. ,
3
4
;
7
;
9
etc. or as
– terminating decimals e.g. 0,5; 0,375; -0,72; -0,24, etc.
or as
– recurring decimals e.g. 0, 5 ; 0,16 ; 0, 37; etc.
5. Irrational Numbers
Irrational numbers are those numbers for which the “exact” value cannot
be determined. and  are examples of irrational numbers.
Examples of irrational numbers:
3
7 ; 9; 𝑒𝑡𝑐.
The elements of rational and irrational numbers cannot be listed.
6. Real Numbers
• The set of rational numbers (terminating and recurring decimals) unified with the
set of irrational numbers (non-recurring decimals) is the set of real numbers.
• R = {real numbers}
• {real numbers} = {rational numbers}  {irrational numbers}
• Numbers that cannot be placed on the number line are non-real, e.g. −2 ;
−4 etc.
7. Division by Zero
• Division by zero is undefined
• i.e. Division by zero is not possible
Conditions for an expression to be
Real, Non real or Undefined
• An expression is
• REAL if the discriminant  0
• NONREAL if the discriminant < 0
• UNDEFINED if the denominator = 0
• The discriminant is the expression under the
sign
Example
For which values of x will
A. Real
B. Nonreal
C. Undefined?
A. Real if x – 3  0
i.e.
x3
B. Nonreal if x – 3 < 0
i.e.
x<3
C. Undefined if x + 1 = 0
i.e.
x = –1
𝑥 −3
𝑥+1
be