Textbook: Number Systems | Mathematics for Grade 10 PDF Download

 Page 1


 chaPTer 2 NUMBER S YSTEMS 23
2 
number Systems
In this chapter, you will:
•	 classify real numbers into sets and learn the symbols for the 
different sets
•	 represent sets of numbers on the number line as sets of 
points
•	 represent sets of numbers in interval notation and set 
notation
•	 revise rounding off
•	 improve your understanding of the relationship between 
rational numbers and irrational numbers
•	 learn how to express integers in binary form and how to do 
arithmetic with binary numbers
•	 be introduced to imaginary numbers, these numbers 
are not real numbers but they are closely related to real 
numbers.
2153 TechMaths Eng G10 LB.indb   23 2015/10/22   3:40 PM
Page 2


 chaPTer 2 NUMBER S YSTEMS 23
2 
number Systems
In this chapter, you will:
•	 classify real numbers into sets and learn the symbols for the 
different sets
•	 represent sets of numbers on the number line as sets of 
points
•	 represent sets of numbers in interval notation and set 
notation
•	 revise rounding off
•	 improve your understanding of the relationship between 
rational numbers and irrational numbers
•	 learn how to express integers in binary form and how to do 
arithmetic with binary numbers
•	 be introduced to imaginary numbers, these numbers 
are not real numbers but they are closely related to real 
numbers.
2153 TechMaths Eng G10 LB.indb   23 2015/10/22   3:40 PM
24 Technical Ma TheMaTicS Grade 10
2.1 real numbers
Real numbers are the numbers we deal with every day. As you have probably seen in the past, we 
classify them in an organised manner as follows:
real numbers
(all Non-Imaginary 
Numbers – See later)
irrational n umbers
(Non-Rational 
Numbers)
rational n umbers
(Numbers that can be 
written as a ratio of 
two whole numbers)
integers
(Whole Numbers)
counting n umbers
(Non-Negative 
Intergers)
non-integer 
rational n umbers
natural n umbers 
(Positive Integers)
Zero
negative intergers
We call each of the ‘collections’ of numbers a set. The sets that are linked to the ones above 
them are called subsets. When a number belongs to a set, it is called an element of the set. 
This arrangement is called a hierarchy or a hierarchical structure. How to understand the 
diagram:
•	 the connecting lines show which sets are linked to each other directly
•	 the numbers in any set are also in all the sets that may lie above as you follow the  
connecting lines
For example, your class can be regarded as a set. Therefore, you and the learners in your class are 
regarded as elements of that set. 
Another example, you are also an element of your school, which is the set of you and all your 
schoolmates. Your class is a subset of the school.
2153 TechMaths Eng G10 LB.indb   24 2015/10/22   3:40 PM
Page 3


 chaPTer 2 NUMBER S YSTEMS 23
2 
number Systems
In this chapter, you will:
•	 classify real numbers into sets and learn the symbols for the 
different sets
•	 represent sets of numbers on the number line as sets of 
points
•	 represent sets of numbers in interval notation and set 
notation
•	 revise rounding off
•	 improve your understanding of the relationship between 
rational numbers and irrational numbers
•	 learn how to express integers in binary form and how to do 
arithmetic with binary numbers
•	 be introduced to imaginary numbers, these numbers 
are not real numbers but they are closely related to real 
numbers.
2153 TechMaths Eng G10 LB.indb   23 2015/10/22   3:40 PM
24 Technical Ma TheMaTicS Grade 10
2.1 real numbers
Real numbers are the numbers we deal with every day. As you have probably seen in the past, we 
classify them in an organised manner as follows:
real numbers
(all Non-Imaginary 
Numbers – See later)
irrational n umbers
(Non-Rational 
Numbers)
rational n umbers
(Numbers that can be 
written as a ratio of 
two whole numbers)
integers
(Whole Numbers)
counting n umbers
(Non-Negative 
Intergers)
non-integer 
rational n umbers
natural n umbers 
(Positive Integers)
Zero
negative intergers
We call each of the ‘collections’ of numbers a set. The sets that are linked to the ones above 
them are called subsets. When a number belongs to a set, it is called an element of the set. 
This arrangement is called a hierarchy or a hierarchical structure. How to understand the 
diagram:
•	 the connecting lines show which sets are linked to each other directly
•	 the numbers in any set are also in all the sets that may lie above as you follow the  
connecting lines
For example, your class can be regarded as a set. Therefore, you and the learners in your class are 
regarded as elements of that set. 
Another example, you are also an element of your school, which is the set of you and all your 
schoolmates. Your class is a subset of the school.
2153 TechMaths Eng G10 LB.indb   24 2015/10/22   3:40 PM
 chaPTer 2 NUMBER S YSTEMS 25
example how the sets are related
•	 The set of integers is a subset of the set of rational numbers. 
•	 The set of rational numbers is a subset of the set of real numbers.
•	 This means that the set of integers is also a subset of the set of real numbers. 
•	 The number 0,56 is a non-integer rational number and also a real number but it is not an 
integer. The counting number 7 is clearly an integer, but 7 is also a rational number and a 
real number. 
•	 A counting number is either zero or a positive integer, e.g. 7 is a positive integer and also  
not zero. The set to which only 0 belongs has only one number in it!
•	 The number pi = 3,141 592 653 589 … is irrational. This means that pi only belongs to 
two sets in the hierarchy: the set of irrational numbers and also the set of real numbers.
We have special symbols for most of these sets. So, e.g. instead of writing the set of real numbers 
we just write R. Here is a full list of the symbols for each of the sets:
the set of real numbers R
the set of rational numbers Q
the set of irrational numbers R - Q or Q' 
the set of integers Z 
the set of non-integer rational numbers Q - Z 
the set of counting numbers (non-negative integers) N
0
the set of positive integers (natural numbers) Z
+
 or N
the set of negative integers Z 
-
  or Z - N
0
the set containing only zero {0}
Writing R - Q is a formal way of saying ‘all the real numbers excluding the rational numbers’.  
We can call this ‘set subtraction’ so long as we understand that it is not the same as the 
subtraction we do with numbers.
Set subtraction is about excluding some elements from a set. In R - Q, we have excluded all the 
rational numbers from the set of real numbers, leaving only the irrational numbers.
This ‘set subtraction’ is normally given a more formal name; ‘the complement of set Q in set R’. 
Many mathematicians prefer to write R - Q as R\Q, which is read, ‘set R excluding set Q’. You 
may do so as well if you prefer.
2153 TechMaths Eng G10 LB.indb   25 2015/10/22   3:40 PM
Page 4


 chaPTer 2 NUMBER S YSTEMS 23
2 
number Systems
In this chapter, you will:
•	 classify real numbers into sets and learn the symbols for the 
different sets
•	 represent sets of numbers on the number line as sets of 
points
•	 represent sets of numbers in interval notation and set 
notation
•	 revise rounding off
•	 improve your understanding of the relationship between 
rational numbers and irrational numbers
•	 learn how to express integers in binary form and how to do 
arithmetic with binary numbers
•	 be introduced to imaginary numbers, these numbers 
are not real numbers but they are closely related to real 
numbers.
2153 TechMaths Eng G10 LB.indb   23 2015/10/22   3:40 PM
24 Technical Ma TheMaTicS Grade 10
2.1 real numbers
Real numbers are the numbers we deal with every day. As you have probably seen in the past, we 
classify them in an organised manner as follows:
real numbers
(all Non-Imaginary 
Numbers – See later)
irrational n umbers
(Non-Rational 
Numbers)
rational n umbers
(Numbers that can be 
written as a ratio of 
two whole numbers)
integers
(Whole Numbers)
counting n umbers
(Non-Negative 
Intergers)
non-integer 
rational n umbers
natural n umbers 
(Positive Integers)
Zero
negative intergers
We call each of the ‘collections’ of numbers a set. The sets that are linked to the ones above 
them are called subsets. When a number belongs to a set, it is called an element of the set. 
This arrangement is called a hierarchy or a hierarchical structure. How to understand the 
diagram:
•	 the connecting lines show which sets are linked to each other directly
•	 the numbers in any set are also in all the sets that may lie above as you follow the  
connecting lines
For example, your class can be regarded as a set. Therefore, you and the learners in your class are 
regarded as elements of that set. 
Another example, you are also an element of your school, which is the set of you and all your 
schoolmates. Your class is a subset of the school.
2153 TechMaths Eng G10 LB.indb   24 2015/10/22   3:40 PM
 chaPTer 2 NUMBER S YSTEMS 25
example how the sets are related
•	 The set of integers is a subset of the set of rational numbers. 
•	 The set of rational numbers is a subset of the set of real numbers.
•	 This means that the set of integers is also a subset of the set of real numbers. 
•	 The number 0,56 is a non-integer rational number and also a real number but it is not an 
integer. The counting number 7 is clearly an integer, but 7 is also a rational number and a 
real number. 
•	 A counting number is either zero or a positive integer, e.g. 7 is a positive integer and also  
not zero. The set to which only 0 belongs has only one number in it!
•	 The number pi = 3,141 592 653 589 … is irrational. This means that pi only belongs to 
two sets in the hierarchy: the set of irrational numbers and also the set of real numbers.
We have special symbols for most of these sets. So, e.g. instead of writing the set of real numbers 
we just write R. Here is a full list of the symbols for each of the sets:
the set of real numbers R
the set of rational numbers Q
the set of irrational numbers R - Q or Q' 
the set of integers Z 
the set of non-integer rational numbers Q - Z 
the set of counting numbers (non-negative integers) N
0
the set of positive integers (natural numbers) Z
+
 or N
the set of negative integers Z 
-
  or Z - N
0
the set containing only zero {0}
Writing R - Q is a formal way of saying ‘all the real numbers excluding the rational numbers’.  
We can call this ‘set subtraction’ so long as we understand that it is not the same as the 
subtraction we do with numbers.
Set subtraction is about excluding some elements from a set. In R - Q, we have excluded all the 
rational numbers from the set of real numbers, leaving only the irrational numbers.
This ‘set subtraction’ is normally given a more formal name; ‘the complement of set Q in set R’. 
Many mathematicians prefer to write R - Q as R\Q, which is read, ‘set R excluding set Q’. You 
may do so as well if you prefer.
2153 TechMaths Eng G10 LB.indb   25 2015/10/22   3:40 PM
26 Technical Ma TheMaTicS Grade 10
example classifying some numbers
A.  2 is a natural number, a counting number, a rational number, and a real number. It is  
neither a negative integer nor an irrational number.
B. 0,578 5 is a rational number (since 0,578 5 can be written as the fraction   
5 785
 
_______
 
10 000
    or as the
   ratio 5 785:10 000) a non-integer rational number and a real number. It is not an integer (or 
anything lower in the hierarchy) or an irrational number.
C.   v
_
 3   is an irrational number (we’ll explain why later) and a real number. It is not a rational 
number (or anything lower in the hierarchy).
D.  Numbers such as  v
_
 2   are called surds. True surds are never rational. ‘Surds’ that are rational 
are ones where the number under the root sign are perfect squares, perfect cubes etc. For 
example,  v
___
 1,21   = 1,1 is rational because 1, 21 is a perfect square (1,1 × 1,1 = 1,21) but   
3
 
v
_____
 1 ,2 1   is 
irrational because 1,21 is not a perfect cube;   
3
 v
___
 27   is rational, but   
3
 v
__
 9   and   
3
 v
__
 3   are irrational.
E.   v
__
 -1   is a non-real number (we’ll look into this at the end of the chapter), and therefore 
it doesn’t fit anywhere into the hierarchy we currently have.  v
__
 - 1   is called an imaginary 
number. This is because there is no real number you can square to get -1. Imaginary 
numbers, together with real numbers, e.g. 3 + 2 ×  v
__
 - 1  , are called complex numbers 
(complex because they are a complex of real and imaginary numbers – think of a housing 
complex made up of different parts).
some more symbols
It is very wordy to say and write the following:
•	 the set of integers is a subset of the rational numbers
•	 49 is an element of the set of counting numbers
We can shorten this by writing the symbols for the sets:
•	 Z is a subset of Q
•	 49 is an element of  N 
0
 
We can shorten this even more with symbol ‘?’ for ‘is a subset of’ and the symbol ‘?’ for ‘is an 
element of’:
•	 Z ? Q
•	 49 ? N0
Sometimes we also need to say that a number is not an element of a particular set. We use the 
symbol ? for this 4,9 ? Z.
2153 TechMaths Eng G10 LB.indb   26 2015/10/22   3:40 PM
Page 5


 chaPTer 2 NUMBER S YSTEMS 23
2 
number Systems
In this chapter, you will:
•	 classify real numbers into sets and learn the symbols for the 
different sets
•	 represent sets of numbers on the number line as sets of 
points
•	 represent sets of numbers in interval notation and set 
notation
•	 revise rounding off
•	 improve your understanding of the relationship between 
rational numbers and irrational numbers
•	 learn how to express integers in binary form and how to do 
arithmetic with binary numbers
•	 be introduced to imaginary numbers, these numbers 
are not real numbers but they are closely related to real 
numbers.
2153 TechMaths Eng G10 LB.indb   23 2015/10/22   3:40 PM
24 Technical Ma TheMaTicS Grade 10
2.1 real numbers
Real numbers are the numbers we deal with every day. As you have probably seen in the past, we 
classify them in an organised manner as follows:
real numbers
(all Non-Imaginary 
Numbers – See later)
irrational n umbers
(Non-Rational 
Numbers)
rational n umbers
(Numbers that can be 
written as a ratio of 
two whole numbers)
integers
(Whole Numbers)
counting n umbers
(Non-Negative 
Intergers)
non-integer 
rational n umbers
natural n umbers 
(Positive Integers)
Zero
negative intergers
We call each of the ‘collections’ of numbers a set. The sets that are linked to the ones above 
them are called subsets. When a number belongs to a set, it is called an element of the set. 
This arrangement is called a hierarchy or a hierarchical structure. How to understand the 
diagram:
•	 the connecting lines show which sets are linked to each other directly
•	 the numbers in any set are also in all the sets that may lie above as you follow the  
connecting lines
For example, your class can be regarded as a set. Therefore, you and the learners in your class are 
regarded as elements of that set. 
Another example, you are also an element of your school, which is the set of you and all your 
schoolmates. Your class is a subset of the school.
2153 TechMaths Eng G10 LB.indb   24 2015/10/22   3:40 PM
 chaPTer 2 NUMBER S YSTEMS 25
example how the sets are related
•	 The set of integers is a subset of the set of rational numbers. 
•	 The set of rational numbers is a subset of the set of real numbers.
•	 This means that the set of integers is also a subset of the set of real numbers. 
•	 The number 0,56 is a non-integer rational number and also a real number but it is not an 
integer. The counting number 7 is clearly an integer, but 7 is also a rational number and a 
real number. 
•	 A counting number is either zero or a positive integer, e.g. 7 is a positive integer and also  
not zero. The set to which only 0 belongs has only one number in it!
•	 The number pi = 3,141 592 653 589 … is irrational. This means that pi only belongs to 
two sets in the hierarchy: the set of irrational numbers and also the set of real numbers.
We have special symbols for most of these sets. So, e.g. instead of writing the set of real numbers 
we just write R. Here is a full list of the symbols for each of the sets:
the set of real numbers R
the set of rational numbers Q
the set of irrational numbers R - Q or Q' 
the set of integers Z 
the set of non-integer rational numbers Q - Z 
the set of counting numbers (non-negative integers) N
0
the set of positive integers (natural numbers) Z
+
 or N
the set of negative integers Z 
-
  or Z - N
0
the set containing only zero {0}
Writing R - Q is a formal way of saying ‘all the real numbers excluding the rational numbers’.  
We can call this ‘set subtraction’ so long as we understand that it is not the same as the 
subtraction we do with numbers.
Set subtraction is about excluding some elements from a set. In R - Q, we have excluded all the 
rational numbers from the set of real numbers, leaving only the irrational numbers.
This ‘set subtraction’ is normally given a more formal name; ‘the complement of set Q in set R’. 
Many mathematicians prefer to write R - Q as R\Q, which is read, ‘set R excluding set Q’. You 
may do so as well if you prefer.
2153 TechMaths Eng G10 LB.indb   25 2015/10/22   3:40 PM
26 Technical Ma TheMaTicS Grade 10
example classifying some numbers
A.  2 is a natural number, a counting number, a rational number, and a real number. It is  
neither a negative integer nor an irrational number.
B. 0,578 5 is a rational number (since 0,578 5 can be written as the fraction   
5 785
 
_______
 
10 000
    or as the
   ratio 5 785:10 000) a non-integer rational number and a real number. It is not an integer (or 
anything lower in the hierarchy) or an irrational number.
C.   v
_
 3   is an irrational number (we’ll explain why later) and a real number. It is not a rational 
number (or anything lower in the hierarchy).
D.  Numbers such as  v
_
 2   are called surds. True surds are never rational. ‘Surds’ that are rational 
are ones where the number under the root sign are perfect squares, perfect cubes etc. For 
example,  v
___
 1,21   = 1,1 is rational because 1, 21 is a perfect square (1,1 × 1,1 = 1,21) but   
3
 
v
_____
 1 ,2 1   is 
irrational because 1,21 is not a perfect cube;   
3
 v
___
 27   is rational, but   
3
 v
__
 9   and   
3
 v
__
 3   are irrational.
E.   v
__
 -1   is a non-real number (we’ll look into this at the end of the chapter), and therefore 
it doesn’t fit anywhere into the hierarchy we currently have.  v
__
 - 1   is called an imaginary 
number. This is because there is no real number you can square to get -1. Imaginary 
numbers, together with real numbers, e.g. 3 + 2 ×  v
__
 - 1  , are called complex numbers 
(complex because they are a complex of real and imaginary numbers – think of a housing 
complex made up of different parts).
some more symbols
It is very wordy to say and write the following:
•	 the set of integers is a subset of the rational numbers
•	 49 is an element of the set of counting numbers
We can shorten this by writing the symbols for the sets:
•	 Z is a subset of Q
•	 49 is an element of  N 
0
 
We can shorten this even more with symbol ‘?’ for ‘is a subset of’ and the symbol ‘?’ for ‘is an 
element of’:
•	 Z ? Q
•	 49 ? N0
Sometimes we also need to say that a number is not an element of a particular set. We use the 
symbol ? for this 4,9 ? Z.
2153 TechMaths Eng G10 LB.indb   26 2015/10/22   3:40 PM
 chaPTer 2 NUMBER S YSTEMS 27
exercises
1 Rewrite all the statements listed in the previous two examples in short form using the 
symbols for sets and the symbols ?, ?, and ?.
2 Classify the following numbers:
 (a) -12 584 (b) -36,36 (c)  
v
__
   
4
 
__
 
5
    
 (d) 1,1 1 1 1... (recurring) (e) 4 ÷ 3 (f) 4 ×  v
_
 3  
 (g)  v
_____
 4 × 3   (h)  v
_
 4   (i) 1 +   
2
 
__
 
3
  
 (j)  v
___
 169    (k) 1 +   
1
 
__
 
2
   +   
1
 
__
 
3
   +   
1
 
__
 
4
   +   
1
 
__
 
5
    (l)  v
_
 7   ×  v
_
 7   
 (m)  v
_
 7   +  v
_
 7   (n)   
3
 v
___
 - 1   (o) pi
 (p) circumference of a circle with radius 5
 (q) the area of a square with sides  v
__
 13  
 (r) the volume of a cube with sides  v
__
 13  
r eal number line: the real numbers as a set of points
Describing real numbers as a set is useful but it is not very helpful. We need a way to represent 
that the set of all the real numbers is an ordered set, because if we compare any two numbers, 
one will always be bigger than the other.
Your class can be made into an ordered set by arranging all the learners alphabetically. This can 
also be done differently, e.g. by date and time of birth, by shoe size, by weight etc. Number sets 
are usually only ordered according to their values.
The way we do this is to imagine that each and every real number is a point on a line, called the 
number line. We show the order of numbers by putting smaller numbers to the left of bigger 
ones. Exactly like your ruler works.
You should be familiar with this representation of real numbers, but here it is again:
We can say that a continuous line is just many, many points very, very close to each other. We 
can pick out points to represent numbers. We always show the position of at least two numbers, 
usually 0 and 1. This allows us to orientate ourselves and also gives us the scale.
Pick out another 
point to represent 1
Pick out one point 
to represent 0
Now we can pick as many points as we wish from the line to represent other numbers. We need 
to do this carefully if we are constructing a number line. We need to measure where the point 
that we need to pick is.
2153 TechMaths Eng G10 LB.indb   27 2015/10/22   3:40 PM