1. Different Numeral Systems

2. Denary System
The denary system is the most common numeral system used in our daily life. Numbers in this system are expressed by using ten numerals, which are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
It groups objects in the following way:

3. Take the following quantity of cubes as an example
Take the following quantity of cubes as an example. They are grouped as:
1 group of hundreds (102)
3 group of tens (101)
4 group of ones (100)
There are 1 group of ‘hundreds’, 3 groups of ‘tens’ and 4 groups of ‘ones’, so the number of cubes is 134(10).
134(10) is a denary number, where the base 10 denotes that the number is in the denary system.

4. In a denary number, the position of each digit has a fixed place value
In a denary number, the position of each digit has a fixed place value. Let’s take as an example: 2 1 7 8 . 5
The value of a binary number can be expressed in an expanded form, for example, 1 2 3
(10) 10
2178.5 - × + = 5 8 7

5. Follow-up question
Express the following denary numbers in the expanded form.
Solution

6. Binary System
Binary system is used to represent data in computers. In this system, only two numerals ‘0’ and ‘1’ are used to represent numbers.
The concept of binary system can be illustrated by grouping objects as follows:

7. Take the following quantity of cubes as an example
Take the following quantity of cubes as an example. They are grouped as:
1 group of eights (23)
0 group of fours (22)
1 group of twos (21)
1 group of ones (20)
There are 1 group of ‘eights’, 0 group of ‘fours’, 1 group of ‘twos’ and 1 group of ‘ones’, so the number of cubes is 1011(2).
1011(2) is a binary number, where the base 2 denotes that the number is in the binary system.

8. The position of each digit of a binary number has a fixed place value
The position of each digit of a binary number has a fixed place value. Let’s take (2) as an example: 1 1 1 . 1
The value of a binary number can be expressed in the expanded form, for example, 1 2 3
(2)
1011.1 - × + =

9. Follow-up question
Express the following binary numbers in the expanded form.
Solution

10. Hexadecimal System
The hexadecimal system uses 16 numerals including 10 digits, which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 6 capital letters, which are A, B, C, D, E, F, to represent numbers, where A, B, C, D, E and F denote 10, 11, 12, 13, 14 and 15 respectively.
The concept of the hexadecimal system can be illustrated by grouping objects in the following way:

11. Take the following quantity of cubes as an example
Take the following quantity of cubes as an example. They are grouped as:
1 group of (163)
2 group of (162)
3 group of (161)
10 group of (160)
From the result of grouping, we can express the number of cubes in a hexadecimal number by writing the digits 1, 2, 3 and A as 123A(16), where the base 16 denotes that the number is in the hexadecimal system.

12. The position of each digit of a hexadecimal number has a fixed place value. Let’s take 137C.A(16) as an example: 1 3 7 C . A
The value of a hexadecimal number can be expressed in the expanded form, for example, 1 2 3
(16) 16
137C.A - × + = A C 7

13. Follow-up question
Express the following hexadecimal numbers in the expanded form.
Solution

14. Conversion between Binary Numbers and Denary Numbers
To convert a binary number into a denary number, we can express the binary number in the expanded form, and the value of the expression gives the required denary number.
For example,

15. To convert a denary number into a binary number, we divide the denary number successively by 2 until the quotient becomes zero. The remainders we get in this process form the required binary number.
For example,
(10) 25
(2)
11001 = 1
Write the digits from the bottom to the top.

16. Follow-up question 1. Convert 11110(2) into a denary number.
2. Convert 10(10) into a binary number.

17. Conversion between Hexadecimal Numbers and Denary Numbers
To convert a hexadecimal number into a denary number, we can express the hexadecimal number in the expanded form with base 16, and write the value of the expression as a denary number.
For example,

18. To convert a denary number into a hexadecimal number, we divide the denary number successively by 16 until the quotient becomes zero. The remainders we get in this process form the required hexadecimal number.
For example,
(10)
691
(16)
2B3 = 2 B 3
Write the digits from the bottom to the top.

19. Follow-up question 1. Convert DEF(16) into a denary number.
2. Convert 50(10) into a hexadecimal number.