1.1.2 | NUMBER SYSTEMS

Topics from the Cambridge IGCSE (9-1) Computer Science 0984 syllabus 2023 - 2025.

OBJECTIVES
1.1.2 (a) Understand the denary, binary and hexadecimal number systems
• Denary is a base 10 system
• Binary is a base 2 system
• Hexadecimal is a base 16 system
1.1.2(b) Convert between (i) positive denary and positive binary (ii) positive denary and positive hexadecimal (iii) positive hexadecimal and positive binary
• Values used will be integers only
• Conversions in both directions, e.g. denary to binary or binary to denary
​• Maximum binary number length of 16-bit

ALSO IN THIS TOPIC
1.1.1 NUMBER SYSTEMS
YOU ARE HERE | 1.1.2 NUMBER SYSTEMS
1.1.3 NUMBER SYSTEMS 
1.1.4 NUMBER SYSTEMS
1.1.5 NUMBER SYSTEMS
1.1.6 NUMBER SYSTEMS
1.2.1 TEXT, SOUND AND IMAGES
1.2.2 TEXT, SOUND AND IMAGES
1.2.3 TEXT, SOUND AND IMAGES
1.3.1 STORAGE AND COMPRESSION
​1.3.2 STORAGE AND COMPRESSION
​1.3.3 STORAGE AND COMPRESSION
​1.3.4 STORAGE AND COMPRESSION
TOPIC 1 KEY TERMINOLOGY
TOPIC 1 ANSWERS
TOPIC 1 TEACHER RESOURCES (CIE)

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DATA REPRESENTATION

1.1.2(A) NUMBER SYSTEMS

BINARY NUMBERS

The number system we all know has Units, Tens, Hundreds, Thousands and so on, we call this traditional number system the Denary number system. The binary number system is different, it has 1s, 2s, 4s, 8s, 16s, 32s and so on. The denery number system is the system you learn in maths and is a base 10 system, the Binary system however, is a base 2 system, this means in Binary that only one of two possible values can go in each column heading, a zero or a one. These ones and zeros are called BITs. 

Like in traditional math if you take the value from each column and multiply it by the column header you get the total value of the number. In the example above, to the left we have (1 x 32) + (1 x 8) + (1 x 4) = 44
So Binary 00101100 is equivalent to Denary 44
Note: If the right most (least significant) binary digit is a 1 then binary value must be representing an odd number.

Eight BITs make a Byte, and this is the most common representation you will see, however for this course you need to be able to work with up to 16 BITs and calculators are not allowed.​

With 16 bits we could uniquely represent 2^16 (65536) different values, which is close to enough to encode every letter, number and character of every language.

HEXADECIMAL NUMBERS

Computers use binary to represent all data and instructions, but binary is difficult for humans to read. Although the computer will always process data in a binary format Hexadecimal is used as an alternative representation.

Hexadecimal is a base 16 system, and each hexadecimal value can only be represented by a single character therefore, to represent all 16 values within the Hex system both number and letters are used as in the chart below.

The number system we all know (denary) has Units, Tens, Hundreds, Thousands and so on. The Hexadecimal number system has 1s, 16s, 256s, 4096s and so on. 

Like in traditional math if you take the value from each column and multiply it by the column header you get the total value of the number. Here we have the HEX value 1 0 4 1 so (1 x 4096 = 4096, 0 x 256 = 0, 4 x 16 = 64, 1 x 1 = 1). 4096 + 0 + 64 + 1 = 4161
So Hex 1041 is equivalent to Denary 4161
The chart below shows the denery value of 255 represented by Binary, Denary and HEX.

1.1.2(B) NUMBER SYSTEMS

CONVERTING BETWEEN NUMBER SYSTEMS - DENARY TO BINARY

Converting binary to denary can also be done using the column headings method, which is a simpler method for converting binary numbers with fewer digits. To use this method, simply write the binary number under the column headings that represent the powers of 2, starting from the rightmost column. Then, add up the values in each column where there is a "1" and ignore the values in columns where there is a "0". For example, to convert the binary number 1011 to denary using column headings, you would write the number under the columns for 2^3, 2^2, 2^1, and 2^0 as follows:

8 4 2 1
1 0 1 1

Then, add up the values in the columns where there is a "1" to get the denary equivalent:

8 + 0 + 2 + 1 = 11

So the denary equivalent of the binary number 1011 using the column headings method is also 11. This method is quick and easy to use for binary numbers with fewer digits and is a useful alternative to the standard multiplication method

Converting denary (decimal) to binary is also a straightforward process. To convert a denary number to binary, you need to repeatedly divide the number by 2 and record the remainder until the result is 0. Then, the binary number is formed by writing down the remainders in reverse order. For example, to convert the denary number 27 to binary, you would start by dividing 27 by 2, which gives you a quotient of 13 with a remainder of 1. You would then divide 13 by 2, which gives you a quotient of 6 with a remainder of 1, and so on, until you reach a quotient of 0.

27 ÷ 2 = 13 R1
13 ÷ 2 = 6 R1
6 ÷ 2 = 3 R0
3 ÷ 2 = 1 R1
1 ÷ 2 = 0 R1

The remainders are 1, 1, 0, 1, and 1, so the binary equivalent of the denary number 27 is 11011. By following this process, any denary number can be converted to binary, allowing it to be represented and processed using binary notation in digital devices and computer systems.

CONVERTING BETWEEN NUMBER SYSTEMS - DENARY TO HEXADECIMAL

Converting denary (decimal) to hexadecimal (HEX) is another common task in digital systems. The HEX system uses a base of 16, allowing a single digit to represent numbers up to 15. To convert a denary number to HEX, you need to divide the number by 16 and record the remainder until the result is less than 16. Then, the HEX number is formed by writing down the remainders in reverse order, with each remainder represented by a single HEX digit. For example, to convert the denary number 428 to HEX, you would start by dividing 428 by 16, which gives you a quotient of 26 with a remainder of 12. You would then divide 26 by 16, which gives you a quotient of 1 with a remainder of 10 (which is represented by the letter A in HEX notation), and so on, until you reach a quotient of less than 16.

428 ÷ 16 = 26 R12 (C)
26 ÷ 16 = 1 R10 (A)
1 ÷ 16 = 0 R1
The remainders are 12, 10, and 1, which are represented by the HEX digits C, A, and 1, respectively.

Lastly, you need to remember to write the number from bottom to top, so the HEX equivalent of the denary number 428 is 1AC.

By following this process, any denary number can be converted to HEX, allowing it to be represented and processed using HEX notation in digital systems.

CONVERTING BETWEEN NUMBER SYSTEMS - HEXADECIMAL TO DENARY

Converting hexadecimal (HEX) to denary (decimal) is a relatively simple process. Each digit in a HEX number represents a power of 16, with the rightmost digit representing 16^0, the next digit representing 16^1, the next representing 16^2, and so on. To convert a HEX number to denary, you need to multiply each digit by its corresponding power of 16 and then add up the results. For example, to convert the HEX number 3F2 to denary, you would write the number under the columns for 16^2, 16^1, and 16^0 as follows:

3 | F | 2
256 | 16 | 1

Then, multiply each digit by its corresponding power of 16 and add up the results:

3 x 256 (768) + 15 x 16 (240) + 2 x 1 (2) = 1010

So the denary equivalent of the HEX number 3F2 is 1010. By following this process, any HEX number can be easily converted to denary, allowing it to be more easily understood and used in other applications.

CONVERTING BETWEEN NUMBER SYSTEMS - HEXADECIMAL TO BINARY

​To convert Binary to Hex and Hex to Binary you can simply split the Binary representation in to 4 bits then use each of the 4 bits to represent the HEX value. The table below shows 254 in binary 11111110 converted to HEX

As can be seen in the table above each HEX digit can be represented by a unique combination of four binary digits, allowing a simple conversion between the two numbering systems. Here is another example, to convert the HEX number 3F2 to binary, you would convert each HEX digit to its 4-bit binary equivalent:

3 -> 0011
F -> 1111
2 -> 0010

So the binary equivalent of the HEX number 3F2 is 001111110010. By following this process, any HEX number can be easily converted to binary, allowing it to be represented and processed using binary notation in digital systems and computer programming.

1: What is the binary equivalent of the denary number 13?
a) 1100
b) 1001
c) 1110
d) 1101

2: What is the denary equivalent of the binary number 101011?
a) 41
b) 43
c) 45
d) 47

3: What is the HEX equivalent of the denary number 1024?
a) 400
b) 3FC
c) 4000
d) 408

4: What is the denary equivalent of the HEX number AB?
a) 165
b) 170
c) 175
d) 180

5: What is the binary equivalent of the HEX number FF?
a) 11011101
b) 11111111
c) 10101010
d) 10011001

6: What is the HEX equivalent of the binary number 101101?
a) 1B
b) 1D
c) 1F
d) 21

7: What is the denary equivalent of the binary number 11100001?
a) 225
b) 229
c) 233
d) 237

8: What is the binary equivalent of the denary number 127?
a) 01111111
b) 10000001
c) 10111111
d) 11111110

ALSO IN THIS TOPIC

1.1.1 NUMBER SYSTEMS | Why Computers use Binary
1.1.2 NUMBER SYSTEMS | Binary, Hex and converting between number systems
1.1.3 NUMBER SYSTEMS | Benefits of Hex
1.1.4 NUMBER SYSTEMS | Binary addition and Overflow
1.1.5 NUMBER SYSTEMS | Binary Shifts
1.1.6 NUMBER SYSTEMS | Negative Number representation, Two's Complement
1.2.1 TEXT, SOUND AND IMAGES | How Computer represent Text
1.2.2 TEXT, SOUND AND IMAGES | How Computers represent Sound
1.2.3 TEXT, SOUND AND IMAGES | How Computers represent Images
1.3.1 STORAGE AND COMPRESSION | How data storage is measured
​1.3.2 STORAGE AND COMPRESSION | Calculating image and sound file size
​1.3.3 STORAGE AND COMPRESSION | Purpose of Data Compression
​1.3.4 STORAGE AND COMPRESSION | Lossy and Lossless, how files are compressed
TOPIC 1 KEY TERMINOLOGY
TOPIC 1 ANSWERS
TOPIC 1 TEACHER RESOURCES (CIE)