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CIE IGCSE Computer Science Your notes
Number Systems
Contents
Computers & Binary
Number Systems
Converting Between Binary & Denary
Converting Between Hexadecimal & Binary
Converting Between Hexadecimal & Denary
Hexadecimal
Binary Addition
Binary Shifts
Two's Complement
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Computers & Binary
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Why Do Computers Use Binary?
All data needs to be converted to binary to be processed by a computer
Binary is a numbering system that uses only 0s (off) and 1s (on)
A computer uses binary data for all of its operations
The 0’s and 1’s are called bits. “Bits” is short for binary digits
A bit is the smallest unit of data a computer can use (a single 0 or 1)
Bits can be grouped to form larger units of data, such as bytes, kilobytes, etc.
By using binary, computers can process and store data using electronic transistors that can be either
on or off
Millions or billions of transistors fit onto a microchip
Any form of data, including text, images, and sound, needs to be converted to binary before it can be
processed by a computer
This conversion process involves assigning a binary code to each character, pixel, or sample in the data
The resulting binary code can then be processed using logic gates and stored in registers
Logic Gates
Logic gates are electronic devices that perform logical operations on binary data
Logic gates are used to process binary data by applying Boolean logic to the input values and
producing a binary output
Registers are temporary storage areas in a computer's CPU (central processing unit) that hold binary
data during processing
Registers are used to store data that needs to be accessed quickly, such as variables in a program or
data being manipulated by logic gates
The size of a register determines the maximum amount of binary data that can be stored in it at one time
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Number Systems
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The Denary, Binary & Hexadecimal Number Systems
In Computer Science there are 3 numbering systems used to represent data:
Denary
Binary
Hexadecimal
The Denary Number System
The denary number system, also known as the decimal system, is a base-10 numbering system that
uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
Denary numbers can be used to represent any quantity, including whole numbers, fractions, and
decimals
Each digit in a denary number represents a power of 10, with the rightmost digit representing 100, the
next representing 101, and so on
It is necessary to convert data from one number system to another eg. denary to binary or denary to
hexadecimal
It is much easier to do conversions with a table:
The number 3268 (three thousand two hundred and sixty-eight) can be represented in the
following table:
(3 x 1000) + (2 x 100) + (6 x 10) + (8 x 1) = 3268
The Binary Number System
The binary number system is a base-2 numbering system that uses only two digits: 0 and 1
Each digit in a binary number represents a power of 2, with the rightmost digit representing 20, the next
representing 21, and so on.
Eg: the number 12 represented in binary is 1100:
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We know this as (1 x 8) + (1 x 4) + (0 x 2) + (0 x 1) =12
The Hexadecimal Number System:
The hexadecimal number system is a base-16 numbering system that uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, A, B, C, D, E, and F where 10 is represented by A, 11 by B and so on up to 15 represented by F
Each digit in a hexadecimal number represents a power of 16, with the rightmost digit representing 160,
the next representing 161, and so on
E.g. the number 146 represented in hexadecimal is 92
(0 x 4096) + (0 x 256) + (9 x 16) + (2 x 1) = 146
Exam Tip
You need to be able to make conversions in both directions, e.g. denary to binary or binary to
denary. Make sure you write the table the correct way round (with 1 in the right hand column) & use
the correct number of bits in your answer (the question should tell you but use multiples of 4 bits if
you’re not sure)
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Converting Between Binary & Denary
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Converting Between Binary & Denary
Converting Denary to Binary Walkthrough:
Write down the powers of 2 in binary from right to left, starting with 20 (1), 21 (2), 22 (4), 23 (8), and so on,
until you reach 128 (as answers must be given in 8 bits)
128 64 32 16 8 4 2 1
0 0 1 0 1 0 1 1
Starting from the leftmost column, write a 1 if the corresponding power of 2 is less than or equal to the
number you’re converting, 171 in this example, otherwise write a 0.
128 64 32 16 8 4 2 1
1 0 1 0 1 0 1 1
Check your working by adding together all column headings with a 1 underneath (128+32+8+2+1=171)
Read the binary digits from left to right to get the binary equivalent of 171.
10101011
Converting Binary to Denary Walkthrough:
Write down the powers of 2 in decimal from right to left, starting with 20 (1), 21 (2), 22 (4), 23 (8), and so
on, until you reach 128 (as answers must be given in 8 bits)
128 64 32 16 8 4 2 1
Starting from the leftmost column, write the binary digit in the column if it is a 1 write 1, and write 0 if it is
a 0.
128 64 32 16 8 4 2 1
0 1 1 0 0 1 1 0
Add up the values in each column where the binary digit is 1.
64 + 32 + 4 + 2 = 102
Exam Tip
If the question tells you to show your working you will get a mark for this so don't forget to write
down yours
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Converting Between Hexadecimal & Binary
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Converting Between Hexadecimal & Binary
Converting Binary to Hexadecimal Walkthrough:
Group the binary digits into groups of 4, starting from the rightmost digit. If there are not enough digits
to make a group of 4, add leading zeros as needed.
0110 1110
Add column headings to work out the value of each nibble
8 4 2 1 8 4 2 1
0 1 1 0 1 1 1 0
4+2 = 6 8+4+2=14 = E in hex
Replace each group of 4 bits with its corresponding hexadecimal value:
0110 1110
6 E
Write down the resulting hexadecimal values to get the final answer:
6E
Converting Hexadecimal to Binary Walkthrough:
To convert a Hex number like A2 into binary write each hex digit in 4-bit binary
A=10
8 4 2 1 8 4 2 1
1 0 1 0 0 0 1 0
The binary representation is the numbers altogether: 10100010
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Converting Between Hexadecimal & Denary
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Converting Between Hexadecimal & Denary
Converting Denary to Hexadecimal Walkthrough:
1. Divide the decimal number (in this example 57) by 16 and write down the answer including the
remainder:
57 ÷ 16 = 3 remainder 9
2. If the remainder is above 9, replace this with the corresponding letter
3. Repeat steps 1 and 2 until the number you’re dividing is zero:
3 ÷ 16 = 0 remainder 3
4. Write the hexadecimal values from step 3 to step 1 in reverse order:
39
Alternatively, you can turn your denary number into binary, and then turn the binary number into
hexadecimal:
1. Work out 57 in binary
128 64 32 16 8 4 2 1
0 0 1 1 1 0 0 1
2. Split it into 2 nibbles
8 4 2 1 8 4 2 1
0 0 1 1 1 0 0 1
3. Turn each nibble into its hex value
2+1=3 8+1=9
Answer is 39
Converting Hexadecimal to Denary Walkthrough:
Write down the place value of each digit in the number, starting from the right and increasing by a
power of 16:
161 160
16 1
If the hex digit is a letter, convert it to its denary equivalent (Using the following table to help you):
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Hexadecimal Decimal
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A 10
B 11
C 12
D 13
E 14
F 15
The hexadecimal value of the leftmost digit E has a decimal value of 14. The hexadecimal value of the
rightmost digit is 5, which has a decimal value of 5
16 1
E 5
Multiply each decimal value by its corresponding place value, and sum the products:
(14 x 16) + (5 x 1) = 224 + 5 = 229
Therefore, the denary equivalent of E5 is 229.
Alternatively, you can turn your hexadecimal number into binary, and then turn the binary number
into denary:
1. Write each hex digit in binary
8 4 2 1 8 4 2 1
1 1 1 0 0 1 0 1
2. Convert the binary to denary
128 64 32 16 8 4 2 1
1 1 1 0 0 1 0 1
128+64+32+4+1=229
Exam Tip
When doing conversions don’t remove any 0s on the right hand side of your answer as this will cost
you marks. E.g. B0 isn’t the same as B, just like 30 isn’t the same as 3.
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Hexadecimal
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Hexadecimal
Why Use Hexadecimal?
Hexadecimal numbers are often used as a beneficial method of data representation in computer
science
It takes fewer digits to represent a given value in hexadecimal than in binary
One hexadecimal digit can represent four bits of binary data
It is beneficial to use hexadecimal over binary because:
The more bits there are in a binary number, the harder it is to read
Numbers with more bits are more prone to errors when being copied
Worked example
The following binary number has 24 bits: 101110110011101011101010. Write the number in hexadecimal
and explain why it is better to write the number in hexadecimal rather than in binary.
This number can be represented with only 6 digits in hexadecimal:
B B 3 A E A Mark for each character [6]
This number is:
Much shorter and a more efficient way of representing numbers [1]
Easier to spot any errors in [1]
How is Hexadecimal used in Computers?
Using hexadecimal allows computer scientists to represent large amounts of binary data in a more
manageable and readable format
There are many uses of hexadecimal in computers:
Memory addresses and other hardware related values
Representation of colour values
Error messages
Memory dump
Debugging
IP addresses
ASCII / Unicode
Assembly language
URLs
Representation of colour values in HTML/CSS and in computer graphics: eg. RGB (red-green-blue)
colour values can be represented in a compact and readable format. E.g. #FF0000
Error messages & debugging: eg. 0x80070643 to give technicians/developers information about
what the error is in relation to
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Memory dump: this is a record of what happened in the computer's memory at the time of an error.
These are often difficult to read because they contain a lot of technical information, but computers can
use hexadecimal values in memory dumps to identify specific memory locations and values. Your notes
Eg. A video game on a computer crashes and a memory dump is created. A technician analyses the
memory dump and looks for hexadecimal values to identify the cause of the crash. They might look
for specific memory addresses and values that relate to the game being played or the graphics
card in the computer.
IP addresses:
Hexadecimal values can be used to represent each set of numbers in an IP address
Each set of numbers can be converted into a two-digit hexadecimal value, giving a total of eight
digits in the IP address
Eg. the IP address "192.168.0.1" could be represented in hexadecimal as "C0A80001"
The hexadecimal values can make IP addresses easier to work with in some cases, such as when
configuring network devices or writing scripts that interact with IP addresses, however, they are
not commonly used in everyday applications or browsing the internet
ASCII / Unicode: eg. The letter "A" is represented in ASCII as the decimal value 65, which is equivalent
to the binary value 01000001. This binary value can also be represented in hexadecimal as the value 41
Assembly language:
Hexadecimal values are used in assembly language to represent the binary code for the
instructions and data
Each instruction or data value is represented by a specific sequence of hexadecimal digits, which
can make it easier for programmers to read and understand the code
Eg. "MOV AX, 5" instruction is represented by the hexadecimal value "B805", which tells the
computer to move the value 5 into register AX
URLs:
To encode a space in a URL using hexadecimal, the space character is replaced by a percent sign
("%") followed by the hexadecimal value of the space character. In ASCII, the space character has
a decimal value of 32, which can be represented in hexadecimal as "20".
So, to encode the URL "my website.com/page one" using hexadecimal, it would look like this:
my%20website.com/page%20one
Exam Tip
When a question is asked which asks you to name a certain number of uses of hexadecimal, ensure
you write the number asked for and no more. E.g. name 3 uses of hexadecimal - if you write more
than 3, the last ones will be ignored by the examiner, even if they’re correct
If you're asked to name any uses of hexadecimal, don't write down any given in the question as you
won't get marks for this
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Binary Addition
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Binary Addition
Adding binary numbers follows a similar process to adding denary numbers
The binary adding rules are:
0+0=0
0+1=1
1+1=10 (The 1 is carried into the next column on the left)
1+1+1=11 (The 1 is carried into the next column on the left)
Adding binary steps:
Step 1: Start by writing the two binary numbers you want to add underneath each other, with the
least significant bit (LSB) on the right.
Step 2: Begin by adding the LSBs together. If the sum is less than or equal to 1, write it down in the
sum column. If the sum is 2 or greater, write the remainder of the sum (i.e., the sum minus 2) in
the sum column and carry over the quotient (i.e., 1) to the next column
Step 3: Repeat this process for the next column to the left, adding the two bits and any carryover
from the previous column. Again, if the sum is less than or equal to 1, write it down in the sum
column; if the sum is 2 or greater, write the remainder of the sum in the sum column and carry
over the quotient to the next column.
Step 4: Continue this process for each subsequent column until you have added all the bits.
Step 5: If the sum of the last two bits produces a carryover, add an additional bit to the left of the
sum to represent the carryover.
Step 6: Check the sum to make sure it fits within 8 bits. If it doesn't, you will need to use more bits to
represent the sum.
Adding binary walkthrough:
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Your notes
In this example, we start by adding the two LSBs: 0 + 0 = 0, which we write down in the sum column
We then move to the next column to the left and add the two bits and the carryover from the previous
column: 1 + 1 + 0 = 10
We write down the remainder of the sum (i.e., 0) in the sum column and carry over the quotient (i.e., 1) to
the next column
We repeat this process for the next two columns, and end up with the sum 101110000
Overflow
An overflow error occurs when the result of a binary addition exceeds the maximum value that can be
represented. In the case of 8-bits, the maximum value is 255
Overflow occurs when the addition of two numbers results in a carry bit that cannot be
accommodated
To avoid overflow errors, it's important to check the result of binary addition to ensure that it doesn't
exceed the maximum value that can be represented
Overflow errors can also occur in other operations besides addition, such as multiplication or division
Exam Tip
You can convert your binary numbers to denary, then perform the calculation and then convert
them back to check you’ve got the right answer. Label this as checking to make sure that the
examiner knows this is a check and not part of your working out
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Binary Shifts
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Binary Shifts
A binary shift is the term used for multiplying or dividing in binary
A binary shift moves all the bits in a binary number a certain number of positions to the left or right
When performing a logical binary shift to the left, the bits shifted from the end of the register are lost,
and zeros are shifted in at the opposite end
When performing a logical binary shift to the right, the bits shifted from the beginning of the register
are lost, and zeros are shifted in at the opposite end
When performing a logical binary shift, the positive binary integer is multiplied or divided according to
the shift performed. A shift to the left is equivalent to multiplication by a power of 2, while a shift to
the right is equivalent to division by a power of 2
When performing a logical binary shift, the most significant bit(s) or least significant bit(s) are lost,
depending on the direction of the shift
Binary shift walkthrough:
E.g. shifting the binary number 11001100 to the left by two positions gives 00110000. The two bits
shifted from the end are lost, and two zeros are shifted in at the opposite end
E.g. shifting the binary number 11001100 to the right by two positions gives 00110011. The two bits
shifted from the beginning are lost, and two zeros are shifted in at the opposite end
E.g. shifting 11001100 to the left by two positions, is multiplying it by 22, or 4. The result is 00110000,
which is equal to 4 times the original value
Shifting the binary number 11001100 to the right by two positions, is dividing it by 22, or 4. The result is
00110011, which is equal to the original value divided by 4
Exam Tip
Make sure you’ve got the same number of bits in your answer as there were in the question. Check
your answer by converting the binary number to denary, working out your answer and converting it
back again. Make sure to label this checking so the examiner knows it isn’t part of your working out
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Two's Complement
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Two's Complement
Two's complement is a method of representing signed integers in binary form, where the leftmost bit
represents the sign (0 for positive and 1 for negative)
To convert a positive number to a two's complement 8-bit integer, first represent the number in binary
form with leading zeros until it is 8 bits long. If the number is positive, the leftmost bit should be 0
To convert a negative number to a two's complement 8-bit integer, first invert all the bits in the binary
representation of the positive equivalent of the number (i.e., flip all the 1's to 0's and all the 0's to 1's).
Then, add 1 to the result to obtain the two's complement representation
Two's complement walkthrough:
The binary number 00101011 represents the positive integer 43 in 8-bit binary form
E.g. to represent the negative integer -43 in two's complement 8-bit form, start by representing the
positive equivalent of 43 in binary form: 00101011. Then invert all the bits to get 11010100, and add 1 to
get 11010101
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