Number Systems and Binary Notes
Number System & Computers & Binary
Computers use binary because data is processed using logic gates with two states.
The binary number system has two digits (1/0), representing different states (1 = on, 0 = off).
All data must be converted to binary for a computer to understand and process it.
Converting to binary allows computers to process data at incredible speeds, perform complex calculations, and efficiently store vast amounts of data.
Secondary storage uses binary; magnetic hard drives use North and South polarity to represent 1 or 0.
Optical disks interpret light hitting a flat area (land) as 1 and light hitting a bump (pit) as 0.
Worked Example
Computers process data using logic gates that have only two states (1/0).
Number Systems: Denary, Binary & Hexadecimal
Denary
Denary is a base-10 number system made up of 10 digits (0-9).
Humans use the denary system for counting, measuring, and performing math calculations.
Combinations of the 10 digits can represent any number.
Example:
Binary
Binary is a base-2 number system made up of two digits (1 and 0).
Each digit has a weight factor of raised to a power.
Each time a new digit is added, the column value is multiplied by .
Combinations of the 2 digits can represent any number.
Example: Binary 1100 =
Hexadecimal
Hexadecimal is a base-16 number system made up of 16 digits: 10 numbers (0-9) and 6 letters (A-F).
Each digit has a weight factor of raised to a power.
One hexadecimal digit can represent four bits of binary data.
Converting Between Binary & Denary
Denary to Binary Conversion
Write out a binary number line.
Find the first column heading with a value larger than the denary value you are converting.
Write down each column heading to the right (not including the largest heading) until you reach 1.
Method 2
Example:
142 divided by 2 = 71, remainder 0
71 divided by 2 = 35, remainder 1
35 divided by 2 = 17, remainder 1
17 divided by 2 = 8, remainder 1
8 divided by 2 = 4, remainder 0
4 divided by 2 = 2, remainder 0
2 divided by 2 = 1, remainder 0
1 divided by 2 = 0, remainder 1
Read the remainders from bottom to top to get the binary number: 10001110.
Examiner Tips and Tricks
Read the question carefully to check for how many bits are expected in your answer.
If the question requires an 8-bit answer and a 6 bit answer has been given, add 2 extra 0s to the start of the answer.
Denary 45 is 101101 in 6-bit binary and 00101101 in 8-bit binary.
Binary to Denary Conversion
Count how many bits make up the value.
Write out the column headings for the number of bits given from right to left.
Add together any column heading with a value of 1 in the column.
Binary to hexadecimal and hexadecimal to binary
Example 1
101111100001
First split this up into groups of 4 bits:
1011 1110 0001
Then, find the equivalent hexadecimal digits:
B E 1
Example 2
10000111111101
First split this up into groups of 4 bits:
10 0001 1111 1101
The left group only contains 2 bits, so add in two 0s:
0010 0001 1111 1101
Now find the equivalent hexadecimal digits:
2 1 F D
Hexadecimal to denary and Denary to hexadecimal
Example 1
Convert the hexadecimal number, 45A, into denary.
Multiply each hex digit by its heading value:
256 16 1
4 5 A
(4 x 256 = 1024) (5 x 16 = 80) (10 x 1 = 10) (NOTE: A = 10)
Then add the three totals together (1024 + 80 + 10) to give the denary number:
1114
Example 2
Convert the hexadecimal number, C8F, into denary.
Multiply each hex digit by its heading value:
256 16 1
C 8 F
(12 x 256 = 3072) (8 x 16 = 128) (15 x 1 = 15) (NOTE: C=12, F = 15)
Then add the three totals together (3072 + 128 + 15) to give the denary number:
3215
Uses of Hexadecimal
Hexadecimal is often preferred when working with large values in Computer Science.
It takes fewer digits to represent a given value in hexadecimal than in binary.
1 hexadecimal digit corresponds 4 bits (one nibble) and can represent 16 unique values (0-F).
It is beneficial to use hexadecimal over binary because:
The more bits there are in a binary number, the harder it makes for a human to read
Numbers with more bits are more prone to errors when being copied.
Uses of Hexadecimal System:
Error codes are often shown as hexadecimal values; numbers refer to the memory location of the error.
A MAC address is usually made up of 48 bits, shown as 6 groups of two hexadecimal digits.
An IPv6 address is a 128-bit number broken down into 16-bit chunks, represented by a hexadecimal number.
A typical hexadecimal color code consists of 6 hexadecimal digits, equivalent to 24 digits in binary.
URLs can only contain standard characters (a-z and A-Z), numbers (0-9), and some special symbols; other characters are converted into a hexadecimal code prefixed with a % sign.
Addition of binary numbers
binary digit | carry | sum |
|---|---|---|
0+0+0 | 0 | 0 |
0+0+1 | 0 | 1 |
0+1+0 | 0 | 1 |
0+1+1 | 1 | 0 |
1+0+0 | 0 | 1 |
1+0+1 | 1 | 0 |
1+1+0 | 1 | 0 |
1+1+1 | 1 | 1 |
Example 1
Add 00100111 and 01001010
Carry values are 111
Sum values are 01110001
Answer: 01110001
Column 1: 1+0= 1 no carry
Column 2: 1+1 = 0 carry 1
Column 3: 1+0+1 = 0 carry 1
Column 4: 0 + 1 + 1 = 0 carry 1
Column 5: 0 + 0 + 1 = 1 no carry
Column 6: 1+0 = 1 no carry
Column 7: 0+1 = 1 no carry
Column 8: 0 +0 = 0 no carry
Overflow & Binary Addition
What is an overflow error?
An overflow error occurs when the result of a binary addition exceeds the available bits
Example 3
*Add 01101110 and 11011110 (using 8 bits)
*Carry: 111111
*Sum: 1 01001100