1. Computer Systems 1 Fundamentals of Computing
Data Representation Decimal, Binary, Hexadecimal

2. Data Representation Decimal (Base 10) Binary (Base 2)
Hexadecimal (Base 16)
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3. Decimal Base 10 Numbers we use in everyday life
10 values then increment to next digit position on left
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Each position represents a value of 10, to the positions power
Units (100), Tens (101), Hundreds (102), etc...
‘Ten fingers & ten toes’
Decimal numbers
E.g.- 5
64281
-3564
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4. Binary Base 2 Commonly used in computers to represent data
Used for all calculations and logic test results
2 values, then increment next digit position on left
0 or 1
Each position represents a value of 2 to the positions power
20 (1), 21(2), 22 (4), 23 (8), 24 (16), etc…
On or off (Digital)
Binary numbers
E.g.- 01
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5. Binary Bits ‘n’ Bytes A bit is a single binary digit (1 or 0)
A kilobit (KB) is 1024 bits
A byte is eight bits
Four bits are sometimes called a nibble
A kilobyte (Kb) is 1024 bytes
A megabyte (Mb) is 1024 kilobytes
A gigabyte (Gb) is 1024 megabytes
A terabyte (Tb) is 1024 gigabytes
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6. Padding, Cladding and Loft Insulation (?!?)
Often when writing binary values we will use a notation method called ‘padding’
This involves padding a value with zeros on the left up to the nearest byte size
Extra zeros become insignificant
Makes reading and working with values easier
The first 1 encountered on the left is most significant bit (MSB)
The last 1 on the furthest right is least significant bit (LSB)
Example
Binary number 1011
Padded to a byte =
Binary 010
Padded to a byte =
This could have originally been just ‘10’ too!
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7. Converting Decimal & Binary
To convert decimal to binary:
If the decimal number is ODD, write a ‘1’ on the far right hand side
OR if the decimal number is EVEN, write a ‘0’ on the far right hand side
Divide the decimal number by TWO
If there is a remainder then ignore it
If the number you have is ODD, write a ‘1’ to the left of the previous digit
OR if the number is EVEN, write a ‘0’ to the left of the previous digit
Repeat until your decimal number is 0
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8. Converting Decimal & Binary
Decimal ’15’ = ODD =
15/2 = 7.5 = 7 = ODD = 11
7/2 = 3.5 = 3 = ODD =
3/2 = 1.5 = 1 = ODD = 1111
Binary ‘1111’ = Decimal ’15’ !!!!
We can ‘pad’ our binary number with 0’s
Eg:
Decimal ’32’ = EVEN =
32/2 = 16 = EVEN =
16/2 = 8 = EVEN =
8/2 = 4 = EVEN =
4/2 = 2 = EVEN =
2/2 = 1 = ODD =
Binary ‘100000’ = Decimal ’32’ !!!!
Eg:
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9. Converting Binary & Decimal
To convert binary to decimal:
Write out the binary number
Add the positional values to each digit
Where there is a ‘1’ below the positional value highlight that positional value
Once all 1’s in the binary number have highlighted their positional values
Add all highlighted values together
Positional values:
These values can of course increase
Doubled with each positional move left
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10. Converting Binary & Decimal
E.g.-
Binary number:
Add positional values:
Highlight the positional values
Add the highlighted values together:
= 27
So Binary = Decimal 27 !!
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11. Hexadecimal Base 16 Another common format for data representation
Easy to map against binary
Used to easily represent large numbers
Common usage is memory locations
16 values then increment next digit on left position
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Each position represents a value of 16 to the positions power
160 (1), 161 (16), 162 (256), 163 (4096), 164 (65536), etc…
Hexadecimal numbers
E.g- 5 10 FF
FB 19
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12. Hexadecimal Values DECIMAL HEXADECIMAL
A B C D E F
DECIMAL
HEXADECIMAL
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13. Converting Decimal & Hexadecimal
To convert decimal to hexadecimal:
Write out the decimal number
Write out the hexadecimal positional values
Find the largest hex positional value that your decimal number will divide into
Divide your decimal number by that hex positional value
Take the answer( ignore the remainder) and add it to the left hand side of your hex answer multiply this number by it’s positional value and then subtract it from your decimal number
Take what’s left of your decimal number and repeat the process until you have nothing left
We can also pad out Hex values with zeros if we want too!
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14. Converting Decimal & Hexadecimal
E.g.-
Decimal 280
Write Hex positional values
Divide 280 by 256 (=1)
Multiply 1 by 256 (=256)
Subtract 256 from 280 ( =24)
Divide 24 by 16(=1)
Multiply 1 by 16 (=16)
Subtract 16 from 24 (24-16=8)
Divide 8 by 1 (=8)
Write in the 8
Hex number = 118
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15. Converting Hexadecimal & Decimal
To convert hexadecimal to decimal:
Write out the hexadecimal number
Add the positional values to each digit
Where there is a value greater than ‘0’ below the positional value highlight that positional value
Once all positional values have been highlighted, multiply the number below, by it’s positional value
Add all results of highlighted values together
Positional values:
These values can of course increase to the power of 16
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16. Converting Hexadecimal & Decimal
E.g.-
Hex number 4C
Add positional values C
Highlight positional values
Multiply positional values by the number below
4*16 = 64
C*1 = 12*1 = 12
Add these together
64+12=76
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17. Notation of Numerical Values
Very often we will have to write many different base value numbers
Performing calculations or comparisons
For easy reading, the following conventions are used:
Decimal
Will either appear with no special notation or with a subscript ‘10’ after the number
e.g or 36510
Binary
Using subscript ‘2’ after the number
e.g
Hexadecimal
Using a subscript ‘16’ after the number, of a prefix ‘0x’
e.g.- 4D16 or 0x4D
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18. CS1: Week 3 What you know now: Hexadecimal Decimal principles Binary
Method
Notation
Bits, bytes, etc.
Conversion
Hexadecimal
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