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# Number Systems PowerPoint PPT Presentation

Number Systems. Computing Theory – F453. Data Representation. Data in a computer needs to be represented in a format the computer understands. This does not necessarily mean that this format is easy for us to understand. Not easy, but not impossible!

Number Systems

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### Number Systems

Computing Theory – F453

### Data Representation

• Data in a computer needs to be represented in a format the computer understands.

• This does not necessarily mean that this format is easy for us to understand.

• Not easy, but not impossible!

• A computer only understand the concept of ON and OFF.

• Why?

• How do we translate this into something WE understand?

• We use a numeric representation (1s and 0s)

### Data Representation

• If a computer can only understand ON and OFF, which is represented by 1 and 0, then which is which?

• 1 = ON

• 0 = OFF

• This is known as the Binary system.

• Because there are only 2 digits involved, it is known as Base 2.

• But what does it MEAN??!

### Denary Numbers

• We use the Denary Number System.

• This is in Base 10, because there are 10 single digits in our number system.

• Why? We are surrounded by things that are divisible by ten.

• Counting in tens is not a new phenomenon…

• Even the Egyptians did it!

### Are there other notations?

• Yes!

• The Mayans used the Vigesimal system (Base 20)

• Nigerians use the Duodecimal system (Base 12)

• The Babylonians used Base 60!!

• So…. ‘Base’ means the number of digits available in the system.

• Denary has 10 individual digits:

• 0 1 2 3 4 5 6 7 8 9

• to make the next number we have to combine two digits: 10

### Binary

• Binary can be easily calculated from denary by using the following steps:

• Let’s try the number 42.

Is 42 > 128?

Is 42 > 64?

Is 42 > 32?

32-42 = 10

Is 10 > 16?

Is 10 > 8?

1

0

1

1

0

0

0

0

10-8 = 2

Is 2 > 4?

Is 2 > 2?

2-2 = 0

### Exam Practice!

• Past paper questions are one of the best ways to help you remember the answers when it matters….

699

### Octal

• The Octal system uses Base 8 (denary is Base 10)

• Which means that Octal numbers are arranged ascending in powers of 8

• So how do I get from Denary to Octal??!

• USE BINARY!

• 00101010 is our binary number

• 000 101 010 is split into three (from the right, adding a leading 0)

• 0 5 2 is the Octal

• 0x82+5x81+2x80= 0 + 40 + 2 = 42

• The 2, 1 & 0 powers relate to the base octal figures in the table above

### Relating Octal to Binary

• From the example shown:

• Octal numbers are creating by taking the sets of three bits from the right of a binary number and creating a leading zero

• This is then converted into it’s Octal format

• This can be converted from Octal to Denary by multiplying each digit by 8 to the power of the Octal unit and adding the total value together.

• Hexadecimal is very similar to Octal in that it has a differing base equivalent to denary, you just need the table!

• Hex, is counted in Base 16, which means it has 16 possible digits before you get into double figures.

• Where in Octal, you needed 3 binary bits to create all possible 8 digits, in Hex we need 4 binary bits.

• This is half a binary byte – also known as a nibble

### Try This:

• Now trying to convert this into Hexadecimal….

Convert 252 in denary into Binary:

4

2

1

8

1

1

1

1

1

1

0

0

1111 = F

1100 = C

252 – 128 = 124

124 – 64 = 60

First split the byte into two nibbles

60 – 32 = 28

Now, convert each of the nibbles into their Hexadecimal equivalent

28 – 16 = 12

So, 252 in denary, must be FC in Hexadecimal!

12 – 8 = 4

4 – 4 = 0

### Binary Coded Decimal

• We have already seen this in action through converting Denary into Octal and Hexadecimal.

• Binary Coded Decimal (BCD) is the binary equivalent of the decimal digits we use.

• Think back to your first years in school where you were taught to count in units, tens, and hundreds. This was teaching you the basis of the denary system.

• BCD takes each of these denary digits and changes them

• Into their binary equivalent….

### Binary Coded Decimal

• Use the table below to write out the denary number 3142 in binary coded decimal.

• BCD can be used to represent large denary numbers.

• Specifically those larger then 255. Why?

0001

0100

0011

0010

3

1

2

4

### But Some Numbers Are Negative!

• Numbers can be represented in a format known as Two’s Complement.

• Think of a car milometer – It starts at 000000.

• Move forward, and it becomes 000001

• But what if it moves back? There is no -1?!

• Instead the clock goes to 999999 which represents -1 mile.

• So, in Two’s complement, we look at the leading number which represents the sign : 1 being negative and 0 being positive.

### Two’s complement

• A few examples:

• 11111101 = -3 why? How?

-128

+64

+32

+16

+8

+4

+1 .

-3 .

If this is a 1, we change 128 to -128

• = -2

= 74

= 82

### Converting Negative Denary into Binary….

• 1. Find the binary value of the equivalent decimal number first

• 2. Change all the 0’s to 1’s and vice versa

• 3. Add 1 to the result.

• OR

• 1. Starting from the right, leave all the digits alone up to and including the first ‘1’

• 2. Now, change all the other digits from 0 to 1 or 1 to 0

### Examples!

= 74

= -74

-128

+32

+16

+4

+2

= - 74!

2. Now, change all the other digits from 0 to 1 or 1 to 0

1. Starting from the right, leave all the digits alone up to and including the first ‘1’

### Enough With The Maths!

• Ok. So how does a computer recognise the alphabet?

• If a computer can recognise numbers through binary, then we can assign a numeric value to a letter which will allow the computer to recognise it!

• A character set is a table of Alphabetical (alpha) characters showing their numeric equivalent.

• There are two major character sets:

• ASCII – American Standard Code for Information Interchange

• UNICODE – Universal Codes

### ASCII / UNICODE Character Sets

• ASCII is pronounced ‘as-ski’

• It is based on the Latin alphabet, that is, it was originally developed to use the English Language.

• Historically ASCII was developed for telegraphic printers (think REALLY old fax machines)

• UNICODE was developed later on to encompass the many different alphabets used across the globe (and now includes the dreaded Wingdings…. And even Klingon!)

### Key Terms:

Character Set

ASCII

Unicode

• Representation

• Denary (Base 10)

• Binary (Base 2)

• Octal (Base 8)