Number Systems

1 / 23

# 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!

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Number Systems' - arwen

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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….
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

More Examples:
• = -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)