Number systems
This presentation is the property of its rightful owner.
Sponsored Links
1 / 23

Number Systems PowerPoint PPT Presentation


  • 61 Views
  • Uploaded on
  • Presentation posted in: General

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!

Download Presentation

Number Systems

An Image/Link below is provided (as is) to download presentation

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

Number Systems

Computing Theory – F453


Data representation

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 representation1

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

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

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

  • 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

Exam Practice!

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


Markscheme answer

Markscheme Answer:

699


Octal

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

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

Hexadecimal

  • 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

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

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 decimal1

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

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

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

More Examples:

  • = -2

= 74

= 82


Converting negative denary into binary

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

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

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 / 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

Key Terms:

Character Set

ASCII

Unicode

  • Representation

  • Denary (Base 10)

  • Binary (Base 2)

  • Octal (Base 8)

  • Hexadecimal (Base 16)

  • Binary Coded Decimal

  • Sign & Magnitude

  • Two’s Compliment


  • Login