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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!

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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….
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