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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….

Markscheme Answer:

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

- 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

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)
- Hexadecimal (Base 16)
- Binary Coded Decimal
- Sign & Magnitude
- Two’s Compliment

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