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Number Systems

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

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  1. Number Systems Computing Theory – F453

  2. 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)

  3. 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??!

  4. 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!

  5. 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

  6. 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

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

  8. Markscheme Answer: 699

  9. 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

  10. 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.

  11. 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

  12. 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

  13. 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….

  14. 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

  15. 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.

  16. 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

  17. More Examples: • = -2 = 74 = 82

  18. 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

  19. 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’

  20. 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

  21. 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!)

  22. 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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