1. Chapter 4: Representation of data in computer systems: Number OCR Computing for GCSE © Hodder Education 2011

2. Denary Numbers can be expressed in many different ways. We usually use decimal or denary. Denary numbers are based on the number 10. We use ten digits: 0,1,2,3,4,5,6,7,8,9. When we put the digits together, each column is worth ten times the one to its right. OCR Computing for GCSE © Hodder Education 2011

3. Denary So, the denary number 1234 is OCR Computing for GCSE © Hodder Education 2011 Place value 1000100101 Digit1234 Place valueDigitValue 100011 × 1000 =1000 1002+ 2 × 100 =200 103+ 3 × 10 =30 14+ 4 × 1 =4 TotalΣ =1234

4. Binary to denary It is simpler to make machines that only need to distinguish two states, not ten. That is why computers use binary numbers. OCR Computing for GCSE © Hodder Education 2011 1286432168421 Each column is worth twice the column to it right. 1286432168421 00000010 Add up the columns that have a 1 on them. In this case it is 2. 1286432168421 10001111 In this case it is 128 + 8 + 4 + 2 + 1 = 143.

5. Denary to binary One technique is to take the denary number and repeatedly divide by 2. Write down the result and the remainder. For example, find the denary number 147: Read from the bottom up: 147 in binary is 10010011. OCR Computing for GCSE © Hodder Education 2011 ResultRemainder 147 ÷ 2 =731 73 ÷ 2 =361 36 ÷ 2 =180 18 ÷ 2 =90 9 ÷ 2 =41 4 ÷ 2 =20 2 ÷ 2 =10 1 ÷ 2 =01

6. Binary addition The rules for binary addition: 0 + 0 = 0 0 + 1 = 1 1 + 1 = 0 carry 1 1 + 1 + 1 = 1 carry 1 Add the binary equivalents of denary 4 + 5 (we know this equals 9). OCR Computing for GCSE © Hodder Education 2011 DenaryBinary 40100 50101 91001 Carry1

7. Binary addition Sometimes we run into problems. Suppose we have eight bits in each location. Add the binary equivalent of denary 150 + 145. We know this equals 295. No room for a carry so it is lost and we get the wrong answer. When there isn’t enough room for a result, this is called overflow and produces an overflow error. OCR Computing for GCSE © Hodder Education 2011 DenaryBinary 15010010110 14510010001 29500100111 Carry11

8. Hexadecimal numbers Programmers often write numbers down in hexadecimal (hex) form. Hexadecimal numbers are based on the number 16. They have 16 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Each column is worth16 times the one on its right. OCR Computing for GCSE © Hodder Education 2011 256161

9. Hexadecimal numbers We can convert denary numbers to hexadecimal by repeated division just as we did to get binary numbers. Take the denary number 141. We have the hexadecimal values 8 and 13 as remainders. 13 in hexadecimal is D. So, reading from the bottom again (where necessary), 141 in hexadecimal is 8D. OCR Computing for GCSE © Hodder Education 2011 ResultRemainder 141 ÷ 16 =813

10. Hexadecimal to denary All we do is multiply the numbers by their place values and add them together. For example, take the hexadecimal number 4F. 64 + 15 = 79 So, 4F is 79 in denary. OCR Computing for GCSE © Hodder Education 2011 Place value256161 Hex digits04F Denary= 0 × 256= 4 × 16= 15 × 1 = 0= 64= 15

11. Binary to hexadecimal This is particularly easy. Simply take each group of four binary digits, starting from the right and translate into the equivalent hex number. OCR Computing for GCSE © Hodder Education 2011 Binary11110011 HexF3

12. Hexadecimal to binary Do the reverse. You may find it easier to go via denary. Treat each hex digit separately. OCR Computing for GCSE © Hodder Education 2011 HexDB Denary1211 Binary11011011

13. Why use hexadecimal? Each hex digit represents four binary digits exactly. This makes it a useful shorthand way for programmers to write numbers. This saves effort and reduces the chance of making mistakes. OCR Computing for GCSE © Hodder Education 2011