1. An introduction to Numbers Dr Andrew French

4. You will need to consult your times table and your tables of integer powers

6. You will find it very useful to learn the powers up to 16 3 = 4,096

7. The DECIMAL system is when numbers are written right to left in powers of ten. Only ten symbols are needed (0,1,2,3,4,5,6,7,8,9) plus a DECIMAL POINT to describe any number, of which there are infinitely many. Not bad eh? In ancient cultures a different symbol is used for each integer, just like the way we say ‘one’, ‘two’, ‘three’ etc. How are numbers ‘best’ written down? Does it matter? So 1234.56 means 1 x 10 3 + 2 x 10 2 + 3 x 10 1 + 4 x 10 0 + 5 x 10 -1 + 6 x 10 -2 1000 + 200 + 30 + 4 + 0.5 + 0.06

8. The decimal system enables us to perform arithmetic calculations on numbers (i.e. addition, subtraction, multiplication and division) in a straightforward, systematic way. You have been practising it for many years now! Note we can use the decimal system to help us work out multiplications using a small set of memorized results (i.e. your times table) 123 x 456 = = 40,000 8000 1200 5000 1000 150 600 120 18 --------- 56,088

9. Use the ‘matrix decimal expansion’ to work out (NO CALCULATOR!) 167 x 394 = 0.15 x 17.2 =

10. Use the ‘matrix decimal expansion’ to work out (NO CALCULATOR!) 167 x 394 = 0.15 x 17.2 = = 30,000 18,000 2100 9000 5400 630 400 240 28 --------- 65,798 = 1.00 0.70 0.02 0.50 0.35 0.01 --------- 2.58

11. We don’t have to use the decimal system. In fact we can use any (integer!) number of symbols from two upwards. A two symbol (0 or 1) system is BINARY (which is used to store and manipulate numbers by computers) Binary numbers 0, 1 DecimalBinary 1710001 1 x 2 4 + 0 x 2 3 + 0 x 2 2 + 0 x 2 1 + 1 x 2 0 = 16 + 1 = 17 123410011010010 Note 1024 + 128 + 64 + 16 + 2 = 1234 1 x 2 10 0 x 2 9 0 x 2 8 1 x 2 7 1 x 2 6 0 x 2 5 1 x 2 4 0 x 2 3 0 x 2 2 1 x 2 1 0 x 2 0 102400128640160020 What are the decimal integers (a) 64 (b) 73 in binary?

12. Binary numbers 0, 1 DecimalBinary 1710001 1 x 2 4 + 0 x 2 3 + 0 x 2 2 + 0 x 2 1 + 1 x 2 0 = 16 + 1 = 17 123410011010010 Note 1024 + 128 + 64 + 16 + 2 = 1234 1 x 2 10 0 x 2 9 0 x 2 8 1 x 2 7 1 x 2 6 0 x 2 5 1 x 2 4 0 x 2 3 0 x 2 2 1 x 2 1 0 x 2 0 102400128640160020 What are the decimal integers (a) 64 is 1000000 since 2 6 = 64 (b) 73 is 1001001 since 64 + 8 + 1 = 73 in binary? 1 x 2 6 0 x 2 5 0 x 2 4 1 x 2 3 0 x 2 2 0 x 2 1 1 x 2 0 64008001 We don’t have to use the decimal system. In fact we can use any (integer!) number of symbols from two upwards. A two symbol (0 or 1) system is BINARY (which is used to store and manipulate numbers by computers)

13. Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F DecimalHexadecimal 1711 1 x 16 1 + 1 x 16 0 = 16 + 1 = 17 12344D2 4 x 16 2 + 13 x 16 1 + 2 x 16 0 = 4 x 256 + 13 x 16 + 2 = 1234 0123456789101112131415 0123456789ABCDEF A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer memory addresses. Decimal Hexadecimal or ‘base 16’ What is (a) 31 (b) 117 in hexadecimal?

14. Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F DecimalHexadecimal 1711 1 x 16 1 + 1 x 16 0 = 16 + 1 = 17 12344D2 4 x 16 2 + 13 x 16 1 + 2 x 16 0 = 4 x 256 + 13 x 16 + 2 = 1234 0123456789101112131415 0123456789ABCDEF A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer memory addresses. Decimal Hexadecimal or ‘base 16’ What is (a) 31is 1F since 1 x 16 1 + 15 x 16 0 = 31 (b) 117 is 75 since 7 x 16 1 + 5 x 16 0 = 112 + 5 = 117 in hexadecimal?

15. Other ‘popular’ bases are: 12Duodecimal0,1,2,3,4,5,6,7,8,9,A,B 60SexagesimalUsed by the ancient Babylonians around 3000BC Note this wasn’t a proper ‘place value’ system as there was no zero! cuneiform digits Although it did appear later as