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Transcript
An introduction to Numbers Dr Andrew French You will need to consult your times table and your tables of integer powers You will find it very useful to learn the powers up to 163 = 4,096 How are numbers ‘best’ written down? Does it matter? 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. So 1234.56 means 1 x 103 + 2 x 102 + 3 x 101 + 4 x 100 + 5 x 10-1 + 6 x 10-2 1000 + 200 + 30 +4 + 0.5 + 0.06 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 = 400 50 6 100 40,000 5000 600 20 8000 1000 120 3 1200 150 18 = 40,000 8000 1200 5000 1000 150 600 120 18 --------56,088 Use the ‘matrix decimal expansion’ to work out (NO CALCULATOR!) 167 x 394 = 0.15 x 17.2 = Use the ‘matrix decimal expansion’ to work out (NO CALCULATOR!) 167 x 394 = 300 90 4 100 30,000 9000 400 60 18,000 5,400 240 7 2,100 630 28 0.15 x 17.2 = 10 7 = 30,000 18,000 2100 9000 5400 630 400 240 28 --------65,798 0 .2 0 .1 1 0.7 0.02 0.05 0.5 0.35 0.01 = 1.00 0.70 0.02 0.50 0.35 0.01 --------2.58 Binary numbers 0, 1 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) Decimal Binary 17 10001 1 x 24 + 0 x 23 + 0 x 22 + 0 x 21 + 1 x 20 = 16 + 1 = 17 1234 10011010010 Note 1024 + 128 + 64 + 16 + 2 = 1234 1 x 210 0 x 29 0 x 28 1 x 27 1 x 26 0 x 25 1 x 24 0 x 23 0 x 22 1 x 21 0 x 20 1024 0 0 128 64 0 16 0 0 2 0 What are the decimal integers in binary? (a) 64 (b) 73 Binary numbers 0, 1 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) Decimal Binary 17 10001 1 x 24 + 0 x 23 + 0 x 22 + 0 x 21 + 1 x 20 = 16 + 1 = 17 1234 10011010010 Note 1024 + 128 + 64 + 16 + 2 = 1234 1 x 210 0 x 29 0 x 28 1 x 27 1 x 26 0 x 25 1 x 24 0 x 23 0 x 22 1 x 21 0 x 20 1024 0 0 128 64 0 16 0 0 2 0 What are the decimal integers in binary? (a) 64 is 1000000 since 26 = 64 (b) 73 is 1001001 since 64 + 8 + 1 = 73 1 x 26 0 x 25 0 x 24 1 x 23 0 x 22 0 x 21 1 x 20 64 0 0 8 0 0 1 Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer memory addresses. Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 A B C D E F Hexadecimal or ‘base 16’ Decimal Hexadecimal 17 11 1 x 161 + 1 x 160 = 16 + 1 = 17 1234 4D2 4 x 162 + 13 x 161 + 2 x 160 = 4 x 256 + 13 x 16 + 2 = 1234 What is in hexadecimal? (a) 31 (b) 117 Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer memory addresses. Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 A B C D E F Hexadecimal or ‘base 16’ Decimal Hexadecimal 17 11 1 x 161 + 1 x 160 = 16 + 1 = 17 1234 4D2 4 x 162 + 13 x 161 + 2 x 160 = 4 x 256 + 13 x 16 + 2 = 1234 What is in hexadecimal? (a) 31 is 1F (b) 117 is 75 since since 1 x 161 + 15 x 160 = 31 7 x 161 + 5 x 160 = 112 + 5 = 117 Other ‘popular’ bases are: 12 Duodecimal 0,1,2,3,4,5,6,7,8,9,A,B 60 Sexagesimal Used by the ancient Babylonians around 3000BC cuneiform digits Note this wasn’t a proper ‘place value’ system as there was no zero! Although it did appear later as