Numbers. Dr Hammadi Nait-Charif. Senior Lecturer Bournemouth University United Kingdom

Transcription

1 Numbers Dr Hammadi Nait-Charif Senior Lecturer Bournemouth University United Kingdom Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 1 / 10

2 Numbers Natural Numbers: The natural numbers 0, 1, 2, 3, 4,. are used for counting, ordering and labelling. We often use natural numbers to subscript a quantity to distinguish one element from another, e.g. x1, x2, x3, x4, etc. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 2 / 10

3 Numbers Natural Numbers: The natural numbers 0, 1, 2, 3, 4,. are used for counting, ordering and labelling. We often use natural numbers to subscript a quantity to distinguish one element from another, e.g. x1, x2, x3, x4, etc. Integers:Integers embrace negative numbers.. 2,10, 1, 2, 3,.. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 2 / 10

4 Numbers Natural Numbers: The natural numbers 0, 1, 2, 3, 4,. are used for counting, ordering and labelling. We often use natural numbers to subscript a quantity to distinguish one element from another, e.g. x1, x2, x3, x4, etc. Integers:Integers embrace negative numbers.. 2,10, 1, 2, 3,.. Rational Numbers:Rational or fractional numbers are numbers that can be represented as a fraction: For example 0.25 = = 4 = 15 = = 30 2 Some rational numbers can be stored accurately inside a computer, whilst many others can only be stored approximately. For example, 4/3 = produces an infinite sequence of threes and has to be truncated when stored as a binary number. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 2 / 10

5 Irrational & Real Numbers Irrational Numbers: Irrational numbers cannot be represented as fractions. Examples being π = and e = Such numbers never terminate and are always subject to a small error when stored within a computer. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 3 / 10

6 Irrational & Real Numbers Irrational Numbers: Irrational numbers cannot be represented as fractions. Examples being π = and e = Such numbers never terminate and are always subject to a small error when stored within a computer. Real Numbers:Real numbers embrace irrational and rational numbers. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 3 / 10

7 Repeating decimals Is an rational number or irrational? Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 4 / 10

8 Repeating decimals Is an rational number or irrational? If x = x = x = Then 99x = 120 (1) Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 4 / 10

9 Repeating decimals Is an rational number or irrational? If x = x = x = Then 99x = 120 (1) Therefore x = 120/99 = is rational. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 4 / 10

10 Number Systems In general, we can define any positive integer as 359 = Where 10 is the base of our decimal system. Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 5 / 10

11 Number Systems In general, we can define any positive integer as 359 = Where 10 is the base of our decimal system. In general, we can define any positive integer as abcd = a b c d 10 0 where a, b, c, d are all 9 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 5 / 10

12 Number Systems In general, we can define any positive integer as 359 = Where 10 is the base of our decimal system. In general, we can define any positive integer as abcd = a b c d 10 0 where a, b, c, d are all 9 And any positive real number as abcd.efg = a b c d e f where a, b, c, d, e, f, g are all 9 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 5 / 10

13 Octal to Decimal Ten is used only because we have ten fingers. If we had 8 fingers we would define an octal positive integer as. here a, b, c, d are all 7 abcd = a b c d 8 0 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 6 / 10

14 Octal to Decimal Ten is used only because we have ten fingers. If we had 8 fingers we would define an octal positive integer as. here a, b, c, d are all 7 table: abcd = a b c d Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 6 / 10

15 Octal to Decimal Ten is used only because we have ten fingers. If we had 8 fingers we would define an octal positive integer as. here a, b, c, d are all 7 table: abcd = a b c d = = = Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 6 / 10

16 Binary Numbers Binary numbers have a base of 2 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 7 / 10

17 Binary Numbers Binary numbers have a base of 2 For a positive binary integer abcd 2 = a b c d 2 0 here a, b, c, d are all 1 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 7 / 10

18 Binary Numbers Binary numbers have a base of 2 For a positive binary integer abcd 2 = a b c d 2 0 here a, b, c, d are all 1 e.g x = = = (2) Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 7 / 10

19 Decimal to Binary Conversion To convert 20 into binary we do what s called successive divisions Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 8 / 10

20 Decimal to Binary Conversion To convert 20 into binary we do what s called successive divisions successive divisions: 20 = 10 Remainder = 5 Remainder 0 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 8 / 10

21 Decimal to Binary Conversion To convert 20 into binary we do what s called successive divisions successive divisions: 20 = 10 Remainder = 5 Remainder = 2 Remainder 1 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 8 / 10

22 Decimal to Binary Conversion To convert 20 into binary we do what s called successive divisions successive divisions: 20 = 10 Remainder = 5 Remainder 0 5 = 2 Remainder = 1 Remainder 0 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 8 / 10

23 Decimal to Binary Conversion To convert 20 into binary we do what s called successive divisions successive divisions: 20 = 10 Remainder = 5 Remainder 0 5 = 2 Remainder = 1 Remainder = 0 Remainder 1 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 8 / 10

24 Decimal to Binary Conversion To convert 20 into binary we do what s called successive divisions successive divisions: 20 = 10 Remainder = 5 Remainder 0 5 = 2 Remainder = 1 Remainder = 0 Remainder = Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 8 / 10

25 Other Number Systems Any base can be used to represent positional numbers abcd x = a x 3 + b x 2 + c x 1 + d x 0 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 9 / 10

26 Other Number Systems Any base can be used to represent positional numbers abcd x = a x 3 + b x 2 + c x 1 + d x = = Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 9 / 10

27 Other Number Systems Any base can be used to represent positional numbers abcd x = a x 3 + b x 2 + c x 1 + d x = = = = Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 9 / 10

28 Other Number Systems Any base can be used to represent positional numbers abcd x = a x 3 + b x 2 + c x 1 + d x = = = = = = Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 9 / 10

29 Hexadecimal Numbers (base 16) Hexadecimal positive Integer can be represented abcd 16 = a b c d 16 0 where a, b, c, d are all 16 Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 10 / 10

30 Hexadecimal Numbers (base 16) Hexadecimal positive Integer can be represented abcd 16 = a b c d 16 0 where a, b, c, d are all 16 but this means we require some extra symbols such as ABCDEF Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 10 / 10

31 Hexadecimal Numbers (base 16) Hexadecimal positive Integer can be represented abcd 16 = a b c d 16 0 where a, b, c, d are all 16 but this means we require some extra symbols such as ABCDEF 2E6 16 = E = = Dr Hammadi Nait-Charif (BU, UK) Numbers NCCA-2011/12 10 / 10