How do computers represent numbers?

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Muriel Norris
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1 How do computers represent numbers? Tips & Tricks Week 1 Topics in Scientific Computing QMUL Semester A 2017/18 1/10

2 What does digital mean? The term DIGITAL refers to any device that operates on discrete quantities (digits) 2/10

3 What does digital mean? The term DIGITAL refers to any device that operates on discrete quantities (digits) Modern computer are digital devices: they work on on a finite amount of finite quantities 2/10

4 What does digital mean? The term DIGITAL refers to any device that operates on discrete quantities (digits) Modern computer are digital devices: they work on on a finite amount of finite quantities DIGITAL often refers also to the way information is represented or encoded 2/10

5 What does digital mean? The term DIGITAL refers to any device that operates on discrete quantities (digits) Modern computer are digital devices: they work on on a finite amount of finite quantities DIGITAL often refers also to the way information is represented or encoded Everything inside a digital computer is represented as a bunch of binary digits, namely sequences of 1s and 0s. 2/10

6 What are integer numbers 3/10

7 What are integer numbers Let us consider the number /10

8 What are integer numbers Let us consider the number 1234 We naturally assume that it is written in base 10, i.e. that: 1234 (10) = /10

9 What are integer numbers Let us consider the number 1234 We naturally assume that it is written in base 10, i.e. that: 1234 (10) = What if we want to represent 1234 in base 8? 3/10

10 What are integer numbers Let us consider the number 1234 We naturally assume that it is written in base 10, i.e. that: 1234 (10) = What if we want to represent 1234 in base 8?Here we go: 1234 (10) = 2322 (8) = /10

11 Integers as your computer sees them Computers work with binary digits, so the number 1234 should be expressed in base 2: 4/10

12 Integers as your computer sees them Computers work with binary digits, so the number 1234 should be expressed in base 2: 1234 (10) = (2) = /10

13 Integers as your computer sees them Computers work with binary digits, so the number 1234 should be expressed in base 2: 1234 (10) = (2) = The sequence (11 bits) is indeed the way your computer will most probably represent the number /10

14 What about non-integer numbers? Not all numbers we deal with are integers 5/10

15 What about non-integer numbers? Not all numbers we deal with are integers Example: how should a computer represent the number 13.37? 5/10

16 What about non-integer numbers? Not all numbers we deal with are integers Example: how should a computer represent the number 13.37? Possible solution: use some bits for the integer part, and some other bits for the decimal part: 5/10

17 What about non-integer numbers? Not all numbers we deal with are integers Example: how should a computer represent the number 13.37? Possible solution: use some bits for the integer part, and some other bits for the decimal part: INTEGER PART DECIMAL PART 5/10

18 What about non-integer numbers? Not all numbers we deal with are integers Example: how should a computer represent the number 13.37? Possible solution: use some bits for the integer part, and some other bits for the decimal part: INTEGER PART DECIMAL PART Problems: how many bits for each part? The precision is fixed a-priori With 6 bits for the decimal part we cannot represent 13.37!!! 5/10

19 A sound solution Note: every real number can be expressed in the form where: s: sign (either +1 or -1) s m b q (1) 6/10

20 A sound solution Note: every real number can be expressed in the form where: s: sign (either +1 or -1) m: significand or mantissa s m b q (1) 6/10

21 A sound solution Note: every real number can be expressed in the form where: s: sign (either +1 or -1) m: significand or mantissa b: base s m b q (1) 6/10

22 A sound solution Note: every real number can be expressed in the form where: s: sign (either +1 or -1) m: significand or mantissa b: base q: exponent s m b q (1) 6/10

23 A sound solution Note: every real number can be expressed in the form where: s: sign (either +1 or -1) m: significand or mantissa b: base q: exponent Example: s m b q (1) = /10

24 How are real numbers stored? IEEE-754 standard 7/10

25 How are real numbers stored? IEEE-754 standard it is a representation based on sign, mantissa, base, exponent 7/10

26 How are real numbers stored? IEEE-754 standard it is a representation based on sign, mantissa, base, exponent For example, single-precision floating point numbers use: base 2 1 bit for sign 24 bits for the mantissa 8 bits for the exponent 7/10

27 How are real numbers stored? IEEE-754 standard it is a representation based on sign, mantissa, base, exponent For example, single-precision floating point numbers use: base 2 1 bit for sign 24 bits for the mantissa 8 bits for the exponent Hence: maximum positive number: O(2 127 ) O(10 38 ) 7/10

28 Double precision Double-precision floating point numbers use: base 2 1 bit for sign 52 bits for the mantissa 11 bits for the exponent 8/10

29 Double precision Double-precision floating point numbers use: base 2 1 bit for sign 52 bits for the mantissa 11 bits for the exponent Hence: maximum positive number: O( ) O( ) 8/10

30 It s a matter of precision a mantissa of 24 bits (single precision), can accurately store only about log 10 (2 24 ) 7 significant digits 9/10

31 It s a matter of precision a mantissa of 24 bits (single precision), can accurately store only about log 10 (2 24 ) 7 significant digits a mantissa of 52 bits (double precision), can accurately store only about log 10 (2 52 ) 15 significant digits 9/10

32 It s a matter of precision a mantissa of 24 bits (single precision), can accurately store only about log 10 (2 24 ) 7 significant digits a mantissa of 52 bits (double precision), can accurately store only about log 10 (2 52 ) 15 significant digits 9/10

33 Common pitfalls Be careful with round-off errors (what is the value of 3 (4/3 1) 300????) Be careful with mixing in the same operation numbers spanning many orders of magnitude Be careful with operations on irrational numbers (what is the value of sin(pi)???) 10/10

Computer Arithmetic. MATH 375 Numerical Analysis. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Computer Arithmetic

Computer Arithmetic. MATH 375 Numerical Analysis. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Computer Arithmetic Computer Arithmetic MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013 Machine Numbers When performing arithmetic on a computer (laptop, desktop, mainframe, cell phone,