How do computers represent numbers?
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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
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