Binary Numbers. Binary Numbers. Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria

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1 Binary Numbers Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria Wolfgang Schreiner RISC

2 Digital Computers Todays computers are digital. Digital: data are represented by discrete pieces. Pieces are denoted by the natural numbers:, 1, 2, 3... Analog: data are represented by continuous signals. For instance, electromagnetic waves. Wolfgang Schreiner 1

3 Character Encodings ASCII: American Standard Code for Information Interchange. 128 = 2 7 characters (letters and other symbols). Also non-printable characters: LF (line-feed). Represented by numbers,...,127. ASCII code Character LineFeed (LF) A Z a z Wolfgang Schreiner 2

4 Character Encodings Text is a sequence of characters. H i, H e a t h e r ISO contains 256 = 2 8 characters (Latin 1). First 128 characters coincide with ASCII standard. Unicode contains = characters. First 256 characters coincide with ISO standard. The ASCII characters are the same in all encodings. Wolfgang Schreiner 3

5 Binary Numbers In which number system are numbers represented? Humans: decimal system. 1 digits:, 1, 2, 3, 4, 5, 6, 7, 8, 9. Computers: binary system. 2 digits:, 1. A bit is a binary digit. Physical representation: e.g. high voltage versus low voltage. Binary numbers are physically easy to represent. Wolfgang Schreiner 4

6 Binary Numbers Decimal number 13: = = 13. Binary number 11111: = = 13 Character g: ASCII code 13. Binary number is computer representation of g. Wolfgang Schreiner 5

7 Conversion of Binary to Decimal = 2999 Result = = = = = = = = = = = 1 Start here Horner s scheme. Wolfgang Schreiner 6

8 Conversion of Decimal to Binary Quotients Remainders = Wolfgang Schreiner 7

9 General Number Systems Any base value (radix) b possible. Decimal system: b = 1. Binary system: b = 2. n digits d n 1... d represent a number m: m = d n 1 b n d b = i<n d i b i. Number bounds: m < b n. Decimal system, 8 digits: m < 1 8 = 1... Binary system, 8 digits: m < 2 8 = 256. Example: How many bit does it take to represent a character in ASCII, in the ISO code, in Unicode? n > log b m. Wolfgang Schreiner 8

10 Other Number Systems Octal system: 8 digits, 1, 2, 3, 4, 5, 6, 7. One octal digit (6) can be represented by 3 bits (11). Conversion of binary number: 11111: Hexadecimal system: digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. One hexadecimal digit (B) can be represented by 4 bits (111). Conversion of binary number 11111: Easy conversion between binary and octal/hexadecimal numbers. Wolfgang Schreiner 9

11 Example Conversions Example 1 Hexadecimal Binary Octal B Example 2 Hexadecimal Binary Octal 7 B A B C Hexadecimal/octal numbers are shorter to write. Wolfgang Schreiner 1

12 Unsigned Binary Numbers Unsigned integers with n bits: from to 2 n 1. Number Unsigned Integer Computer representation of finite-precision integers. Wolfgang Schreiner 11

13 Signed Binary Numbers Signed integers with n bits: from 2 n 1 to 2 n 1 1. Two s complement representation. Number Unsigned Integer Signed Integer Wolfgang Schreiner 12

14 Signed Binary Numbers Why this representation? Arithmetic independent of interpretation. Computation of representation: Determine representation of 3: Representation of +3: 11. Invert representation: 1. Add 1: 11. Binary: = 111 Unsigned: = 7 Signed: 2 3 = 1 Simple implementation in arithmetic hardware. Wolfgang Schreiner 13

15 Binary Arithmetic Overflow: Addend 1 1 Augend Sum 1 1 Carry 1 Carry generated by addition of left-most bits is thrown away. Addend and augend are of same sign, result is of opposite sign. All hardware arithmetic is finite-precision. Wolfgang Schreiner 14

16 Floating-Point Numbers How to represent 1.375? Representation: (s, m, e) the sign bit s denotes +1 or 1, the mantissa m is a n bit binary number representing the value m/2 n (< 1), the exponent e is a binary number. Value: s m/2 n 2 e Example: Eight bit floating point: the first bit represents the sign +1, the five bit mantissa 111 represents the fraction 11/32 (why?), the two bit exponent is 2. Value: +1 11/ = Wolfgang Schreiner 15

17 Reals and Floating Points 1 Negative overflow 3 Negative underflow 2 Expressible negative numbers 4 Zero 5 Positive underflow 6 Expressible positive numbers 7 Positive overflow Example: fraction with 3 decimal digits, exponent with 2 digits. 1. Numbers between and Zero. 3. Numbers between and Real values are rounded to the closest floating point value. Overflows and underflows may occur. Wolfgang Schreiner 16

18 Binary Numbers Normalized Floating Point Numbers Unnormalized: Sign Excess 64 + exponent is = Example 1: Exponentiation to the base Fraction is = 2 2 ( ) = 432 To normalize, shift the fr action left 11 bits and subtr act 11 from the e xponent Normalized: Sign Excess 64 + exponent is = 9. Fraction is Example 2: Exponentiation to the base 16 = 2 9 ( ) = Unnormalized: = 16 5 ( B 16 4 ) = 432 Sign Excess 64 Fraction is B exponent is = 5 To normalize, shift the fr action left 2 he xadecimal digits, and subtr act 2 from the e xponent. Normalized: = 16 3 ( B 16 2 ) = 432 Sign Excess 64 + exponent is = Fraction is B Left-most digit of mantissa is always non-zero. Wolfgang Schreiner 17

19 IEEE Floating-Point Standard 754 IEEE standard for floating point representation (1985). 1. Single precision: 32 bits (= ). 2. Double precision: 64 bits (= ). 3. Extended precision: 8 bits (inside hardware units only). Bits 1 Sign 8 23 Exponent Fraction (a) Bits Exponent Fraction Sign (b) Wolfgang Schreiner 18

20 IEEE Floating Point Characteristics Item Single Precision Double Precision Smallest normalized number Largest normalized number Decimal range 1 38 to to 1 38 Smallest denormalized number Denormalized numbers: first bit of mantissa. Distinguished from normalized numbers by exponent. Used to represent very small floating point numbers. Avoid underflows by giving up precision of mantissa. Smallest value: 1 in the rightmost bit, rest. Wolfgang Schreiner 19

21 IEEE Numerical Types Normalized ± < Exp < Max Any bit pattern Denormalized ± Any nonzero bit pattern Zero ± Infinity ± Not a number ± Any nonzero bit pattern Sign bit Two zeros (positive and negative). Plus and minus infinity (overflows). NaN (Not a Number = infinity / infinity). Wolfgang Schreiner 2