Number Systems - Part I CS 215 Lecture # 5

Motivation The representation of numbers is made difficult because of the discrete nature of computer representation Not all numbers can be represented in a computer The chosen method of representation affects the ease by which arithmetic operations are performed

Unary Representation In a unary representation, the number one is represented by a single mark, the number two by two marks, etc. Unary has limitations: the number of symbols necessary to represent a number grows directly with the value negative numbers can not be represented

Roman Numerals Roman numerals are sorted by decreasing values A symbol representing a smaller value preceding a symbol representing a larger value means we have to subtract the smaller value from the larger value Only a single smaller symbol may precede a larger one Only symbols representing powers of ten (I,X,C) may be placed out of order I 1 V 5 X 10 L 50 C 100 D 500 M 1000

Addition in Roman Numerals? A possible algorithm Convert all position sensitive symbols into position insensitive form Sort by values of symbols Merge the two position-insensitive representations into a single position-insensitive representation Combine lowest-value types into higher value-types Create legal Roman numerals by converting back to the unique format

Example 5.1 XIX (19) + MXMV (1995) XIX = XVIIII MXMV = MDCCCCLXXXXV XVIIII + MDCCCCLXXXXV = MDCCCCLXXXXXVVIIII MDCCCCLXXXXXVVIIII = MDCCCCLXXXXXXIIII = MDCCCCLLXIIII = MDCCCCCXIIII = MDCCCCCXIIII = MDDXIIII = MMXIIII MMXIIII = MMXIV (2014)

Weighted Positional Notation Use the position of the symbol to indicate the value By assigning each position the appropriate power of the base, we can get a unique representation of numbers in that base Decimal System (Base 10) Given a sequence of n digits d0,d1,…,dn-1 dn-1.10n-1 + … + d1.101 + d0.100

In general, given a sequence of n digits s0,s1,…,sn-1 and a base b, sn-1.bn-1 + … + s1.b1 + s0.b0 yields a unique integer N s0 is the least significant digit sn-1 is the most significant digit The most significant symbol can not be zero

Any positive integer value can be used as the base or radix for a weighted positional notation For bases less than or equal to 10, a subset of the ten decimal digits can be used, e.g., binary (base 2) and octal (base 8) For bases more than 10, new symbols will have to be introduced, e.g., hexadecimal (base16)

We use a subscript to indicate the base of the number when we are writing. In programming languages, octal representation may be preceded with a zero (0), e.g., 033. A hexadecimal representation may be preceded with 0x, e.g., 0x1b SAL and MAL use these for hexadecimal and decimal numbers; octal is not recognized

Representations in different radices binary octal hexa decimal 000 0 0 0 001 1 1 1 010 2 2 2 011 3 3 3 100 4 4 4 1111 17 F 15

Hexadecimal and Octal Hex Octal Decimal Binary 1 1 1 0001 2 2 2 0010 1 1 1 0001 2 2 2 0010 3 3 3 0011 4 4 4 0100 5 5 5 0101 6 6 6 0110 7 7 7 0111 8 10 8 1000 9 11 9 1001 A 12 10 1010 B 13 11 1011 C 14 12 1100 D 15 13 1101 E 16 14 1110 F 17 15 1111

Discriminating only between two symbols would be far cheaper in digital circuitry than three, four, etc. thus binary is the chosen representation for computers. Since bits can be grouped into three to obtain octal value and into four to obtain hexadecimal value, it is much easier for humans to work with octal and hexadecimal representations.

hexadecimal, octal, and binary To convert the octal or hexadecimal representation from binary simply group them starting from the least significant bit into three for octal and into four for hexadecimal Example octal hexa binary 11011 011 011 0001 1011 3 3 1 B hence: 11011 is 338 or 1B16 or 0x1B or 033

Example 5.2 Convert 11101 to octal _________ Convert 11101 to hexadecimal _________ Convert 100110 to octal _________ Convert 11001100 to hexadecimal _________

Zero and negative numbers To represent zero, we violate the rule that says the most significant symbol can not be zero, and use the single symbol ‘0’ We can specify, a symbol ‘-’ that is placed only at the beginning of the sequence to signify negative numbers The absence of or presence of the symbol ‘+’ signify positive numbers

Conversion into Decimal Given a sequence of base-r digits sn-1...s1s0 convert the sequence into a decimal number N using: N = sn-1.bn-1 + … + s1.b1 + s0.b0

Example 5.3 Convert 10 01102 to decimal. = 3810 Convert 3b216 to decimal. N = 3*162 + 1110*161 + 2*160 = 94610 Convert 3708 to decimal. N = 3*82 + 7*81 + 0*80 = 192 + 56 + 0 = 24810

Example 5.4 Convert the following to decimal: 1. 3467 _______ 1. 3467 _______ 2. 123 _______ 4. 1448 _______ 5. 6A16 _______ 6. 1002 _______