Logical Bases: Representation of Numbers Decimal number Positional notation Binary numbers

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1 Logical Bases: Representation of Numbers Decimal numbers most commonly used by humans are in base 0 and consist of the ten decimal digits 0,,, 3, 4, 5, 6, 7, 8, 9. For example, the decimal number 373 is 373 = Positional notation describes the numbers according to position. For example, the two 3s in the decimal number 373 are in different positions, the 3 in the first position (at left) has a value which is 00 times more than the other 3. This notation can be described by a data flow diagram as follows at the left below. Binary numbers are most commonly used in digital computing; they have base. The data flow diagram at the right shows the binary number 0 which is the decimal number 3. 0 = = 3 in decimal Decimal numbers (base 0) Binary numbers (base )

2 Binary Addition To add the ten decimal digits requires a table of 00 = 00 entries; but if the digits are binary a table of only = 4 rows is needed. The binary add definition follows: 0 0 = 0 0 = 0 = = = 0 = 0 with carry of A table of such a simple adder (half adder) follows, showing the output of a sum s and a carry c. A logical gate diagram shows that the carry is simply the And of a, b. The sum is actually the exclusive-or, which is formed in detail below. in out a b s c a b HA = Half Adder a b a & b! & carry s c

3 Longer Binary Addition Addition of two numbers, say A=3= and B=6=0 to yield a third D is done in stages. At each stage there is input a bit value a, and b, and a carry from the previous stage. Then there is an output of a sum s, and a carry to the next stage. This process begins with the least significant digits (at the right) and proceeds to the left. The previous Half Adders can be used as shown to create a Full Adder. A = 3 = 0 0 FA = Full Adder B = 6 = 0 0 c c s HA a b A B Carry D = 9 = 0 FA FA FA 0 FA c s HA a b c sum 3

4 General Numbers: to any base b In general, a number can be represented in any base b as shown. The base may be any integer; the bases, 8, 0, 6 are most common. However, there may be unusual bases, negative values, fractions, etc. The coefficients c0, c, c,.. are usually integers from some small set, usually 0,,, etc up to the value of the base. cn General numbers (base b)... c c c0 b b n b b b 0 cnb n.. cb cb c0b 0 4

5 Numbers in base 3 Numbers in base 3 have the 3 coefficients, 0, and. Some numbers, from 0 to, are shown in this base Note the beauty of this table; group them 3 at a time. Ternary numbers (base 3) 7 Example: 9 0 (0) 3 = = 9 03 = 8 0 = 9 i base (0) 3 = (9) 0 5

6 Numbers in base 8 (octal) Numbers in base 8 have the 8 coefficients, 0,,,.. 7. Some numbers, from 0 to are shown in this base 8. cn Octal numbers (base 8) 5... c 64 c cn8 n.. c8 c8 c c0 Example: (0) 8 = = = 8 0 = 9 i octal (0) 8 = (9) 0 6

7 Numbers in base 6 (Hexadecimal) Numbers in base 6 have the 6 coefficients, 0,,, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Some numbers, from 0 to 6 are shown in this base This hexadecimal base (called Hex) is very common in computing. cn Octal numbers (base 8) c 56 c 6 c0 cn6 n.. c6 c6 c Example: (0) 6 = = = 5 0 = 53 i hex A B C 3 D 4 E 5 F (0) 6 = (53)

8 Numbers in base - Bases can be negative;the following one is binary base (-). The coefficients are still binary, 0s and s only. Some numbers, both positive and negative are shown in this base This is not a common way to create negative numbers. Binary numbers (base -) cn... c c c cnb n.. cb cb c0b 0 i base i base

9 Numbers in base (binary) with negative coefficients Bases can be positive with negative coefficients; The following one has a binary base with coefficients, 0 and -. The coefficient - can be written as 7, or as a with a bar over it. Some numbers, both positive and negative are shown in this base This is also not a common way to create negative numbers. cn Binary numbers (base )... c c 8 4 c0 i base i base cnb n.. cb cb c0b 0 9

10 Common Binary Procrustean representation Most computers are binary with a fixed number of bits (8, 6, 4, 3, 64). They are called procrustean, after the Greek legend of Procrustes, or PolyDemon, who had an iron bed, placed people on it, and if they were too tall he chopped them off to the fixed size; if they were too short he stretched them to this fixed size. Computer registers, words, etc are also fixed size. Signed Magnitude (s&m) A simple way of representing binary numbers which are positive or negative is to specify the sign as the first binary digit (or bit). A negative sign is ; a positive sign is 0. For example, with 4 bits, there are 6 possible integers as shown. Notice however that there are two zeros, a positive and a negative one. Not good! Binary numbers (base ) /- c Sign 0 is is - 4 c c0 i s&m i s&m

11 Ones Complement Another way of representing negative numbers is to do a complement; to complement or Not each bit of the positive number to get the negative. For example, with 4 bits, the 6 possible integers are shown below. Notice again however that there are two zeros, a positive and a negative one. Not good. The conversion of a number to its negative is shown in the data diagram below. cn... c!! cn c c! c c0! c0 i comp i comp

12 Twos Complement Yet another way of representing negative numbers is to do another kind of complement; to view the negative numbers as a cyclic odometer as on a vehicle. A 4 bit system, goes forward, from 0000 to and then repeats starting with Going backwards from 0000, the next value (which is ) corresponds to -. Similarly for -, -3, etc to -8. There is only one value for zero, and the values range from -8 to 7. In general for n digits the values range from - n to n - Such a s complement is most commonly used in computers, with more than 4 bits. The s complement negative numbers are just one more than the s complement, so to create the s complement, simply invert all the bits and add to the result. i comp i comp = 000 = = = 0 = = 0-5 = 0 - = 0-7 = 00 The sum of two numbers in s complement is shown to the right, in three examples. Note the last example, adding bit by bit yields a final carry of, which is ignored. Recall that these complement systems are procrustean; they hold for a fixed size only.

13 Questions involving Number Representations.. Drill and Kill a. Represent the decimal number 49 in all the various bases, 3, 4, 5, 6, 7,8 9, 0,,6, 0. b. What decimal number is represented by 0000 in bases:, 3, 4, 5, 6, 7,8 9, 0,,6, 0. c. Show the addition of 66 and 99 in s complement. d. Add the hex numbers ABC and DEF.. How many integers can you represent with your 0 fingers? 3. Octal tøfrom Binary Relate a number such as 49, in binary and octal; Show how to convert from one to the other very conveniently. 4. Create a 3-valued integer system with negative coefficients, and represent a dozen values in this system. 5. Create a tens-complement system. 6. Create a subtractor of positive integers. 7. Create your own number system; Can you use a base such as /, /3, 0.7, or PI? 3

14 More Questions involving Number Representations.. Base 3 Consider a base 3 radix system with coefficients ranging from 0 tdo 9 and including JS, K and L (for 0,, ) respectively. Convert the following numbers into base 0 equivalent numbers a. K9 b. KKK Convert the following decimal numbers into base 3 c. 49 d. 00. Pex: base 3 Consider a system with base 3, represented by 5 bit cominations With the 3 pex digits ranging from 0 dto 9 and then A, B, C.. W excluding the letter 0 which could be confused by the digot 0. Write the following binary sequence in thiss pex notation, by bunches of 5 bits Then convert this pex integer into a decimal integer. Check your answer in another way. 3. Balanced Ternary Radix Suppose that the coefficients of a radix system are -, 0 and And the base is also negative at -3. Show the 5 decimal digits (7 to -7) when converted to this system (I.e. count from 7 down to 0 and then continue to -7. 4

15 Even More Questions involving Number Representations.. Odd Radix If the coefficients of a radix system are the positive odd integers, 3, 5, 7, represent the integers 0 to 6 by this system. Observe the squares, 4, 9, 6. What is the smallest set of digits needed to represent all positive integers?. Non 9 Decimal System An integer can be represented in a decimal system with the ten digits: -, 0,,, 3, 4, 5, 6, 7, 8 (but not 9). Use this system to represent the decimals 0, 9, and Many Complements Write the negative number -7 in binary twos complement form For the following size of register: 4, 5, 7, 9. 5

CMPE223/CMSE222 Digital Logic Design. Positional representation

CMPE223/CMSE222 Digital Logic Design Number Representation and Arithmetic Circuits: Number Representation and Unsigned Addition Positional representation First consider integers Begin with positive only