Managed web hosting - 194 ARITHMETIC 4.1 5. (001 Explain why a
194 ARITHMETIC 4.1 5. (001 Explain why a negative integer in nines complement notation has a repre- sentation in ten s complement notation that is always one greater, if the representations are regarded as positive. 6. [16] What are the largest and smallest pbit integers that can be represented in (a) signed-magnitude binary notation (including one bit for the sign), (b) two s complement notation, (c) ones complement notation? 7. [M.Z0] The text defines ten s complement notation only for integers represented in a single computer word. Is there a way to define a ten s complement notation for all real numbers, having infinite precision, analogous to the text s definition? Is there a similar way to define a nines complement notation for all real numbers? 8. [MIo] Prove Eq. (5). b 9. [15] Change the following octal numbers to hexadecimal notation, using the hexadecimal digits 0, 1, . . . , F: 12; 5655; .2550.276; 76545336; 3726755. 10. [M.%?] Generalize Eq. (5) to mixed-radix notation. 11. [z?] Design an algorithm that uses the -2 number system to compute the sum of (a,. . alao)- and (b, . . blbo)-l, obtaining the answer (c,+z . c~co)-~. 12. [~3] Specify algorithms that convert (a) the binary signed magnitude number f(arl . . . ao)z to its negabinary form (&+I . . . bo)-2; and (b) the negabinary number Pn+l . . . bo)-z to its signed magnitude form &(a,+1 . . . a0)2. b 13. [A&1] In the decimal system there are some numbers with two infinite decimal expansions; e.g., 2.3599999.. . = 2.3600000.. . Does the negadecimal (base -10) system have unique expansions, or are there real numbers with two different infinite expansions in this base also? 14. [14] Multiply (11321)2i by itself in the quater-imaginary system using the method illustrated in the text. 15. [A4.24] What are the sets s={~m~yk~ ak an allowable digit } , - analogous to Fig. 1, for the negative decimal and for the quater-imaginary number systems? 16. [A&.24] Design an algorithm to add 1 to (a,. . . ala~)~-l in the i-l number system. 17. [A&N] It may seem peculiar that i -1 has been suggested as a number-system base, instead of the similar but intuitively simpler number i + 1. Can every complex number a+ bi, where a and b are integers, be represented in a positional number system to base i f 1, using only the digits 0 and l? 18. [HiV3.2] Show that the set S of Fig. 1 is a closed set that contains a neighborhood of the origin. (Consequently, every complex number has a binary representation to base i -1.) b 19. [Z3] (David W. Matula.) Let D be a set of b integers, containing exactly one solution to the congruence 2 3 j (modulo b) for 0 5 j < b. Prove that all integers m (positive, negative, or zero) can be represented in the form m = (a,. . . aO)b, where all the a3 are in D, if and only if all integers in the range 1 5 m < u can be so represented, where 1 = -max{ a ( a E D}/(b -l), u = -min{ a 1 a E D}/(b -1). For example, D = {-l,O,. . , b -2} satisfies the conditions for all b 2 3. [Hint: Design an algorithm that constructs a suitable representation.]