c. First convert to base-10 by expanding out the powers of 2:

Transcription

1 EGR4 F 18 Assignment #1 Solution Due Date: Monday (Section 10), September 10, Beginning of class Tuesday (Section 0), September 11, Beginning of class Assigned Readings: Lecture notes pages [0 pts] Determine the base-n representation of the following numbers, with N as given in each part. Show your work! Submit this problem on paper (not by ). a. 4310, N = b. 636, N = 4 c , N = 16 d , N = 8 e , N = 10 f. 5748, N = g. CE916, N = 5 h , N = 4 i. 3738, N = 10 j. 3637, N = 10 a. Using the direct approach (i.e., fit in powers of ) we get: 4310 = = = = = = (101011) b. Not possible as 6 is not a valid base-6 number. c. First convert to base-10 by expanding out the powers of : = = = (45)10 Let s use the informal approach : 4510 = = D 16 0 = (D)16 d. Remember that converting between base- and base-8 is easy: just group 3 bits at a time and map 0 -> 000, 1 -> 001, etc. up to 7 -> 111. Append 0 in the front, if group of 3 bits is not formed. Thus we can write: = = (11766)8 1

2 e. Here s something new to try. Rather than futzing with lots of powers of, let s convert to base 16 first (easy!) and then only worry about a few powers of = = 95F616 = = = (38390)10 f. This is same as part (d) but in reverse, mapping octal digits to groups of 3 bits. Thus: 5748 = = ( ) g. First, we find that CE916 = = Now let s use the informal approach by fitting in powers of 5: = = = = = = (10110)5 h. Using the direct-map approach (i.e., one base is power of of other base 4 = ): Base Base Map the base- number to corresponding base-4 number (by grouping digits together) = ( ) 10 = 4 11 = = 14 (3113)4 i. First, we find that 3738 = = (010)10 j. Just expand it out: 3637 = = = (1346)10

3 . [15 pts] Solve the following puzzles. Show your work! a. What is the four-digit number (Base-10) in which the first digit is one-third the second, the third is the sum of the first and second, and last is three times the second. After obtaining the result in Base-10, represent the result in Base-6 and Base-1. First digit is one-third the second digit: Let first digit be x, therefore second digit is 3x Third digit is sum of the first and second: So third digit is x + 3x = 4x Last digit is three times the second: So fourth digit is 3*3x = 9x So the four digits are x, 3x, 4x, 9x Let x = 1 so the four-digit number in base-10 is (1349)10 To convert from base-10 to base-6, use divide-by-base approach (1349)10 = 1349/6 = 4 5 = 4/6 = 37 5 = 37/6 = 6 15 = 6/6 = = 1/6 = = (1015)6 To convert from base-10 to base-1, use divide-by-base approach (1349)10 = 1349/1 = 11 5 = 11/1 = 9 45 = 9/1 = = (945)1 b. The statement below is correct in at least one number system (besides base-1). That is, the statement is correct if we assume the numbers are expressed in a base other than 10. It is up to you to find out which number base makes each statement correct. You need to justify your answer by converting the numbers in each operation to base 10 and showing that the statement is correct. For example, 36/6 = 7 is clearly not correct in base 10 but it is correct in base 1 because 361 = 410 and 410/610 = 710. Thus, 361/61 = 71 is true. 140 x 3 / 10 = x 3 / 10 = 10 The key is to remember that in any base B, the symbol 140 means 1 B + 4 B B 0. Similarly, the symbol 3 means 3 B 0 and symbol 10 means 1 B + 0 B 1 + B 0. Having converted these symbols to base-10, you can now use familiar mathematical technique to solve the problem. (B + 4B) x 3 = B (B + ) Therefore, 3B + 1B = B(B + ) 3

4 3B + 1B = B 3 + B 0 = B 3 3B - 10B 0 = B(B - 3B - 10) Therefore, B - 3B - 10 = 0 Solving the quadratic equation, we get: (B-5)(B+) = 0 Therefore, B = 5 or -. We will ignore B = - The resulting number system is B = 5 c. Perform the following steps to uncover the hidden word. Start with the number ( )8 Convert the number to base-10 (315848)10 Add the following number (183,000,000)10 to the result obtained in the step above (183,315,848)10 Convert the result obtained in the step above to base-16 (AEDD88)16 Add the following number (4C7C3EBC)16 to the result obtained in the step above (57696C44)10 Divide the result up into digit chunks = C 44 Look up each two-digit hex number in a table of ASCII codes and write the word it spells W i l D 4

5 3. [15 pts] Write a C program that asks the user for an integer N (in Base-10 format) that must be in the range 0 through The program then prints the representation of the number N entered by the user in base-16 (hexadecimal format). The user s input must be validated and appropriate error messages displayed if necessary. The user must be prompted as many times as necessary until a valid integer in the range 0 through is obtained. The program is expected to work. A non-working program, a program that does not validate its input, or a program that does not loop until a valid integer is obtained will receive a grade of 0. Proper program structure, modularity, and documentation are required. Submit your source code by by the due date and time indicated above. Ensure you follow the submission guidelines specified by your instructor. /* This program is Assignment #1, Problem #3. The program asks the user to type in an integer between 0 and The program prints the representation of N in base 16. */ #include <stdio.h> int main(void) int dividend=-1,i,num,rem,j,ans[4]=0,temp; /* **** Error checking for number between 0 and **** */ while((dividend<0) (dividend>65535)) printf("enter Number between 0 and 65535: "); scanf("%d",&dividend); if((dividend<0) (dividend>65535)) printf("number %d is out of range... Please Enter again\n",dividend); /* **** Using Divide-by-base approach **** */ num = 16; i=0; while(dividend!=0) temp = dividend/num; rem = dividend%num; dividend = temp; ans[i] = rem; // Storing the result in array i++; printf("ans in BASE-16 is "); for(j=i-1;j>=0;j--) if(ans[j]>9) printf("%c",ans[j]+55); // Print values between 10 and 15 else printf("%d",ans[j]); // Print values between 0 and 9 printf("\n"); return(0); 5