Add, subtract, and multiply in binary: (a) 1111 and 1010

Convert to hexadecimal and then to binary: (a) \(757.25_{10}\) (b) \(123.17_{10}\) (c) \(356.89_{10}\) (d) \(1063.5_{10}\)

Convert to octal. Convert to hexadecimal. Then convert both of your answers to decimal, and verify that they are the same. (a) \(111010110001.011_2\) (b) \(10110011101.11_2\)

Convert to base \(\text { 6: 3??.}25_{14}\) (do all of the arithmetic in decimal).

(a) Convert to hexadecimal: \(1457.11_{10}\). Round to two digits past the hexadecimal point. (b) Convert your answer to binary, and then to octal. (c) Devise a scheme for converting hexadecimal directly to base 4 and convert your answer to base 4. (d) Convert to decimal: \(\mathrm{DEC.A_{16}}\).

Add, subtract, and multiply in binary: (a) 1111 and 1010 (b) 110110 and 11101 (c) 100100 and 10110

Subtract in binary. Place a 1 over each column from which it was necessary to borrow. (a) 11110100 ? 1000111 (b) 1110110 ? 111101 (c) 10110010 ? 111101

Add the following numbers in binary using 2’s complement to represent negative numbers. Use a word length of 6 bits (including sign) and indicate if an overflow occurs. (a) 21 + 11 (b) (?14) + (?32) (c) (?25) + 18 (d) (?12) + 13 (e) (?11) + (?21) Repeat (a), (c), (d), and (e) using 1’s complement to represent negative numbers.

A computer has a word length of 8 bits (including sign). If 2’s complement is used to represent negative numbers, what range of integers can be stored in the computer? If 1’s complement is used? (Express your answers in decimal.)

Construct a table for 7-3-2-1 weighted code and write 3659 using this code.

Convert to hexadecimal and then to binary. (a) \(1305.375_{10 }\) (b) \(111.33_{10}\) (c) \(301.12_{10}\) (d) \(1644.875_{10}\)

Convert to octal. Convert to hexadecimal. Then convert both of your answers to decimal, and verify that they are the same. (a) \(101111010100.101_{2}\) (b) \(100001101111.01_2\)

(a) Convert to base \(\text { 3: 375.}54_{8}\) (do all of the arithmetic in decimal). (b) Convert to base \(\text { 4: 384.}74_{10}\). (c) Convert to base \(\text { 9: A52.A} 4_{11}\) (do all of the arithmetic in decimal).

Convert to hexadecimal and then to binary: \(544.1_9\).

Convert the decimal number \(97.7_{10}\) into a number with exactly the same value represented in the following bases. The exact value requires an infinite repeating part in the fractional part of the number. Show the steps of your derivation. (a) binary (b) octal (c) hexadecimal (d) base 3 (e) base 5

Devise a scheme for converting base 3 numbers directly to base 9. Use your method to convert the following number to base \(\text { 9: 1110212.}20211_3\)

Convert the following decimal numbers to octal and then to binary: (a) \(2983^{63}/_{64}\) (b) 93.70 (c) \(2983^{31}/_{32}\) (d) 109.30

Add, subtract, and multiply in binary: (a) 1111 and 1001 (b) 1101001 and 110110 (c) 110010 and 11101

Subtract in binary. Place a 1 over each column from which it was necessary to borrow. (a) 10100100 ? 01110011 (b) 10010011 ? 01011001 (c) 11110011 ? 10011110

Divide in binary: (a) \(11101001 \div 101 \) (b) \(110000001 \div 1110 \) (c) \(1110010 \div 1001\) Check your answers by multiplying out in binary and adding the remainder.

Divide in binary: (a) \(10001101 \div 110\) (b) \(110000011 \div 1011 \) (c) \(1110100 \div 1010\)

Assume three digits are used to represent positive integers and also assume the following operations are correct. Determine the base of the numbers. Did any of the additions overflow? (a) 654 + 013 = 000 (b) 024 + 043 + 013 + 033 = 223 (c) 024 + 043 + 013 + 033 = 201

What is the lowest number of bits (digits) required in the binary number approximately equal to the decimal number \(0.6117_{10}\) so that the binary number has the same or better precision?

Convert \(0.363636 \ldots _{10}\) to its exact equivalent base 8 number.

(a) Verify that a number in base b can be converted to base \(b^3\) by partitioning the digits of the base b number into groups of three consecutive digits starting at the radix point and proceeding both left and right and converting each group into a base \(b^3\) digit. (Hint: Represent the base b number using the power series expansion.) (b) Verify that a number in base \(b^3\) can be converted to base b by expanding each digit of the base \(b^3\) number into three consecutive digits starting at the radix point and proceeding both left and right.

(a) Show how to represent each of the numbers (5 ? 1), \((5^2 - 1)\), and \((5^3 - 1)\) as base 5 numbers. (b) Generalize your answers to part (a) and show how to represent \((b^n ? 1)\) as a base b number, where b can be any integer larger than 1 and n any integer larger than 0. Give a mathematical derivation of your result.

(a) Show that the number \(121_b\), where b is any base greater than 2, is a perfect square (i.e., it is equal to the square of some number). (b) Repeat part (a) for the number \(12321_b\), where b > 3. (c) Repeat part (a) for the number \(14641_b\), where b > 6. (d) Repeat part (a) for the number \(1234321_{b}\), where b > 4.

(a) Convert \((0.12)_3\) to a base 6 fraction. (b) Convert \((0.375)_{10}\) to a base 8 fraction. (c) Let \(N = (0.a_{-1}a_{-2} \cdots a_{-m})_R\) be an any base R fraction with at most m nonzero digits. Determine the necessary and sufficient conditions for N to be representable as a base S fraction with a finite number of nonzero digits; say \(N=\left(0 . b_{-1} b_{-2} \cdots b_{-n}\right)_{S}\). (Hint: Part (a) gives an example. Note that \(\left(a_{-1} R^{-1}+a_{-2} R^{-2}+\cdots a_{-m} R^{-m}\right) S^{n}\) must be an integer.) (d) Generalize part (a) to determine necessary and sufficient conditions for a specific, but not every, base R fraction, \(N=\left(0 . a_{-1} a_{-2} \cdots a_{-m}\right)_{R}\), to be representable as a base S fraction with a finite number of nonzero digits.

Construct a table for 4-3-2-1 weighted code and write 9154 using this code.

Is it possible to construct a 5-3-1-1 weighted code? A 6-4-1-1 weighted code? Justify your answers.

Is it possible to construct a 5-4-1-1 weighted code? A 6-3-2-1 weighted code? Justify your answers.

Construct a 6-2-2-1 weighted code for decimal digits. What number does 1100 0011 represent in this code?

Construct a 5-2-2-1 weighted code for decimal digits. What numbers does 1110 0110 represent in this code?

Construct a 7-3-2-1 code for base 12 digits. Write B4A9 using this code.

(a) It is possible to have negative weights in a weighted code for the decimal digits, e.g., 8, 4, ?2, and ?1 can be used. Construct a table for this weighted code. (b) If d is a decimal digit in this code, how can the code for 9 ? d be obtained?

Convert to hexadecimal, and then give the ASCII code for the resulting hexadecimal number (including the code for the hexadecimal point): (a) \(222.22_{10}\) (b) \(183.81_{10}\)

Repeat 1.7 for the following numbers: (a) (?10) + (?11) (b) (?10) + (?6) (c) (?8) + (?11) (d) 11 + 9 (e) (?11) + (?4)

Because A ? B = A + (?B), the subtraction of signed numbers can be accomplished by adding the complement. Subtract each of the following pairs of 5-bit binary numbers by adding the complement of the subtrahend to the minuend. Indicate when an overflow occurs. Assume that negative numbers are represented in 1’s complement. Then repeat using 2’s complement. (a) \(\begin{array}{r} 01001 \\ -11010 \\ \hline \end{array} \) (b) \(\begin{array}{r} 11010 \\ -11001 \\ \hline \end{array}\) (c) \(\begin{array}{r} 10110 \\ -01101 \\ \hline \end{array}\) (d) \(\begin{array}{r} 11011 \\ -00111 \\ \hline \end{array}\) (e) \(\begin{array}{r} 11100 \\ -10101 \\ \hline \end{array}\)

Work Problem 1.37 for the following pairs of numbers: (a) \(\begin{array}{r} 11010 \\ -10100 \\ \hline \end{array}\) (b) \(\begin{array}{r} 01011 \\ -11000 \\ \hline \end{array}\) (c) \(\begin{array}{r} 10001 \\ -01010 \\ \hline \end{array}\) (d) \(\begin{array}{r} 10101 \\ -11010 \\ \hline \end{array}\)

(a) A = 101010 and B = 011101 are 1’s complement numbers. Perform the following operations and indicate whether overflow occurs. (i) A + B (ii) A ? B (b) Repeat part (a) assuming the numbers are 2’s complement numbers.

(a) Assume the integers below are 1’s complement integers. Find the 1’s complement of each number, and give the decimal values of the original number and of its complement. (i) 0000000 (ii) 1111111 (iii) 00110011 (iv) 1000000 (b) Repeat part (a) assuming the numbers are 2’s complement numbers and finding the 2’s complement of them.

An alternative algorithm for converting a base 20 integer, \(d_{n-1} d_{n-2} \cdots d_{1} d_{0}\), into a base 10 integer is stated as follows: Multiply \(d_i\) by \(2^i\) and add i 0’s on the right, and then add all of the results. (a) Use this algorithm to convert \(GA7_{20}\) to base 10. (\(G_{20}\) is \(16_{10}\).) (b) Prove that this algorithm is valid. (c) Consider converting a base 20 fraction, \(0 . d_{-1} d_{-2} \cdots d_{-n+1} d_{-n}\), into a base 10 fraction. State an algorithm analogous to the one above for doing the conversion. (d) Apply your algorithm of part (c) to \(0.FA7_{20}\).

Let A and B be positive integers, and consider the addition of A and B in an n-bit 2’s complement number system. (a) Show that the addition of A and B produces the correct representation of the sum if the magnitude of (A + B) is \(< 2^{n-1} - 1\) but it produces a representation of a negative number of magnitude \(2^n - (A + B)\) if the magnitude of (A + B) is \(> 2^{n-1} - 1\). (b) Show that the addition of A and (?B) always produces the correct representation of the sum. Consider both the case where \(A \geq B\) and the case A < B. (c) Show that the addition of \((2^n - A) + (2^n - B)\), with the carry from the sign position ignored, produces the correct 2’s complement representation of ?(A + B) if the magnitude of A + B is less than or equal to \(2^{n-1}\). Also, show that it produces an incorrect sum representing the positive number \(2^n - (A + B)\) if the magnitude of \((A + B) > 2^{n-1}\).

Let A and B be integers and consider the addition of A and B in an n-bit 1’s complement number system. Prove that addition of A and B using the end-around carry produces the correct representation of the sum provided overflow does not occur. Consider the four cases: A and B both positive, A positive and B negative with the magnitude of A greater than the magnitude of B, A positive and B negative with the magnitude of A less than or equal to the magnitude of B, and A and B both negative.

Prove that in a 2’s complement number system addition overflows if and only if the carry from the sign position does not equal the carry into the sign position. Consider the three cases: adding two positive numbers, adding two negative numbers, and adding two numbers of opposite sign.

Restate the method for detecting overflow of Problem 1.44 so that it applies to 1’s complement numbers.

Let \(B=b_{n-1} b_{n-2} \cdots b_{1} b_{0}\) be an n-bit 2’s complement integer. Show that the decimal value of B is \(-b_{n-1} 2^{n-1}+b_{n-2} 2^{n-2}+b_{n-3} 2^{n-3}+\cdots+b_{1} 2+b_{0}\). (Hint: Consider positive \(\left(b_{n-1}=0\right)\) and negative \(\left(b_{n-1}=1\right)\) numbers separately, and note that the magnitude of a negative number is obtained by subtracting each bit from 1 (i.e., complementing each bit) and adding 1 to the result.)