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Chapter 1: Problem 32

Use 8-bit representations to compute the sums in 31-36. $$ 62+(-18) $$

Short Answer

Expert verified
First, we convert 62 and -18 to 8-bit binary representations: 62 is 00111110, and -18 is 11101110 using the two's complement method. Next, we add the binary representations together: $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c@{}c@{}c@{}c} & & & & \stackrel{1}{}& \stackrel{1}{}& & & \\ & 0 & 0\!1 & 1 & 1 \!1 & \!1 \!0 & 1 & 0 \\ +\!\! & \!1 & 1 \!1 & 0 & 1 \!1 & 0 \!1 & 1 & 0\\ \cline{1-8} & 1 & 0 \!0 & 1 & 0 \!0 & 1 & 0 & 0 \end{array} $$ Ignore the extra bit at the beginning, and the result is 0010100. Convert it back to decimal form: \(2^5 + 2^2 = 32 + 4 = 44\). Therefore, using 8-bit representations, the sum of 62 and -18 is 44.

Step by step solution

01

Convert numbers to 8-bit binary representation

First, we need to convert 62 and -18 into their 8-bit binary representations. 62 in binary is 00111110. For -18, we first find 18 in binary: 00010010 Next, we'll use the two's complement method to find the binary representation of -18: (Invert all bits and add 1) Inverting: 11101101 Adding 1: 11101110 So, 62 in 8-bit binary representation is 00111110, and -18 in 8-bit binary representation is 11101110.
02

Perform binary addition

Now that we have the binary representations, we can add them together: 1 (carry) 00111110 (62) + 11101110 (-18) ---------- 100100100 (result with an extra bit) However, since we are using 8-bit representations, we will ignore the extra bit at the beginning: 0010100 (5th to 1st bit)
03

Convert result to decimal

The final step is to convert the resulting binary number (0010100) back into decimal form: $$ (2^5) + (2^2) = 32 + 4 = 44 $$ So the sum of 62 and -18 using 8-bit representations is 44.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

8-bit binary representations
When working with binary numbers, each digit is a bit that can be either 0 or 1. An 8-bit binary representation uses 8 bits to represent every number. This is a way to handle both positive and negative numbers in computing, providing a total of 256 different values (from 0 to 255 for unsigned values).
For instance, to represent the number 62, we convert its decimal form to binary:
  • Start with the highest power of 2 less than or equal to the number, in this case, 32 (which is \(2^5\)).
  • Continue subtracting to find the next bits: 62 - 32 = 30, with the next highest being \(2^4 = 16\), and so forth until you completely break down the number.
62 in binary becomes 00111110, filling up the 8 bits.
Negative numbers require a slightly different approach, using something called 'two's complement' to fit into an 8-bit structure.
two's complement
Two's complement is a method used in computing to represent negative numbers. With 8-bit representation, it allows for encoding negative numbers seamlessly into binary systems. The process involves a couple of straightforward steps:
  • First, convert the positive value of the number into binary. For example, for 18, the binary is 00010010.
  • Invert all the bits, turning 0s into 1s and vice versa. In this example, it becomes 11101101.
  • Add 1 to the inverted binary number. When you add 1 to 11101101, it becomes 11101110, which is the two's complement for -18.
This method makes subtraction operations unnecessary in binary arithmetic, streamlining calculations involving negative numbers.
binary addition
Binary addition is quite similar to decimal addition but is simpler because it only involves two digits: 0 and 1. Let's look at how to add 62 (00111110) and -18 (11101110) using binary arithmetic:
  • Write the numbers underneath each other, aligning the bits correctly.
  • Start adding from the rightmost bit to the left, just like in decimal addition.
  • If you get a sum of 2 (binary 10), you write down 0 and carry over 1 to the next left bit.
During the addition of 62 and -18, a carry is produced which extends past the 8 bits, resulting in 100100100. Since only the first 8 bits are considered, and you discard the overflow, you're left with 0010100 for the result.
decimal conversion
Converting binary numbers back to decimal is a crucial step to making sense of binary arithmetic in day-to-day terms. After performing binary addition, you're left with a binary number like 0010100, and here鈥檚 how you would convert it to decimal:
  • Identify the positions of the 1s in the binary number. In 0010100, there is a 1 at the positions representing \(2^5\) and \(2^2\).
  • Calculate the value for each of these positions: \(2^5 = 32\) and \(2^2 = 4\).
  • Add these values together to get the decimal equivalent: 32 + 4 = 36.
This shows your binary result corresponds to the decimal number 36. Hence, understanding binary and how to interconvert with decimal forms is essential for working with computer arithmetic.

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Most popular questions from this chapter

In addition to binary and hexadecimal, computer scientists also use octal notation (base 8) to represent numbers. Octal notation is based on the fact that any integer can be uniquely represented as a sum of numbers of the form \(d \cdot 8^{n}\), where each \(n\) is a nonnegative integer and each \(d\) is one of the integers from 0 to 7 . Thus, for example, \(5073_{8}=5 \cdot 8^{3}+0 \cdot 8^{2}+7 \cdot 8^{1}+3 \cdot 8^{0}=2619_{10} .\) a. Convert \(61502_{8}\) to decimal notation. b. Convert \(20763_{8}\) to decimal notation. c. Describe methods for converting integers from octal to binary notation and the reverse that are similar to the methods used in Examples 1.5.12 and 1.5.13 for converting back and forth from hexadecimal to binary notation. Give examples showing that these methods result in correct answers.

Let \(p\) be the statement "DATAENDFLAG is off,"' \(q\) the statement "ERROR equals 0 ," and \(r\) the statement "SUM is less than 1,000." Express the following sentences in symbolic notation. a. DATAENDFLAG is off, ERROR equals 0 , and SUM is less than \(1,000 .\) b. DATAENDFLAG is off but ERROR is not equal to \(0 .\) c. DATAENDFLAG is off; however ERROR is not 0 or SUM is greater than or equal to 1,000 . d. DATAENDFLAG is on and ERROR equals 0 but SUM is greater than or equal to 1,000 . e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM is less than 1,000 .

In 38-41 (a) use the logical equivalences \(p \rightarrow q \equiv \sim p \vee q\) and \(p \leftrightarrow q \equiv(\sim p \vee q) \wedge(\sim q \vee p)\) to rewrite the given statement forms without using the symbol \(\rightarrow\) or \(\leftrightarrow\), and (b) use the logical equivalence \(p \vee q \equiv \sim(\sim p \wedge \sim q)\) to rewrite each statement form using only \(\wedge\) and \(\sim\). $$ (p \rightarrow(q \rightarrow r)) \leftrightarrow((p \wedge q) \rightarrow r) $$

Represent the decimal integers in 1-6 in binary notation. 1609

Rewrite the statements in \(1-4\) in if-then form. Fix my ceiling or I won't pay my rent.
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