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Chapter 1: Problem 44

Convert the integers in \(44-46\) from binary to hexadecimal notation. $$ 00101110_{2} $$

Short Answer

Expert verified
The binary number \(00101110_2\) can be converted to hexadecimal by grouping the digits into sets of four, starting from the right, resulting in the groups \(0010, 1011, 1010\). Converting each group to its corresponding hexadecimal digit using the relationship table gives us \(2, B, A\). Therefore, the equivalent hexadecimal notation is \(2BA\).

Step by step solution

01

Understand the relationship between binary and hexadecimal representations

Binary numbers are base-2 representations, while hexadecimal numbers are base-16 representations. The conversion from binary to hexadecimal can be done by grouping the binary digits into sets of four, starting from the right-most digit. Each group of four binary digits can be represented as a single hexadecimal digit, using the following relationships: - 0000 = 0 - 0001 = 1 - 0010 = 2 - 0011 = 3 - 0100 = 4 - 0101 = 5 - 0110 = 6 - 0111 = 7 - 1000 = 8 - 1001 = 9 - 1010 = A - 1011 = B - 1100 = C - 1101 = D - 1110 = E - 1111 = F We will use these relationships to convert the given binary number into its hexadecimal equivalent.
02

Group the binary digits into sets of four

We start by grouping the given binary number (00101110) into groups of four, starting from the right: $$ 00 \; \underline{1011} \; \underline{1010} $$ Since there are only two digits in the first group, we can add two leading zeros to make it a group of four: $$ \underline{0010} \; \underline{1011} \; \underline{1010} $$
03

Convert each group into hexadecimal digits

Now, we refer to the relationship table mentioned earlier, and convert each group of four binary digits into its corresponding hexadecimal digit: - 0010 = 2 - 1011 = B - 1010 = A
04

Combine the hexadecimal digits

Finally, we combine the hexadecimal digits to form the final equivalent hexadecimal number: $$ 2BA $$ So, the binary number \(00101110_2\) can be represented as \(2BA\) in hexadecimal notation.

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Key Concepts

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

Base-2 to Base-16
When converting from binary (Base-2) to hexadecimal (Base-16), it's important to understand the relationship between these two number systems. Each digit in binary, which only uses 0 and 1, represents exponential values of 2. Meanwhile, hexadecimal uses a larger range of digits: 0-9 and the letters A-F to denote values 10-15.
This means one hexadecimal digit can represent four binary digits (or bits). For example:
  • Binary 0000 converts to Hexadecimal 0.
  • Binary 1001 converts to Hexadecimal 9.
  • Binary 1010 converts to Hexadecimal A.
To convert binary to hexadecimal, break down the binary number into groups of four bits. Each group translates directly into a single hexadecimal digit. This direct mapping makes conversion straightforward. Adding or interpreting these sets creatively maintains accuracy and simplifies calculations.
Hexadecimal Representation
In the hexadecimal system, numbers are represented using a combination of 16 distinct symbols. These are the digits 0 through 9 and the letters A through F. Here's what each letter represents:
  • A = 10
  • B = 11
  • C = 12
  • D = 13
  • E = 14
  • F = 15
Hexadecimal is often used in computing and digital electronics because it efficiently represents binary data. Since each hexadecimal digit corresponds to four binary digits, it reduces the complexity while dealing with large binary numbers.
For example, if you have a long sequence of binary numbers, translating it into hexadecimal form not only simplifies representations but also makes it easier to read and debug large amounts of data.
Number Systems
Number systems are fundamental to understanding how we represent and handle numbers in different bases. The most common number systems include binary (Base-2), decimal (Base-10), and hexadecimal (Base-16). Each system has its own applications and uses in technology and general arithmetic.
  • **Binary** is based on two symbols, 0 and 1. It's primarily used in digital circuits and computing.
  • **Decimal** is our everyday number system with ten symbols (0 through 9), used widely in general counting and arithmetic.
  • **Hexadecimal** is crucial in computing for simplifying binary code, as each hex digit represents four binary bits.
Understanding these systems helps not only in academic contexts but also in practical scenarios, such as programming, data analysis, and more. Recognizing how different bases convert and relate can aid in learning various computer science and engineering concepts.

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Most popular questions from this chapter

A conditional statement is not logically equivalent to its converse.

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 .

Indicate which of the following sentences are statements. a. 1,024 is the smallest four-digit number that is a perfect square. b. She is a mathematics major. c. \(128=2^{6}\) d. \(x=2^{6}\)

Assume \(x\) is a particular real number and use De Morgan's laws to write negations for the statements in 35-38. $$ -2

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 \(\sim\). $$ (p \rightarrow(q \rightarrow r)) \leftrightarrow((p \wedge q) \rightarrow r) $$
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