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Chapter 4: Problem 8

Convert (BADFACED) \(_{16}\) from its hexadecimal expansion to its binary expansion.

Short Answer

Expert verified
The binary expansion of \( (BADFACED)_{16} \) is \( 1011 1010 1101 1111 1010 1100 1110 1101_2 \).

Step by step solution

01

- Write each hexadecimal digit

First, write down each digit of the hexadecimal number separately: B, A, D, F, A, C, E, D.
02

- Convert each digit to binary

Convert each hexadecimal digit to its 4-bit binary equivalent:\(B_{16} = 1011_2\)\(A_{16} = 1010_2\)\(D_{16} = 1101_2\)\(F_{16} = 1111_2\)\(A_{16} = 1010_2\)\(C_{16} = 1100_2\)\(E_{16} = 1110_2\)\(D_{16} = 1101_2\)
03

- Combine binary digits

Combine all the binary representations from each hexadecimal digit in order:\(B A D F A C E D_{16} = 1011 1010 1101 1111 1010 1100 1110 1101_2\)

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Key Concepts

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

hexadecimal system
The hexadecimal system, also known as base-16, is a numeral system that uses sixteen distinct symbols. These symbols include the digits from 0 to 9, and the letters A to F. Hexadecimal is widely used in computing and digital electronics because it can represent large binary numbers in a more readable and concise form.
For example, the hexadecimal number (BADFACED) \(_{16}\) is easier to read and write than its corresponding binary number.
Each hexadecimal digit represents four binary digits (bits), which makes conversion between these systems straightforward.
binary system
The binary system, or base-2, is a numeral system that uses only two symbols: 0 and 1. It is the foundation of all modern computing systems.
Each position in a binary number represents a power of 2, with the rightmost position representing 2^0, the next representing 2^1, and so on.
The binary system is efficient for computers because it matches the digital architecture of electronic circuits, which have two states: on and off.
number conversion steps
Converting a number from hexadecimal to binary involves a few clear steps:
  • Step 1: Write down each hexadecimal digit separately.
  • Step 2: Convert each hexadecimal digit to its 4-bit binary equivalent. For example, B in hexadecimal is 1011 in binary.
  • Step 3: Combine the binary equivalents of all the hexadecimal digits in order. This gives you a single binary string representing the original hexadecimal number.

Let's revisit the example: Convert (BADFACED) \(_{16}\) to binary.
Step 1: Write down the digits: B, A, D, F, A, C, E, D.
Step 2: Convert each digit to binary:
\(B_{16} = 1011_2\)
\(A_{16} = 1010_2\)
\(D_{16} = 1101_2\)
\(F_{16} = 1111_2\)
\(A_{16} = 1010_2\)
\(C_{16} = 1100_2\)
\(E_{16} = 1110_2\)
\(D_{16} = 1101_2\)
Step 3: Combine all the binary pieces: \(BADFACED_{16} = 1011 1010 1101 1111 1010 1100 1110 1101_2\).
This binary number represents the hexadecimal number (BADFACED) \(_{16}\) in the binary system.

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Most popular questions from this chapter

Find the sequence of pseudorandom numbers generated by the power generator with \(p=11, d=2,\) and seed \(x_{0}=3 .\)

Prove that the set of positive rational numbers is countable by showing that the function \(K\) is a one-to- one correspondence between the set of positive rational numbers and the set of positive integers if \(K(m / n)=p_{1}^{2 a_{1}} p_{2}^{2 a_{2}} \cdots \cdots p_{s}^{2 a_{s}} q_{1}^{2 b_{1}-1} q_{2}^{2 b_{2}-1} \ldots \cdots q_{t}^{2 b_{t}-1}\) where gcd \((m, n)=1\) and the prime-power factorizations of \(m\) and \(n\) are \(m=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots \cdot p_{s}^{a_{s}}\) and \(n=q_{1}^{b_{1}} q_{2}^{b_{2}} \cdots q_{t}^{b_{t}}\)

What sequence of pseudorandom numbers is generated using the linear congruential generator \(x_{n+1}=\) \(\left(4 x_{n}+1\right)\) mod 7 with seed \(x_{0}=3 ?\)

Show that 2 is a primitive root of 19

Show that every positive integer can be represented uniquely as the sum of distinct powers of 2 . (Hint: Consider binary expansions of integers.)
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