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Chapter 4: Problem 12

Convert \((1100001100011)_{2}\) from its binary expansion to its hexadecimal expansion.

Short Answer

Expert verified
1863

Step by step solution

01

- Group the Binary Digits

Begin by grouping the binary digits into sets of four, starting from the right: (1, 1000, 0110, 0011).
02

- Add Leading Zeros

If a group has fewer than four digits, add leading zeros to make it a complete group of four digits: (0001, 1000, 0110, 0011).
03

- Convert Each Group to Hexadecimal

Next, convert each 4-bit binary group to its hexadecimal equivalent: 0001 鈫� 1 1000 鈫� 8 0110 鈫� 6 0011 鈫� 3.
04

- Combine the Results

Combine all the hexadecimal digits obtained in the previous step to get the final answer: (1, 8, 6, 3), which can be written as 1863.

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Key Concepts

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

binary number
A binary number is a number expressed in the base-2 numeral system. The system uses only two digits, 0 and 1. Each binary digit is known as a bit.
Computers use binary numbers because they can easily represent the two states of a switch (on and off) with 0s and 1s.
Understanding binary numbers is fundamental in fields like computer science and electrical engineering.
The binary numeral system works similarly to the decimal (base-10) system, but with only two digits. Each bit in a binary number represents an increasing power of 2, starting from the rightmost bit, which represents 2^0.
hexadecimal number
A hexadecimal number is a number expressed in the base-16 numeral system. This system uses sixteen distinct symbols: 0-9 to represent values zero to nine, and A-F to represent values ten to fifteen.
Hexadecimal numbering is often used in computing as a more human-friendly representation of binary-coded values. It's easier to read and write large binary numbers when they are expressed in hexadecimal.
For example, the binary number 11111111 can be expressed more conveniently as FF in hexadecimal.
The hexadecimal system works similarly to the decimal and binary systems, but it uses base-16. Each digit represents an increasing power of 16, starting from the rightmost digit: \((16^0, 16^1, 16^2, ...)\).
number system conversion
Number system conversion involves changing a number from one base to another. In particular, converting between binary and hexadecimal is common in computing.
To convert a binary number to hexadecimal:
  • Group the binary digits into sets of four, starting from the right.
  • Add leading zeros if the leftmost group has fewer than four digits.
  • Convert each group to its hexadecimal equivalent.

For example, to convert the binary number 1100001100011 to hexadecimal:
  • Group the digits: (1, 1000, 0110, 0011).
  • Add leading zeros: (0001, 1000, 0110, 0011).
  • Convert each group: 0001 鈫� 1, 1000 鈫� 8, 0110 鈫� 6, 0011 鈫� 3.
  • Combine the results: 1863.
Number system conversions are crucial for understanding different data representations and their uses in technology and mathematics.

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Most popular questions from this chapter

What are the quotient and remainder when a) 44 is divided by 8\(?\) b) 777 is divided by 21\(?\) c) \(-123\) is divided by 19\(?\) d) \(-1\) is divided by 23\(?\) e) \(-2002\) is divided by 87\(?\) f) 0 is divided by 17\(?\) g) \(1,234,567\) is divided by 1001\(?\) h) \(-100\) is divided by 101\(?\)

Prove or disprove that \(p_{1} p_{2} \cdots p_{n}+1\) is prime for every positive integer \(n,\) where \(p_{1}, p_{2}, \ldots, p_{n}\) are the \(n\) smallest prime numbers.

How many divisions are required to find gcd(34, 55) using the Euclidean algorithm?

What are the greatest common divisors of these pairs of integers? $$ \begin{array}{l}{\text { a) } 2^{2} \cdot 3^{3} \cdot 5^{5}, 2^{5} \cdot 3^{3} \cdot 5^{2}} \\ {\text { b) } 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13,2^{11} \cdot 3^{9} \cdot 11 \cdot 17^{14}} \\ {\text { c) } 17,17^{17} \quad \text { d) } 2^{2} \cdot 7,5^{3} \cdot 13} \\ {\text { e) } 0,5 \quad \text { f) } 2 \cdot 3 \cdot 5 \cdot 7,2 \cdot 3 \cdot 5 \cdot 7}\end{array} $$

a) Use Fermat's little theorem to compute \(3^{302}\) mod 5 \(3^{302} \bmod 7,\) and \(3^{302} \bmod 11 .\) b) Use your results from part (a) and the Chinese remainder theorem to find \(3^{302}\) mod \(385 .\) (Note that \(385=5 \cdot 7 \cdot 11 . )\)
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