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Chapter 5: Problem 5

How many bits are needed to represent each integer. $$ 128 $$

Short Answer

Expert verified
8 bits are needed to represent 128.

Step by step solution

01

Identify the number

The number given is 128.
02

Convert to Binary

Converting the number to its binary form, we follow the process of continuously halving the number and recording the remainder. The binary equivalent for 128 is 10000000.
03

Count the Bits

Counting the places (0's and 1's tallied in binary representation), there are 8 bits in 128.

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Key Concepts

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

Integer Representation
In computing and digital electronics, integer representation is a crucial concept. It's all about how numbers are expressed within a computer system. Unlike the decimal system, which we use in everyday life, computers use the binary system. This means numbers are represented using only two digits: 0 and 1.

These binary digits, or bits, correspond to the presence or absence of a voltage signal in a computer's circuitry. Each digit in a binary number represents a power of two, starting with 2 raised to the power of 0 at the rightmost bit. This method allows computers to efficiently process and store data.
Bit Count
Bit count refers to the number of bits used to represent a specific integer in binary form. Each bit in a binary number can be a 0 or a 1 and contributes to the overall value.

Counting bits is essential to understand how much data storage a particular number requires or how a number will be processed in computational operations. More bits can represent larger numbers. For instance, if an integer requires 8 bits in binary form, this means that any number up to 255 (2^8 - 1) can be represented using 8 bits.
Binary Conversion
Binary conversion is the process of changing a number from the decimal system into its equivalent in the binary system. This process is essential for working with numbers in computers.

To convert a decimal integer to binary:
  • Continuously divide the number by 2.
  • Record the remainder of each division.
  • Read the remainders in reverse order to get the binary equivalent.
For example, converting 128 to binary involves dividing by 2 until the quotient is zero, resulting in a binary representation of 10000000.
Number of Bits
The number of bits required to represent an integer in binary is important for understanding storage and processing requirements in computing. It's determined by the highest power of two that is equal to or less than the number itself.

Here's how you can determine it:
  • Calculate the binary of the number.
  • Count the total number of digits (bits) in the binary format.
For instance, the number 128 is represented as 10000000 in binary, which consists of 8 bits. Understanding the number of bits required helps optimize data storage and ensure efficient computations.

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Most popular questions from this chapter

Add the binary numbers. $$ 101101+11011 $$

Use the following notation and terminology. We let \(E\) denote the set of positive, even integers. If \(n \in E\) can be written as a product of two or more elements in \(E\), we say that \(n\) is \(E\) -composite; otherwise, we say that \(n\) is \(E\) -prime. As examples, 4 is \(E\) -composite and 6 is \(E\) -prime. Is \(10 E\) -prime or \(E\) -composite?

Use the following definition: \(A\) subset \(\left\\{a_{1}, \ldots, a_{n}\right\\}\) of \(\mathbf{Z}^{+}\) is \(a^{*}\) -set of size \(n\) if \(\left(a_{i}-a_{j}\right) \mid a_{i}\) for all \(i\) and \(j,\) where \(i \neq j, 1 \leq i \leq n,\) and \(1 \leq j \leq n .\) These exercises are due to Martin Gilchrist. Prove that for all \(n \geq 2,\) there exists a \(*\) -set of size \(n .\) Hint: Use induction on \(n .\) For the Basis Step, consider the set \\{1,2\\} For the Inductive Step, let \(b_{0}=\prod_{k=1}^{n} a_{k}\) and \(b_{i}=b_{0}+a_{i}\) for \(1 \leq i \leq n\).

Show another way to prove that if a and \(b\) are nonnegative integers, not both zero, there exist integers sand t such that $$ \operatorname{gcd}(a, b)=s a+t b $$ However, unlike the Euclidean algorithm, this proof does not lead to a technique to compute s and \(t\). Show that \(g\) is the greatest common divisor of \(a\) and \(b\).

Find the prime factorization of \(11 !\).
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