Problem 7 - Discrete Mathematics with Applications

Question

Find the number of bytes that: Contain exactly eight 0's.

Step-by-Step Solution

Answer

There is only \(1\) byte that contains exactly eight 0's, and that is \(00000000\).

Step 1: Understand Byte and Bit Structure

A byte consists of 8 bits, where each bit can be either 0 or 1. In this case, we are looking for bytes that have exactly eight 0's, which means all the bits in the byte are 0's.

Step 2: Determine the Possible Byte Values

Since we are looking for bytes containing exactly eight 0's, there is only one possible value for the byte: 00000000. All other bytes will contain at least one 1.

Step 3: Calculate the Number of Bytes With Eight 0's

As we determined in Step 2, there is only one possible byte value containing exactly eight 0's: 00000000. Therefore, the number of bytes containing exactly eight 0's is 1.

Key Concepts

Byte and Bit Structure

In digital computing, the building blocks of data are bits and bytes. A bit is the most basic unit and can have a value of either 0 or 1. Bits are the DNA of digital data, dictating the binary language that computers understand. When we combine eight of these bits, we form what is known as a byte. The structure of a byte is significant because each bit within the byte can vary independently, leading to different possible combinations.

For example, a single byte can be structured as follows:

• 00000000
• 00000001
• 00000010
• ... all the way up to
• 11111111

As you can notice, the leftmost bit is the least significant, and the rightmost bit is the most significant. This is crucial to understand because the position of the bit within a byte affects its value when calculating the overall value of the byte.

Calculating Byte Values

To comprehend how byte values are calculated, one must first understand the binary number system. In binary, each bit position represents a power of 2, with the rightmost position being the zeroth power. Each bit's value is calculated as 2 to the power of its position, multiplied by the bit value (0 or 1).

Let's calculate the value of a byte:

• The rightmost bit (least significant bit) has a value of \(2^0 = 1\).
• The next bit to the left has a value of \(2^1 = 2\).
• This continues until the leftmost bit (most significant bit), which has a value of \(2^7 = 128\).

Therefore, to find a byte's total value, one would add up the values of each bit. For instance, the byte 00000001 has a value of 1, and the byte 10000000 has a value of 128.

Discrete Mathematics Problems

Discrete mathematics explores topics and problems that involve discrete structures, rather than continuous. This branch of mathematics is particularly relevant to computer science because it deals with distinct values, much like how computers operate with binary digits. When working on discrete mathematics problems, we often use combinatorics, logic, set theory, and graph theory.

In the context of finding the number of bytes containing exactly eight 0's, we're really looking at a combinatorial problem. We have eight positions to fill with zeroes, and we're exploring all possible combinations. Since we're looking for the special case where all bits are 0, it doesn't matter how we arrange them—there is only one way. This simplifies to the problem of counting the number of ways we can arrange eight identical objects (0's), which is, of course, just one way.