Question

Suppose you want to encrypt the message 10101111 by encrypting the decimal number that corresponds to the message. What is the decimal number?

Short answer

The decimal number is 175.

Step-by-step solution

Understand Binary to Decimal Conversion

The given message is a binary number: 10101111. Each digit in this binary number represents a power of 2, starting from the rightmost digit (least significant bit). The rightmost digit is 2^0, the next is 2^1, and so on.

Identify the Binary Digits and Powers of 2

Write down the binary digits and their corresponding powers of 2: - Leftmost digit (1) corresponds to 2^7 - Second digit (0) corresponds to 2^6 - Third digit (1) corresponds to 2^5 - Fourth digit (0) corresponds to 2^4 - Fifth digit (1) corresponds to 2^3 - Sixth digit (1) corresponds to 2^2 - Seventh digit (1) corresponds to 2^1 - Rightmost digit (1) corresponds to 2^0

Calculate Each Term for the Conversion

Convert each binary digit to its decimal equivalent using its power of 2: - 1 * 2^7 = 128 - 0 * 2^6 = 0 - 1 * 2^5 = 32 - 0 * 2^4 = 0 - 1 * 2^3 = 8 - 1 * 2^2 = 4 - 1 * 2^1 = 2 - 1 * 2^0 = 1

Sum Up All the Calculated Terms

Add all the calculated terms together to get the decimal number:\[128 + 0 + 32 + 0 + 8 + 4 + 2 + 1 = 175\]

Conclusion

The decimal equivalent of the binary number 10101111 is 175. This is the decimal number that corresponds to the message.

Key concepts

Binary Number System

The binary number system is one of the simplest forms of numeral systems and is fundamental in digital electronics and computing. It consists of only two digits: 0 and 1. These are known as bits, which is short for binary digits. Each bit in a binary number represents a power of 2, making it a base-2 numeral system. For example, in the binary number 101, the rightmost digit represents \(2^0\), the next digit represents \(2^1\), and the leftmost digit represents \(2^2\). This structure allows binary numbers to efficiently represent all types of data in electronic devices.

• Binary numbers can express any quantity that can also be represented in the decimal or any other system.
• The simplicity of using only 0 and 1 is ideal for digital systems, which can represent these binary states as on (1) and off (0).

This efficiency is why binary is used in almost all contemporary technology like computers and digital communication.

Decimal Number System

The decimal number system, also known as the base-10 system, is the most widely used number system in the world. It uses 10 different digits: 0 through 9. Just as binary is a base-2 system, the decimal system is a base-10 system. Each position in a decimal number is a power of 10, with the digits gaining weight from right to left. The number 175 in the decimal system, for example, is calculated as \(1 \times 10^2 + 7 \times 10^1 + 5 \times 10^0\).

• Decimal is intuitive for human use because we naturally use a ten-finger counting system.
• It is widely used in everyday arithmetic and transactions.

Although digital devices operate in binary, many user interfaces convert outputs to decimal for easier human interpretation.

Number Conversion Techniques

Number conversion techniques are essential for translating numbers from one numeral system to another, which is vital especially in computer science and electronics. The conversion from binary to decimal, for example, involves interpreting each binary digit as a power of 2.
This process requires you to:

• Identify each bit in the binary number and its corresponding power of 2.
• Convert each bit to its decimal value by multiplying the bit with the power of 2.
• Sum all the decimal values to arrive at the final decimal number.

Illustrated with the binary number 10101111, you convert it to its decimal form (175) by calculating the sum of the powers of 2 that correspond to each bit of 1. Understanding conversion techniques enhances the ability to work across different numeral systems, ensuring effective communication between digital systems and human users.