What Is The Base 10 Representation Of The Number $1100110_2$?A) 94 B) 102 C) 112 D) 126
Understanding Binary and Decimal Representations
In mathematics, we often encounter numbers in different bases, such as binary (base 2) and decimal (base 10). Binary is a base-2 number system that uses only two digits: 0 and 1. Decimal, on the other hand, is the base-10 number system that we commonly use in everyday life. In this article, we will explore how to convert a binary number to its decimal representation.
What is the Binary Number $1100110_2$?
The binary number $1100110_2$ is a sequence of 7 digits, each representing a power of 2. To convert this number to decimal, we need to understand the positional value of each digit.
Converting Binary to Decimal: A Step-by-Step Process
To convert a binary number to decimal, we need to multiply each digit by the corresponding power of 2 and then sum the results. Here's a step-by-step process to convert $1100110_2$ to decimal:
- Identify the positional value of each digit: The binary number $1100110_2$ has 7 digits, each representing a power of 2. The positional values are:
- 2^6 = 64
- 2^5 = 32
- 2^4 = 16
- 2^3 = 8
- 2^2 = 4
- 2^1 = 2
- 2^0 = 1
- Multiply each digit by its positional value: Multiply each digit by its corresponding positional value:
- 1 × 64 = 64
- 1 × 32 = 32
- 0 × 16 = 0
- 0 × 8 = 0
- 1 × 4 = 4
- 1 × 2 = 2
- 0 × 1 = 0
- Sum the results: Add up the results of the multiplications:
- 64 + 32 + 0 + 0 + 4 + 2 + 0 = 102
Conclusion
In conclusion, the base 10 representation of the binary number $1100110_2$ is 102. This is the correct answer among the options provided.
Why is this Important?
Converting binary to decimal is an essential skill in computer science and mathematics. It helps us understand how computers represent numbers and perform arithmetic operations. In addition, it is a fundamental concept in programming languages, where binary numbers are used to represent data and instructions.
Real-World Applications
Converting binary to decimal has numerous real-world applications, including:
- Computer programming: Binary numbers are used to represent data and instructions in programming languages.
- Digital electronics: Binary numbers are used to represent digital signals and perform arithmetic operations in digital circuits.
- Data storage: Binary numbers are used to represent data in storage devices, such as hard drives and solid-state drives.
Common Mistakes to Avoid
When converting binary to decimal, it's essential to avoid common mistakes, such as:
- Misinterpreting the positional value of each digit: Make sure to understand the positional value of each digit and multiply it by the correct power of 2.
- Failing to sum the results: Add up the results of the multiplications carefully to avoid errors.
Conclusion
Frequently Asked Questions
In this article, we will address some of the most common questions related to converting binary to decimal.
Q: What is the binary number $1100110_2$?
A: The binary number $1100110_2$ is a sequence of 7 digits, each representing a power of 2.
Q: How do I convert a binary number to decimal?
A: To convert a binary number to decimal, you need to multiply each digit by the corresponding power of 2 and then sum the results.
Q: What is the positional value of each digit in a binary number?
A: The positional value of each digit in a binary number is determined by the power of 2 it represents. For example, the first digit from the left represents 2^6, the second digit represents 2^5, and so on.
Q: How do I multiply each digit by its positional value?
A: To multiply each digit by its positional value, you need to multiply the digit by the corresponding power of 2. For example, if the digit is 1 and the positional value is 2^6, you would multiply 1 by 64 (2^6).
Q: What is the sum of the results of the multiplications?
A: The sum of the results of the multiplications is the decimal representation of the binary number.
Q: Why is it essential to understand the positional value of each digit?
A: Understanding the positional value of each digit is crucial in converting binary to decimal. If you misinterpret the positional value of each digit, you may end up with an incorrect decimal representation.
Q: What are some common mistakes to avoid when converting binary to decimal?
A: Some common mistakes to avoid when converting binary to decimal include:
- Misinterpreting the positional value of each digit
- Failing to sum the results of the multiplications
- Not understanding the power of 2 represented by each digit
Q: What are the real-world applications of converting binary to decimal?
A: Converting binary to decimal has numerous real-world applications, including:
- Computer programming
- Digital electronics
- Data storage
Q: Why is it essential to understand how computers represent numbers?
A: Understanding how computers represent numbers is essential in computer science and programming. Computers use binary numbers to represent data and instructions, and understanding how to convert binary to decimal is crucial in programming languages.
Q: Can you provide an example of converting a binary number to decimal?
A: Yes, let's consider the binary number $1100110_2$. To convert this number to decimal, we need to multiply each digit by its positional value and then sum the results:
- 1 × 64 = 64
- 1 × 32 = 32
- 0 × 16 = 0
- 0 × 8 = 0
- 1 × 4 = 4
- 1 × 2 = 2
- 0 × 1 = 0
The sum of the results is 102, which is the decimal representation of the binary number $1100110_2$.
Conclusion
In conclusion, converting binary to decimal is a fundamental concept in mathematics and computer science. By understanding the positional value of each digit and multiplying it by the correct power of 2, we can convert binary numbers to their decimal representation. This skill is essential in programming languages, digital electronics, and data storage.