Convert The Binary Number $101101_2$ To Base 10.

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Introduction

In the world of mathematics, numbers can be represented in various bases, including binary, decimal, and hexadecimal. Binary is a base-2 number system that uses only two digits: 0 and 1. It's a fundamental concept in computer science and is used in the design of digital circuits, computer programming, and data storage. In this article, we'll explore how to convert a binary number to its decimal equivalent.

Understanding Binary Numbers

A binary number is a sequence of 0s and 1s that represents a numerical value. Each digit in a binary number is called a bit. The position of each bit in the sequence determines its value. The rightmost bit is the least significant bit (LSB), and the leftmost bit is the most significant bit (MSB).

The Binary Number $101101_2$

Let's take the binary number $101101_2$ as an example. This number consists of 6 bits: 1, 0, 1, 1, 0, and 1. To convert this number to decimal, we need to understand the value of each bit.

Converting Binary to Decimal

To convert a binary number to decimal, we need to multiply each bit by its corresponding power of 2 and then sum the results. The power of 2 for each bit is determined by its position in the sequence. The rightmost bit has a power of 2^0, the next bit has a power of 2^1, and so on.

Step-by-Step Conversion

Let's convert the binary number $101101_2$ to decimal using the step-by-step process:

  1. Identify the bits: The binary number $101101_2$ consists of 6 bits: 1, 0, 1, 1, 0, and 1.
  2. Determine the powers of 2: The powers of 2 for each bit are:
    • 2^5 (32) for the leftmost bit (1)
    • 2^4 (16) for the next bit (0)
    • 2^3 (8) for the next bit (1)
    • 2^2 (4) for the next bit (1)
    • 2^1 (2) for the next bit (0)
    • 2^0 (1) for the rightmost bit (1)
  3. Multiply each bit by its corresponding power of 2: Multiply each bit by its corresponding power of 2:
    • 1 × 2^5 = 1 × 32 = 32
    • 0 × 2^4 = 0 × 16 = 0
    • 1 × 2^3 = 1 × 8 = 8
    • 1 × 2^2 = 1 × 4 = 4
    • 0 × 2^1 = 0 × 2 = 0
    • 1 × 2^0 = 1 × 1 = 1
  4. Sum the results: Add up the results of the multiplications:
    • 32 + 0 + 8 + 4 + 0 + 1 = 45

Conclusion

In this article, we've explored how to convert a binary number to its decimal equivalent. We've taken the binary number $101101_2$ as an example and used a step-by-step process to convert it to decimal. The result is 45, which is the decimal equivalent of the binary number $101101_2$.

Common Binary to Decimal Conversions

Here are some common binary to decimal conversions:

  • 1010_2$ = 10

  • 1101_2$ = 13

  • 1001_2$ = 9

  • 1111_2$ = 15

  • 1100102110010_2

  • 1010112101011_2

  • 1110012111001_2

  • 1001012100101_2

  • 1011012101101_2

Conclusion

Introduction

In our previous article, we explored how to convert binary numbers to their decimal equivalents. In this article, we'll answer some frequently asked questions about converting binary to decimal.

Q: What is the difference between binary and decimal numbers?

A: Binary numbers are base-2 numbers that use only two digits: 0 and 1. Decimal numbers, on the other hand, are base-10 numbers that use 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Q: How do I convert a binary number to decimal?

A: To convert a binary number to decimal, you need to multiply each bit by its corresponding power of 2 and then sum the results. The power of 2 for each bit is determined by its position in the sequence.

Q: What is the most significant bit (MSB) and the least significant bit (LSB)?

A: The most significant bit (MSB) is the leftmost bit in a binary number, and the least significant bit (LSB) is the rightmost bit.

Q: How do I determine the power of 2 for each bit?

A: The power of 2 for each bit is determined by its position in the sequence. The rightmost bit has a power of 2^0, the next bit has a power of 2^1, and so on.

Q: Can I use a calculator to convert binary to decimal?

A: Yes, you can use a calculator to convert binary to decimal. Most calculators have a built-in binary-to-decimal conversion function.

Q: What are some common binary to decimal conversions?

A: Here are some common binary to decimal conversions:

  • 1010_2$ = 10

  • 1101_2$ = 13

  • 1001_2$ = 9

  • 1111_2$ = 15

  • 1100102110010_2

  • 1010112101011_2

  • 1110012111001_2

  • 1001012100101_2

  • 1011012101101_2

Common Mistakes to Avoid

Here are some common mistakes to avoid when converting binary to decimal:

  • Not understanding the value of each bit
  • Not multiplying each bit by its corresponding power of 2
  • Not summing the results correctly
  • Not using the correct power of 2 for each bit

Conclusion

Converting binary to decimal is a fundamental concept in mathematics and computer science. By understanding the value of each bit and multiplying it by its corresponding power of 2, we can easily convert binary numbers to their decimal equivalents. With practice, you'll become proficient in converting binary to decimal and be able to solve problems with ease.