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Binary to Decimal Converter



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Converting a Binary Number to Decimal

Binary numbers are the foundation of all modern computing systems. The binary number system, also known as base-2, uses only two digits: 0 and 1. Each digit in a binary number is referred to as a bit, and the position of each bit determines its value in the overall number, similar to the way place value works in the decimal system (base-10).

To convert a binary number to its decimal equivalent, you must understand the positional value of each bit. Starting from the rightmost bit (the least significant bit), each position in a binary number represents a power of 2, beginning with \(2^0\) and increasing as you move to the left. The value of the binary number is the sum of these powers of 2, for each position where there is a 1.

For example, the binary number 1101 can be converted to decimal by calculating:

\(1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\)

This method is the standard approach to converting binary to decimal.

Examples

Let’s explore some examples to understand how to convert binary numbers to their decimal equivalents.

1. Convert the binary number 1101 to decimal.

Answer

First, we identify the given number:

Given:

  • Binary: 1101

Next, we calculate the decimal value by applying the powers of 2 to each bit:

Steps:

  • Starting from the rightmost bit, we assign powers of 2: \(2^3\), \(2^2\), \(2^1\), \(2^0\).
  • Calculate each term where the binary digit is 1:
    • \(1 \times 2^3 = 8\)
    • \(1 \times 2^2 = 4\)
    • \(0 \times 2^1 = 0\)
    • \(1 \times 2^0 = 1\)
  • Sum these values to get the decimal equivalent:
    • \(8 + 4 + 0 + 1 = 13\)

The final decimal value is:

Decimal Equivalent:

13

Result:

∴ The decimal equivalent of the binary number 1101 is 13.

2. Convert the binary number 1010 to decimal.

Answer

We start by identifying the number:

Given:

  • Binary: 1010

Next, we calculate the decimal value by applying the powers of 2 to each bit:

Steps:

  • Assign powers of 2: \(2^3\), \(2^2\), \(2^1\), \(2^0\).
  • Calculate each term where the binary digit is 1:
    • \(1 \times 2^3 = 8\)
    • \(0 \times 2^2 = 0\)
    • \(1 \times 2^1 = 2\)
    • \(0 \times 2^0 = 0\)
  • Sum these values to get the decimal equivalent:
    • \(8 + 0 + 2 + 0 = 10\)

The final decimal value is:

Decimal Equivalent:

10

Result:

∴ The decimal equivalent of the binary number 1010 is 10.

3. Determine the decimal equivalent of the binary number 11111.

Answer

First, we identify the given number:

Given:

  • Binary: 11111

Next, we calculate the decimal value by applying the powers of 2 to each bit:

Steps:

  • Assign powers of 2: \(2^4\), \(2^3\), \(2^2\), \(2^1\), \(2^0\).
  • Calculate each term where the binary digit is 1:
    • \(1 \times 2^4 = 16\)
    • \(1 \times 2^3 = 8\)
    • \(1 \times 2^2 = 4\)
    • \(1 \times 2^1 = 2\)
    • \(1 \times 2^0 = 1\)
  • Sum these values to get the decimal equivalent:
    • \(16 + 8 + 4 + 2 + 1 = 31\)

The final decimal value is:

Decimal Equivalent:

31

Result:

∴ The decimal equivalent of the binary number 11111 is 31.

4. Convert the binary number 10001 to decimal.

Answer

We start by identifying the number:

Given:

  • Binary: 10001

Next, we calculate the decimal value by applying the powers of 2 to each bit:

Steps:

  • Assign powers of 2: \(2^4\), \(2^3\), \(2^2\), \(2^1\), \(2^0\).
  • Calculate each term where the binary digit is 1:
    • \(1 \times 2^4 = 16\)
    • \(0 \times 2^3 = 0\)
    • \(0 \times 2^2 = 0\)
    • \(0 \times 2^1 = 0\)
    • \(1 \times 2^0 = 1\)
  • Sum these values to get the decimal equivalent:
    • \(16 + 0 + 0 + 0 + 1 = 17\)

The final decimal value is:

Decimal Equivalent:

17

Result:

∴ The decimal equivalent of the binary number 10001 is 17.