The binary number 1110001 converts exactly to the decimal number 113.
Converting a binary number to its decimal equivalent is a fundamental concept in digital systems and computer science. Binary (base-2) numbers use only two digits, 0 and 1, while decimal (base-10) numbers use digits from 0 to 9. The conversion process relies on the positional value of each digit in the binary number, which corresponds to powers of 2.
Understanding Binary to Decimal Conversion
Each digit in a binary number, called a bit, holds a specific positional weight. These weights are powers of 2, starting from 20 (which is 1) for the rightmost digit, and increasing by one for each position to the left.
To convert a binary number to decimal, you multiply each binary digit by its corresponding positional weight and then sum the results.
Step-by-Step Conversion of 1110001
Let's break down the conversion of the binary number 1110001 into its decimal form:
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Identify Positions: Starting from the rightmost digit, assign a position number (exponent) to each binary digit, beginning with 0.
- 1 (rightmost) is at position 0 (20)
- 0 is at position 1 (21)
- 0 is at position 2 (22)
- 0 is at position 3 (23)
- 1 is at position 4 (24)
- 1 is at position 5 (25)
- 1 (leftmost) is at position 6 (26)
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Calculate Positional Values: Determine the value of 2 raised to the power of each position.
- 20 = 1
- 21 = 2
- 22 = 4
- 23 = 8
- 24 = 16
- 25 = 32
- 26 = 64
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Multiply and Sum: Multiply each binary digit by its corresponding positional value and then add all the products together.
Here’s a table illustrating the process for 1110001:
| Binary Digit | Position | Power of 2 | Positional Value (Decimal) | Calculation |
|---|---|---|---|---|
| 1 | 6 | 26 | 64 | 1 * 64 = 64 |
| 1 | 5 | 25 | 32 | 1 * 32 = 32 |
| 1 | 4 | 24 | 16 | 1 * 16 = 16 |
| 0 | 3 | 23 | 8 | 0 * 8 = 0 |
| 0 | 2 | 22 | 4 | 0 * 4 = 0 |
| 0 | 1 | 21 | 2 | 0 * 2 = 0 |
| 1 | 0 | 20 | 1 | 1 * 1 = 1 |
Finally, sum all the results:
64 + 32 + 16 + 0 + 0 + 0 + 1 = 113
Thus, the binary number 1110001 is equal to the decimal number 113. This method can be applied to any binary number to find its decimal equivalent.
For a deeper dive into number systems and their conversions, you can explore various online resources such as Understanding Binary Numbers.
