The binary number 10101010 is equivalent to the denary (decimal) number 170.
Understanding Binary and Denary Numbers
Number systems are fundamental to computing and mathematics. While humans commonly use the denary (base-10) system, computers primarily operate using the binary (base-2) system.
- Denary (Decimal) System: This system uses ten distinct digits (0 through 9) and is based on powers of 10. Each position in a denary number represents a power of 10 (e.g., units, tens, hundreds, etc.).
- Binary System: This system uses only two distinct digits (0 and 1) and is based on powers of 2. Each position in a binary number represents a power of 2.
Converting Binary to Denary
To convert a binary number to its denary equivalent, you multiply each binary digit by the corresponding power of 2, starting from the rightmost digit (which is position 0). Then, you sum up all the results.
Let's break down the conversion of 10101010:
- Identify Positional Values: Assign powers of 2 to each digit, starting from 20 for the rightmost digit and increasing by one for each position to the left.
- Multiply and Sum: Multiply each binary digit by its corresponding power of 2. Sum these products to get the denary value.
Here's a step-by-step breakdown for 10101010:
| Binary Digit | Position (from right) | Power of 2 (2n) | Calculation | Result |
|---|---|---|---|---|
| 1 | 7 | 27 = 128 | 1 × 128 | 128 |
| 0 | 6 | 26 = 64 | 0 × 64 | 0 |
| 1 | 5 | 25 = 32 | 1 × 32 | 32 |
| 0 | 4 | 24 = 16 | 0 × 16 | 0 |
| 1 | 3 | 23 = 8 | 1 × 8 | 8 |
| 0 | 2 | 22 = 4 | 0 × 4 | 0 |
| 1 | 1 | 21 = 2 | 1 × 2 | 2 |
| 0 | 0 | 20 = 1 | 0 × 1 | 0 |
| Total | Sum | 170 |
As demonstrated, when you sum the results (128 + 0 + 32 + 0 + 8 + 0 + 2 + 0), the total is 170. This process is a standard method for converting any binary number to its denary equivalent, highlighting how 10101010 represents 170 in the decimal system. Understanding these conversions is crucial for anyone working with digital systems or computer science fundamentals.
For further exploration of number systems and their conversions, you can refer to educational resources on binary to decimal conversion.
