Binary 110110101 is equal to decimal ________.
437
Understanding Binary to Decimal Conversion
Converting a binary number to its decimal equivalent involves understanding place values. In the binary system (base 2), each digit's position corresponds to a power of 2, starting from the rightmost digit at position 0 (representing \(2^0\)).
Step-by-Step Binary Conversion Process
Let's convert the binary number 110110101 to decimal. We will assign a power of 2 to each digit, starting from \(2^0\) for the rightmost digit and increasing the power by one for each position to the left.
The binary number is 110110101. It has 9 digits. The positions and corresponding powers of 2 are:
- Position 8: \(2^8\)
- Position 7: \(2^7\)
- Position 6: \(2^6\)
- Position 5: \(2^5\)
- Position 4: \(2^4\)
- Position 3: \(2^3\)
- Position 2: \(2^2\)
- Position 1: \(2^1\)
- Position 0: \(2^0\)
Now, we multiply each binary digit by its corresponding power of 2 and sum the results:
| Binary Digit | Position | Power of 2 | Calculation |
|---|---|---|---|
| 1 | 8 | \(2^8 = 256\) | \(1 \times 256 = 256\) |
| 1 | 7 | \(2^7 = 128\) | \(1 \times 128 = 128\) |
| 0 | 6 | \(2^6 = 64\) | \(0 \times 64 = 0\) |
| 1 | 5 | \(2^5 = 32\) | \(1 \times 32 = 32\) |
| 1 | 4 | \(2^4 = 16\) | \(1 \times 16 = 16\) |
| 0 | 3 | \(2^3 = 8\) | \(0 \times 8 = 0\) |
| 1 | 2 | \(2^2 = 4\) | \(1 \times 4 = 4\) |
| 0 | 1 | \(2^1 = 2\) | \(0 \times 2 = 0\) |
| 1 | 0 | \(2^0 = 1\) | \(1 \times 1 = 1\) |
Summing the results from the calculation column:
Decimal value = \(256 + 128 + 0 + 32 + 16 + 0 + 4 + 0 + 1\)
Decimal value = $437$
Therefore, the binary number 110110101 is equal to the decimal number 437.
Revision Table: Binary to Decimal Conversion
| Concept | Description |
|---|---|
| Binary Number System | A base-2 system using only digits 0 and 1. |
| Decimal Number System | A base-10 system using digits 0 through 9. |
| Place Value (Binary) | Each position represents a power of 2 (\(2^0, 2^1, 2^2\), etc., from right to left). |
| Conversion Method | Multiply each binary digit by its corresponding place value (power of 2) and sum the results. |
Additional Information: Number Systems
Number systems are ways of representing numbers. The decimal system is the most common in everyday life, but binary is fundamental in computing because electronic circuits can easily represent two states (like on/off or high/low voltage), corresponding to 1 and 0.
Other common number systems include:
- Octal (Base 8): Uses digits 0-7. Each position represents a power of 8 (\(8^0, 8^1, 8^2\), etc.).
- Hexadecimal (Base 16): Uses digits 0-9 and letters A-F (where A=10, B=11, ..., F=15). Each position represents a power of 16 (\(16^0, 16^1, 16^2\), etc.). Hexadecimal is often used in computing as a shorthand for binary because groups of 4 binary digits can be represented by a single hexadecimal digit.
Conversions between these number systems (binary, decimal, octal, hexadecimal) are important skills in computer science and related fields.
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