Binary number 101110110 is equal to decimal number _______.
374
Binary to Decimal Conversion of 101110110
Converting a binary number to a decimal number involves multiplying each digit by the corresponding power of 2 and summing the results. The powers of 2 start from \(2^0\) for the rightmost digit and increase by one for each position to the left.
Let's convert the binary number $101110110$ to its decimal equivalent.
The binary number is $101110110$. We can write out the digits and their positions, starting from position 0 on the right:
- The rightmost digit (0) is at position 0, corresponding to \(2^0\).
- The next digit (1) is at position 1, corresponding to \(2^1\).
- The next digit (1) is at position 2, corresponding to \(2^2\).
- The next digit (0) is at position 3, corresponding to \(2^3\).
- The next digit (1) is at position 4, corresponding to \(2^4\).
- The next digit (1) is at position 5, corresponding to \(2^5\).
- The next digit (1) is at position 6, corresponding to \(2^6\).
- The next digit (0) is at position 7, corresponding to \(2^7\).
- The leftmost digit (1) is at position 8, corresponding to \(2^8\).
Now, we multiply each binary digit by the value of the power of 2 for its position:
| Binary Digit | Position | Power of 2 | Calculation | Result |
|---|---|---|---|---|
| 1 | 8 | \(2^8 = 256\) | \(1 \times 256\) | 256 |
| 0 | 7 | \(2^7 = 128\) | \(0 \times 128\) | 0 |
| 1 | 6 | \(2^6 = 64\) | \(1 \times 64\) | 64 |
| 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 |
| 1 | 1 | \(2^1 = 2\) | \(1 \times 2\) | 2 |
| 0 | 0 | \(2^0 = 1\) | \(0 \times 1\) | 0 |
Finally, we sum the results from the last column:
Decimal value = \(256 + 0 + 64 + 32 + 16 + 0 + 4 + 2 + 0\)
Decimal value = $374$
Therefore, the binary number $101110110$ is equal to the decimal number $374$.
Revision Table: Binary Number Conversion
| Concept | Description | Example |
|---|---|---|
| Binary System | Base-2 number system using only digits 0 and 1. | \(101_2\) |
| Decimal System | Base-10 number system using digits 0 through 9. | \(5_{10}\) |
| Binary to Decimal Conversion | Sum of (digit * \(2^{\text{position}}\)) for each digit. | \(101_2 = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5_{10}\) |
Additional Information on Number Systems
Number systems are ways of representing numbers. The most common system we use daily is the decimal system (base 10). Computers and digital electronics primarily use the binary system (base 2).
- Base: The base of a number system indicates the number of unique digits used (including zero) and is the value of the base raised to the power of the position. For example, the decimal system has a base of 10.
- Place Value: In any positional number system, the position of a digit determines its value. This value is the digit multiplied by the base raised to the power of the position index. For example, in \(123_{10}\), the '2' is in the tens place (\(10^1\)), so its value is \(2 \times 10^1 = 20\). In \(101_2\), the middle '0' is in the \(2^1\) place, so its value is \(0 \times 2^1 = 0\).
- Other Systems: Besides binary and decimal, other number systems like Octal (base 8) and Hexadecimal (base 16) are also used, particularly in computing and digital electronics. Octal uses digits 0-7, while Hexadecimal uses digits 0-9 and letters A-F (representing 10-15).
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