To convert a number from any base to base 10 (decimal), you multiply each digit by its corresponding place value (which is the base raised to the power of the digit's position) and then sum the results.
Here's a breakdown with examples:
The Process:
- Identify the Base: Know the base of the number you are converting (e.g., base 2, base 8, base 16).
- Identify the Digits and their Positions: Start from the rightmost digit, which is position 0. Move left, incrementing the position by 1 for each digit.
- Multiply and Sum: Multiply each digit by (base ^ position). Sum up all the results.
Examples:
Example 1: Converting Binary (Base 2) to Decimal (Base 10)
Let's convert the binary number 1011012 to base 10:
| Digit | Position | Calculation | Result |
|---|---|---|---|
| 1 | 5 | 1 * (25) | 32 |
| 0 | 4 | 0 * (24) | 0 |
| 1 | 3 | 1 * (23) | 8 |
| 1 | 2 | 1 * (22) | 4 |
| 0 | 1 | 0 * (21) | 0 |
| 1 | 0 | 1 * (20) | 1 |
| Total: | 45 |
Therefore, 1011012 = 4510.
Example 2: Converting Octal (Base 8) to Decimal (Base 10)
Let's convert the octal number 3728 to base 10:
| Digit | Position | Calculation | Result |
|---|---|---|---|
| 3 | 2 | 3 * (82) | 192 |
| 7 | 1 | 7 * (81) | 56 |
| 2 | 0 | 2 * (80) | 2 |
| Total: | 250 |
Therefore, 3728 = 25010.
Example 3: Converting Hexadecimal (Base 16) to Decimal (Base 10)
Hexadecimal uses digits 0-9 and letters A-F, where A=10, B=11, C=12, D=13, E=14, and F=15.
Let's convert the hexadecimal number 2A16 to base 10:
| Digit | Position | Calculation | Result |
|---|---|---|---|
| 2 | 1 | 2 * (161) | 32 |
| A (10) | 0 | 10 * (160) | 10 |
| Total: | 42 |
Therefore, 2A16 = 4210.
Summary
Converting from any base to base 10 involves understanding place values and using the formula: (digit * baseposition), then summing all the calculated values. This straightforward method allows you to easily translate numbers from different bases into the familiar decimal system.
