Binary to Decimal
Translate base-2 binary strings into standard base-10 decimal numbers using positional powers of 2. Designed for software developers, electrical engineers, students, and computer architecture researchers requiring deterministic client-side accuracy.
Enter binary digits (0 and 1 only).
Calculated Output
Step-by-Step Mathematical Proof
Active DerivationWhat Is Binary to Decimal?
Binary to decimal conversion calculates the real base-10 numerical value of a binary sequence. In base 2, each column represents an increasing power of two ($2^0, 2^1, 2^2, 2^3, \dots$) moving from right to left.
How Does It Work?
To convert binary to decimal, multiply each binary digit by its corresponding positional weight ($2^i$, where $i$ is the 0-indexed column from right to left) and sum all the products.
Mathematical Algorithm
Step-by-Step Example
Important Rules & Edge Cases
- Input must contain only 0 and 1 digits.
- Leading zeros do not change the numerical value (e.g., 00101 equals 101).
- Each position to the left doubles in mathematical weight.
Practical Applications in Engineering
- Reading raw digital sensor values and converting them to human-readable units.
- Translating CPU program counters and machine codes into decimal instruction numbers.
- Decoding bitwise flags and permission sets in system software.
Common Mistakes to Avoid
- Caution: Starting positional powers from 1 instead of 0 for the rightmost bit.
- Caution: Confusing powers of 2 (e.g., calculating 2^3 as 6 instead of 8).
- Caution: Entering digits other than 0 and 1.
Frequently Asked Questions
What is 1010 in decimal?
1010 in binary equals 10 in decimal: (1×8) + (0×4) + (1×2) + (0×1) = 10.
How do you check if a binary number is odd or even?
Look at the least significant bit (rightmost bit). If it is 1, the number is odd; if it is 0, the number is even.
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