NumberTally Binary Calculator
Convert binary to decimal and decimal to binary instantly, add binary numbers with full carry working shown, and see the hexadecimal equivalent. Results update instantly. Free, no sign-up.
Advanced options — binary arithmetic
Arithmetic is performed on the binary digits with full working shown step by step.
How Binary Works
Binary is base 2: every position is worth twice the one to its right (…, 8, 4, 2, 1). A 1 means that place value counts; a 0 means it doesn't. So 1010 = 8 + 0 + 2 + 0 = 10. This is the native language of computers — every bit of memory and every CPU operation works in binary.
Binary addition follows the same column rules as decimal, but simpler: 0+0=0, 1+0=1, and 1+1=0 with a carry of 1 to the next column. The carry row in the working below shows exactly where each carry lands.
Binary, Decimal and Hexadecimal Together
Programmers constantly translate between three bases: binary (base 2, the machine), hexadecimal (base 16, compact for humans — one hex digit is exactly 4 bits), and decimal (base 10, everyday counting). The conversion row in the results shows all three side by side so you can see how they line up.
Frequently Asked Questions
How do I convert binary to decimal?
Add up the place values (8, 4, 2, 1, …) wherever there is a 1. For 1010: 8 + 2 = 10.
How do I convert decimal to binary?
Repeatedly divide by 2 and read the remainders bottom-up. 10 → remainders 0, 1, 0, 1 → 1010.
How does binary addition handle carries?
1 + 1 = 10 in binary (0, carry 1). If a carry arrives too, 1 + 1 + 1 = 11 (1, carry 1). The working below traces every column.
Why do programmers use hexadecimal too?
One hex digit maps exactly to 4 bits, so long binary strings become short and readable: 11111111₂ = FF₁₆ = 255₁₀.
What characters are valid in binary?
Only 0 and 1. Anything else is rejected with an error — binary has just two digits by definition.
Can I subtract, multiply or divide in binary?
Yes — open "Advanced options" and pick the operation. Subtraction shows borrows, multiplication shows shift-and-add partial products, and division shows full binary long division with quotient and remainder.