Binary Addition Calculator

Add two binary (base-2) numbers using bit-by-bit addition with carry propagation, then see the sum in binary, decimal, and hexadecimal.

Quick Facts

Method
Bit-by-bit addition with carry propagation
Same carry rule as decimal addition: 1 + 1 = 10 in binary, so you write 0 and carry the 1 into the next column.

Your Results

Calculated
Binary sum
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A + B in base 2
Decimal equivalent
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Sum converted to base 10
Hexadecimal equivalent
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Sum converted to base 16
Result length
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Bits in the sum

Ready

Enter two binary numbers, then press Calculate.

How Binary Addition Works

Binary addition works exactly like the addition you already know from grade school, except it uses only two digits, 0 and 1, instead of ten. You line the two numbers up by their rightmost (least significant) digit and add column by column, carrying into the next column whenever a column's total is 2 or more.

The bit-by-bit carry method

Each column can only produce one of five outcomes, and every binary addition — no matter how long the numbers are — reduces to applying these rules repeatedly from right to left:

  • 0 + 0 = 0 — no carry.
  • 0 + 1 = 1 and 1 + 0 = 1 — no carry.
  • 1 + 1 = 10 — write 0, carry 1 into the next column.
  • 1 + 1 + 1 (two bits plus an incoming carry) = 11 — write 1, carry 1 into the next column.

If a carry is left over after the leftmost column, it becomes a brand-new leading bit, so the sum of two n-bit numbers can be as long as n + 1 bits.

Worked example

Add 1011 (decimal 11) and 1101 (decimal 13):

  • Column 1 (rightmost): 1 + 1 = 10 → write 0, carry 1.
  • Column 2: 1 + 0 + carry 1 = 10 → write 0, carry 1.
  • Column 3: 0 + 1 + carry 1 = 10 → write 0, carry 1.
  • Column 4: 1 + 1 + carry 1 = 11 → write 1, carry 1.
  • Final carry: 1 becomes a new leading bit.

Reading the results bottom-to-top gives 11000, which is decimal 24 — matching 11 + 13.

Converting the result to decimal

Each position in a binary number represents a power of two, doubling from right to left starting at 2⁰ = 1. To convert, add up the place values wherever there's a 1 bit. For 11000: the 1s sit in the 2⁴ (16) and 2³ (8) positions, so 16 + 8 = 24.

Common errors

  • Forgetting to carry: any column that totals 2 or 3 must carry a 1 into the next column, just like carrying a ten in decimal addition.
  • Misaligning the digits: both numbers must be right-aligned at the last digit before adding, the same way you'd align decimal numbers by their ones place.
  • Using invalid digits: binary numbers contain only 0 and 1 — any 2 through 9 means the input isn't actually binary.

Where binary addition is used

Binary addition is the operation computer processors are built to perform in hardware: chains of full-adder circuits (ripple-carry adders) implement exactly the carry logic above to add integers at the electronic level. It's also a core topic in computer science and digital logic courses, and a quick way to hand-check conversions between binary, decimal, and hexadecimal.

Frequently Asked Questions

How does binary addition work?
Binary addition adds two base-2 numbers column by column from right to left, carrying a 1 into the next column whenever a column's total reaches 2 or 3 - the same carrying idea as decimal addition, just with only two digits, 0 and 1.
What is 1 + 1 in binary?
1 + 1 equals 10 in binary, read as one-zero, not ten. You write down 0 in that column and carry a 1 into the next column. It is the binary way of writing the decimal number 2.
Why can the sum have more digits than either input?
If the leftmost column produces a carry, that carry becomes a new leading bit. Adding two 4-bit numbers can produce a 5-bit sum, for example 1011 + 1101 = 11000.
How do I check a binary sum by hand?
Convert each binary number to decimal by summing the powers of two marked with a 1 bit, add the two decimal values normally, then convert that decimal total back to binary (or convert your binary sum to decimal) and confirm the two match.

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