Add two binary numbers with a full column-by-column carry trace – see exactly where 1+1 writes 0 and carries 1, for your actual inputs.
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Carries, column by column
Binary addition is four rules applied rightward-to-leftward: 0+0=0, 0+1=1, 1+1=0-carry-1, 1+1+1=1-carry-1. This page traces which rule fired in every column of YOUR sum — the fastest route from memorised rules to fluent hand arithmetic.
The trace is also a circuit diagram in disguise: each line is one full adder of a ripple-carry chain, carry-out wired to the next carry-in.
Worked example
10110 + 1101 col 1: 0+1 = 1 col 2: 1+0 = 1 col 3: 1+1 = 0 carry 1 col 4: 0+1+1 = 0 carry 1 col 5: 1+0+1 = 0 carry 1 carry out → 1 Sum: 100011 (22 + 13 = 35 ✓)
| Column a+b+carry | Write | Carry |
|---|---|---|
| 0+0+0 | 0 | 0 |
| 0+1+0 | 1 | 0 |
| 1+1+0 | 0 | 1 |
| 1+1+1 | 1 | 1 |
Reading a wrong hand-sum
Dropped carries make answers too small by a power of two; phantom carries make them too large the same way. Compare your working against the trace column-by-column: the first divergence names the guilty column, which is far faster than re-adding blind.
Length growth and overflow
A final carry grows the sum by one digit — 5-bit operands, 6-bit result. In unbounded arithmetic (here) that is just a new digit; at fixed register width it is the carry flag. Both views describe the same event.
Privacy
The trace is computed locally from your operands and discarded when you leave.
How to use the Binary Addition Calculator
- Enter the two binary addends.
- Click "Add".
- Read the column-by-column trace – each line shows the bits, any carry in, what is written and what carries out.
- The decimal cross-check confirms the sum.
Frequently asked questions
What are the four single-column rules?
0+0=0; 0+1=1; 1+1=0 carry 1; 1+1+1=1 carry 1. Every binary addition of any length is just these four rules repeated right to left – the trace shows which rule fired in each of your columns.
Why does the sum sometimes gain a digit?
A carry out of the leftmost column: 1011 + 110 = 10001 grows to five bits. In fixed-width registers that final carry is the overflow/carry flag – here, with unbounded integers, it simply becomes a new digit.
How is this the same as a hardware adder?
The trace IS a ripple-carry adder on paper: each column is a full adder taking two bits and carry-in, producing sum and carry-out. Faster circuits (carry-lookahead) compute the same answers with rearranged wiring.
Can I add more than two numbers?
Chain the results: add the first pair, then add the third to the sum. Column addition generalises, but two-at-a-time keeps each carry trace readable.
Do the operands need equal length?
No – the shorter is left-padded with zeros automatically, which the first trace line shows. Value is unaffected.
What does a wrong hand-sum usually look like?
A dropped carry – the answer is too small by a power of two. Compare your working column-by-column with the trace and the first divergent column is exactly where the carry vanished.
Is my input transmitted?
No – the trace is computed in the page from your operands and discarded when you leave.
