Binary Addition Rules: 0+0, 1+1 & Carry-Over

Quick answer

0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0 in the column, carry 1 left).

Introduction

Confirm homework with the calculator after you apply these rules on paper.

The rules are short enough to memorize in one sitting. The skill is applying them under carry pressure when several columns fire in a row.

If you have not read the overview yet, start with what binary addition is so the column layout and place values feel familiar.

Instructors often test the four rules in isolation before multi-digit sums. Treat this page as your reference card during practice sets.

The four rules and carry-over

0 + 0 = 0 with no carry-out. Both inputs are low; the column stays quiet.

0 + 1 = 1 and 1 + 0 = 1 with no carry-out. Exactly one input is high; the sum bit is 1.

1 + 1 = 10 in binary notation: result bit 0 under the column, carry 1 into the next column left.

Carry-over rules: when carry-in is 1, you are effectively adding three bits. Sums of 2 or 3 still write mod-2 under the column and may emit carry-out 1.

Write carries above the next column before you compute that column. Dropped carries are the top mistake in student work.

Rules as a decision table

• carry_out = 1 when (bit_a + bit_b + carry_in) is 2 or 3
• result_bit = (bit_a + bit_b + carry_in) mod 2
• carry_in for column k comes from carry_out of column k-1

The table is the human-friendly view; the formulas are what calculators and adder circuits implement.

Ripple carries happen when many columns see 1+1 or 1+1+1 in sequence. Our dedicated binary addition with carry article walks through chains and overflow.

Once the table is automatic, you can express the same outcomes with XOR and carry formulas when you study logic gates.

Step-by-step guide

1. Write the four lines at the top of your paper. Keep them visible until you no longer pause to recall 1+1.
2. Mark carry-in before each column. Use a small 1 above the column or a running tally in the margin.
3. Compute result bit with mod-2 logic. Even count of 1s yields 0; odd count yields 1 when you include carry-in.
4. Emit carry-out when the column sum is 2 or 3. That carry becomes carry-in for the column to the left.
5. Re-add if a carry was skipped. Every column to the left of a mistake will disagree with the correct answer.

Rule checks with and without carry-in

Ones column of 11 + 01: 1 + 1 = 10, so write 0 and carry 1.

Next column with carry-in 1: if both operand bits are 0, you still write 1 because 0+0+1 = 1.

Practice 0111 + 0001 to see every column generate or absorb a carry.