Quick answer

A carry-out of 1 is passed left when a column sum is 2 or 3; carry-in for the next column equals that carry-out.

Rules

  • Generate: 1+1 or 1+1+1 in a column
  • Propagate: carry_out becomes carry_in for the column on the left
  • Overflow: final carry-out may exceed fixed register width

Introduction

Use the calculator to see overflow when carries exit your selected bit width.

Students who know 1+1=10 still lose points by dropping carry-in in later columns. This page targets that gap.

Review binary addition rules if you need the four local outcomes before studying chains.

Hardware readers can connect carries to binary adder circuits after finishing the manual examples here.

Carry generation and propagation

Carry generation happens when a column sum reaches 2 or 3. Without generation, addition would reduce to bitwise OR in each column.

Propagation means carry_out at bit index i feeds carry_in at index i+1. Errors compound leftward if any link breaks.

Ripple-carry delay in hardware is the time for the worst chain to settle. In homework, you have unlimited time; in CPUs, depth matters.

Common mistakes: forgetting carry-in, misaligning columns so carries apply to wrong powers of two, and ignoring final carry-out in fixed-width answers.

Logic gate applications use full adders with explicit carry_in and carry_out pins that implement the same propagation story.

Worked propagation patterns

  • 0111 + 0001: carries in every column through the low three bits
  • 1111 + 0001 = 10000: fifth bit is the final carry-out
  • 1010 + 1010: carries appear in non-adjacent columns depending on bits

Mark each carry above the column it enters. Color coding helps in notebooks during review sessions.

When multiple carries occur, read the result from least to most significant bit only after the full chain is stable.

Compare ripple patterns with decimal addition of the same values to build intuition for why carries fire more often in binary.

Step-by-step guide

  1. Identify columns that generate carries. Circle any column where both operand bits are 1.
  2. Track carry_in separately from operand bits. A quiet-looking 0+0 column may still output 1 if carry_in is 1.
  3. Follow propagation to the MSB. Stop only when all operand columns and the final carry-out are accounted for.
  4. Interpret overflow for fixed width. Decide whether extra carry-out is discarded, flagged, or stored as an extended bit per assignment rules.
  5. Verify with decimal or calculator. Mismatch often points to the first column where carry_in was wrong.

Worked carry chain

1111 + 0001 = 10000 with carry-out into a fifth column. In 4-bit storage, discuss wrap versus mathematical sum.

1011 + 1011 = 10110 in unlimited width; trace carries through ones, twos, fours, and eights columns step by step.

Simulate incrementing a counter from 0110 to 0111 to 1000 to see how carry patterns change one bit at a time.