Building Arithmetic

Ripple-Carry Addition

Four one-bit adders make a small binary addition.

logic-computation Every row is intentionally ordered and pinned to the lesson specification.

Example

Four one-bit adders make a small binary addition.

highlighted = computed this step

Build the concrete table

Compute the highlighted logic-table value.

bit 0: a=1 b=1 carry_in=0total=2 sum=0 carry_out=1\begin{array}{c|c}\text{bit 0: a=1 b=1 carry\_in=0}&\hlmath{\text{total=2 sum=0 carry\_out=1}}\end{array}

Build the concrete table

Compute the highlighted logic-table value.

bit 0: a=1 b=1 carry_in=0total=2 sum=0 carry_out=1bit 1: a=1 b=0 carry_in=1total=2 sum=0 carry_out=1\begin{array}{c|c}\text{bit 0: a=1 b=1 carry\_in=0}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 1: a=1 b=0 carry\_in=1}&\hlmath{\text{total=2 sum=0 carry\_out=1}}\end{array}

Build the concrete table

Compute the highlighted logic-table value.

bit 0: a=1 b=1 carry_in=0total=2 sum=0 carry_out=1bit 1: a=1 b=0 carry_in=1total=2 sum=0 carry_out=1bit 2: a=1 b=1 carry_in=1total=3 sum=1 carry_out=1\begin{array}{c|c}\text{bit 0: a=1 b=1 carry\_in=0}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 1: a=1 b=0 carry\_in=1}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 2: a=1 b=1 carry\_in=1}&\hlmath{\text{total=3 sum=1 carry\_out=1}}\end{array}

Build the concrete table

Compute the highlighted logic-table value.

bit 0: a=1 b=1 carry_in=0total=2 sum=0 carry_out=1bit 1: a=1 b=0 carry_in=1total=2 sum=0 carry_out=1bit 2: a=1 b=1 carry_in=1total=3 sum=1 carry_out=1bit 3: a=0 b=0 carry_in=1total=1 sum=1 carry_out=0\begin{array}{c|c}\text{bit 0: a=1 b=1 carry\_in=0}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 1: a=1 b=0 carry\_in=1}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 2: a=1 b=1 carry\_in=1}&\text{total=3 sum=1 carry\_out=1}\\\text{bit 3: a=0 b=0 carry\_in=1}&\hlmath{\text{total=1 sum=1 carry\_out=0}}\end{array}

Build the concrete table

Compute the highlighted logic-table value.

bit 0: a=1 b=1 carry_in=0total=2 sum=0 carry_out=1bit 1: a=1 b=0 carry_in=1total=2 sum=0 carry_out=1bit 2: a=1 b=1 carry_in=1total=3 sum=1 carry_out=1bit 3: a=0 b=0 carry_in=1total=1 sum=1 carry_out=0final carrycarry_out=0\begin{array}{c|c}\text{bit 0: a=1 b=1 carry\_in=0}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 1: a=1 b=0 carry\_in=1}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 2: a=1 b=1 carry\_in=1}&\text{total=3 sum=1 carry\_out=1}\\\text{bit 3: a=0 b=0 carry\_in=1}&\text{total=1 sum=1 carry\_out=0}\\\text{final carry}&\hlmath{\text{carry\_out=0}}\end{array}

Read the table verdict

Compute the highlighted logic-table value.

bit 0: a=1 b=1 carry_in=0total=2 sum=0 carry_out=1bit 1: a=1 b=0 carry_in=1total=2 sum=0 carry_out=1bit 2: a=1 b=1 carry_in=1total=3 sum=1 carry_out=1bit 3: a=0 b=0 carry_in=1total=1 sum=1 carry_out=0final carrycarry_out=0verdict0111 + 0101 = 1100\begin{array}{c|c}\text{bit 0: a=1 b=1 carry\_in=0}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 1: a=1 b=0 carry\_in=1}&\text{total=2 sum=0 carry\_out=1}\\\text{bit 2: a=1 b=1 carry\_in=1}&\text{total=3 sum=1 carry\_out=1}\\\text{bit 3: a=0 b=0 carry\_in=1}&\text{total=1 sum=1 carry\_out=0}\\\text{final carry}&\text{carry\_out=0}\\\text{verdict}&\hlmath{\text{0111 + 0101 = 1100}}\end{array}