A half adder.

Step 1 of 5

Adding two bits

Adding two bits is a small truth table: $0 + 0 = 0$⁠, $0 + 1 = 1$⁠, $1 + 0 = 1$⁠, and $1 + 1 = 2$⁠. But base 2 has no digit 2, so $2$⁠ is written $10_2$⁠: two digits. Every sum of two bits gives two bits, a sum digit and a carry digit, just as $5 + 5$⁠ in base ten gives 0 and carries 1 into the next column.

The board has rewired the same gates, plus one more AND, into a half adder. The two lights are the carry and the sum, and together they read $A + B$⁠ in base 2. Predict, and then both inputs will go up.

Step 2 of 5

A sum with no carry

The sum column and the carry column differ. Find a setting where only the sum is on.

Step 3 of 5

The carry

Write out the carry column for the four rows, and compare it with the columns you know.

Step 4 of 5

The sum

The sum column is 0, 1, 1, 0: XOR. The board has no XOR gate, so the sum digit $S$⁠ is built from the others, just as in the last Lesson:

$S = (A \text{ OR } B) \text{ AND NOT } (A \text{ AND } B)$$S =$$(A \text{ OR } B) \text{ AND NOT } (A \text{ AND } B)$⁠

Follow the traces: the AND’s output goes to the carry light and also to the NOT, and the NOT and the OR go together into the second AND, which lights the sum. Predict what will block the sum when B goes up.

Step 5 of 5

Half an adder

A half adder adds any two bits correctly. But to add long numbers, column by column, every column except the first also receives a carry from the column to its right. That needs something more.