Students will need to be able to add together up to three binary numbers…

The idea

Binary addition works column by column from the right, exactly like decimal addition but with a base of 2. Each column can hold only 0 or 1, so when a column total reaches 2 you write 0 and carry 1 to the next column; a total of 3 writes 1 and carries 1. With up to three 8-bit addends, a column can receive carries from the column to its right, so its total can be as high as 5 (three 1s plus a carry of 2). Work right to left, record each column total, and keep every carry. The largest possible sum is 255 + 255 + 255 = 765, which needs 10 bits, so the answer may be wider than 8 bits.

What earns the marks

• Aligns the addends so that the least significant bits are in the same column.
• Adds each column from right to left, including any carry from the previous column.
• Applies the rule that a column total of 2 gives 0 with a carry of 1, and a total of 3 gives 1 with a carry of 1.
• Handles a column total of up to 5 when three addends plus an incoming carry meet in one column.
• Records the final carry as an extra bit rather than discarding it.
• Recognises that the maximum sum of three 8-bit numbers, 765, requires 10 bits.

Common mistakes and how to put them right

• Forgetting to add the carry into the next column.

  Put it right: Include the carry in the column total before deciding the digit and the new carry.

• Assuming a column total can never exceed 4.

  Put it right: With three addends and an incoming carry of 2, a column total can reach 5.

• Discarding the final carry so the answer is truncated to 8 bits.

  Put it right: Keep the extra bit, since 255 + 255 + 255 = 765 needs 10 bits.

Exam tips

• Write the addends in a column grid with place values labelled so carries have a clear home.
• Work strictly right to left and jot each carry above the next column before adding it.
• Check the result by converting to denary and adding, then converting back.