Binary arithmetic: Be able to add together up to three binary numbers.

The idea

Binary addition works column by column from the least significant bit (rightmost), carrying into the next column when a column total reaches 2 or more. With two numbers a column can total 0, 1 or 2; with three numbers it can total 0, 1, 2 or 3. Record the total modulo 2 (the bit that stays) and carry the total divided by 2 (the carry). For example, adding 0110 and 0011: rightmost column 0+1=1, write 1 carry 0; next 1+1=2, write 0 carry 1; next 1+0+1=2, write 0 carry 1; leftmost 0+0+1=1, write 1, giving 1001. Always check the result by converting to denary: 6+3=9, and 1001₂=9. Extend the same carry method to three addends, tracking the running carry carefully.

What earns the marks

• Adds column by column starting at the least significant bit and works leftwards.
• Applies the rule: column total 0 or 1 gives that bit with carry 0; total 2 gives bit 0 carry 1; total 3 gives bit 1 carry 1.
• Correctly handles a carry into a column that already contains two or three addend bits.
• Produces a correct final binary sum and can verify it by converting addends and result to denary.
• Manages three addends by summing all three bits plus any incoming carry in each column.

Common mistakes and how to put them right

• Adding left to right like denary reading order.

  Put it right: Begin at the least significant (rightmost) bit and carry leftwards.

• Forgetting the incoming carry when a column already totals 2 or 3.

  Put it right: Include the carry as an extra value in the column total.

• Writing a column total of 2 as the digit 2.

  Put it right: Binary digits are only 0 or 1, so write 0 and carry 1.

Exam tips

• Lay the numbers out in aligned columns and mark carries above the next column to avoid losing them.
• Convert to denary to check your answer quickly before moving on.
• For three addends, total all bits plus carry in one step, then split into result bit and carry.