Binary addition

Watch a scoreboard tick from nine up to ten, and you have just watched a carry happen. The nine ran out of digits, so a one stepped left into a brand new column. A computer has only two digits to work with, so that running-out moment arrives much sooner, at one plus one. Binary addition is the whole business of handling a column with nowhere left to put its own answer.

Video six of nine here in Binary and number bases. Everything today rests on writing a number in binary columns, so if that part is wobbly, go and sort out C S oh one, oh two, Denary to binary and back, first.

Four sums cover the whole of binary addition, and three of them you already know. Zero and zero make zero, zero and one make one, and one and zero make one as well. Nothing surprising there, because those three behave exactly as they do in ordinary arithmetic. The fourth sum is one plus one, and in binary that answer cannot be written as a two. Binary has no digit two, in the same way that ordinary numbers have no single digit for ten. So you write a zero in the column you are standing in, and send a one into the column on its left. That one is not lost anywhere. It arrives in a column worth double, and double one is exactly the two you needed. Those four lines are the entire arithmetic of this topic, and the fourth is the only new idea in it.

Once a carry is moving, some columns end up adding three digits rather than two. Two ones from the numbers themselves, plus a one carried in from the right, comes to three. Three in binary is written one one, so you put down a one and pass a one to the left. Which points at your line for this video: count the ones in the column, then write that count in binary. Two in binary is one zero, so down goes the zero and a one goes left. Three in binary is one one, so down goes a one and a one still goes left. One idea covers all five of those lines, including the awkward one, because all of it is counting. Take a column of your own: a one from the top number, a one from the bottom number, and a one carried in. Is it write one carry zero, or write zero carry one, or write one carry one? Take your pick, and hold on to your reason. I'll wait. Three ones counted up make three, and three in binary is one one, so it is write one and carry one. Write one carry zero and the carry has gone; write zero carry one and this column's own one has gone. Get that single column right and the rest of this is the same move, done eight times over.

A real one next: two eight-bit numbers, added with every carry written down. The two numbers are on screen, one above the other, with their columns lined up carefully. Above them sits an empty carry row, with a slot for every column and one spare slot out at the far left. That row is not just decoration. It is the actual working, and it stays right there on the page the whole way through. Start at the right-hand end, in the ones column, exactly as you would with ordinary numbers. One and zero make one, then zero and one make one, then one and zero make one again, three quiet columns in a row. The eights column holds two ones, so that is a zero written down and a one going up into the carry row. In the sixteens column both digits are zero, but the carried one is waiting there, so the answer is a single one and nothing moves on. Pause the video here and finish the last three columns yourself, including the carry row. The thirty-twos column has two ones, so write a zero there and carry a one into the sixty-fours. The sixty-fours column then holds one digit and that carried one, so it is zero again with a one carried on. The last carry lands in the one hundred and twenty-eights column, where both digits are zero, and it comes down as a one. The answer reads one, zero, zero, one, zero, one, one, one, and that pattern is one hundred and fifty-one. One hundred and nine plus forty-two is one hundred and fifty-one, so the pattern and the value agree. The spare slot out on the far left stayed empty, which is how you know the answer still fits inside eight bits. When that slot does not stay empty, you have met overflow, and C S oh one, oh seven, Overflow, is the video that deals with it.

If you are sitting A Q A, there is one extra case to know here. A Q A can ask for three binary numbers added at once. On O C R and Edexcel this next part is a bonus. Three eight-bit numbers go up on screen, still with one carry row above them all. The ones column has a one at the top and a one at the bottom, which is zero carry one. The twos column then holds two ones and the carried one, counting up to three, so it is one carry one. Every column after that works the same way, counting the ones and including the carry each time. The answer comes out as zero one zero one zero zero one zero, which reads as eighty-two. Fifty-one, twenty-two and nine do add to eighty-two, so pattern and value agree again. A Q A caps it there: three numbers at most, and no more than three ones in any single column. Nothing new was needed for that, which is the reward for counting ones instead of memorising cases.

Converting to denary first feels like the safe route, and it is where this topic quietly goes wrong. An examiner's report on O C R paper one, from summer twenty twenty-five, looked at a binary addition question. It opens with good news, because candidates often gained the correct final answer on it. Then comes the line that matters, quoted exactly. Some candidates did this by converting the binary numbers into denary and then converting the result back, which did not gain the working mark for binary addition. So the final answer can be right while the working mark is gone, because the working asked for was binary addition. The detour is slower as well: three separate conversions, against eight columns of counting ones. The same report names a second habit, this time inside the working itself. Quoted exactly. Some candidates did not include the final carry in their working, only showing two carries and writing the final carry direct into the position of the most significant bit. So fill in every slot of the carry row, the last one included, and then still add that final column properly. In the sum we worked through, that final carry is the one that produced the leading one of the answer. The working here is not a courtesy. It is part of the answer, and the carry row is where it lives.

Recap, and it starts with your line for this video: count the ones in the column, then write that count in binary. No ones gives zero, one one gives one, two ones give zero carry one, and three ones give one carry one. Work right to left, and give the carry row a slot for every column, filled in to the very last one. Converting to denary and back can reach the right answer while the working mark for binary addition goes missing. And if you are with A Q A, three numbers at once is on your paper, using that same counting.

One piece of housekeeping now, and it works entirely in your favour. A thumb on a video is you closing that topic off, so give one to anything here you could explain out loud without notes, and whatever stays bare is your revision plan, already written. If carrying has not settled yet, do not fake the tick. Bookmark this one and come back to it fresh, because the columns get much easier once you trust the count.

Next in the chapter: Overflow, which is what a computer does when the answer no longer fits inside eight bits.

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