Adding and subtracting decimals by aligning place value

0:00Here are two lengths of cloth, cut from the same roll. One of them is two point seven metres. The other is three point five. Two questions. How much cloth is there altogether, and how much longer is the second than the first? You could reach for a rule about lining the decimal points up. Instead, let us do the sum twice over, and watch what is actually going on.

0:27First, forget the point entirely. Two point seven metres is twenty seven tenths of a metre. And three point five metres is thirty five tenths. So the question has become: what is twenty seven tenths plus thirty five tenths? And that is a sum of whole numbers. Twenty seven plus thirty five. Seven and five is twelve. Write the two, carry the one. Two and three and the carried one is six. Sixty two.

0:59Sixty two tenths, and nothing about that was new to you. Now write the very same sum with the point, one number above the other. Seven tenths and five tenths is twelve tenths. Write the two, carry the one. Two units and three units and the carried one is six units. Six point two. Now look at those two workings side by side. Not similar. Identical. The same column totals, the same digit written down, the same carry, in the same place.

1:34So what is that carried one actually doing? After the first column you have five units and twelve tenths. And twelve tenths is more tenths than will fit inside one unit. So take ten of them and trade them for one whole unit. Ten tenths, gathered up, become one unit, and that unit goes into the units column. Which leaves six units and two tenths. Six point two metres of cloth.

2:03The carried one is not a trick that somebody invented. It is the ten for one trade, the same one running through every place in this system. It happens to be written very small above the column, but that is all that is small about it. Now the second question. How much longer is three point five than two point seven? Start at the tenths. Five tenths take away seven tenths.

2:32There are not enough tenths there, so go and get some more. Break one whole unit of the three point five open into ten tenths. Now it reads two units and fifteen tenths, which is the same length written differently. Fifteen tenths take away seven tenths is eight tenths. Two units take away two units is nothing at all. Nought point eight metres. And the borrow is exactly the carry, run backwards. One unit broken into ten tenths, instead of ten tenths gathered into one.

3:09Which brings us to the point of all this, and it is a strange thing to have to say out loud. Nothing new has been taught. The method you use for whole numbers is the method you use for decimals. It was not modified. It was reused. And it works for exactly the reason it always worked. Column arithmetic adds like to like. Tenths to tenths, units to units, hundredths to hundredths.

3:38The digits have to be in the right columns for that, and nothing else about it matters. So the only real question left is this. What puts a digit in the right column? The answer is the point, and here that is the only job it has. It marks where the units place ends, which tells you which columns are alike. Line the two points up, one directly above the other, and every column becomes a column of one size of part.

4:10Draw that vertical line and leave it there. Both points sit on it, and so does the answer's. Now here is what happens if you line the numbers up at their right hand ends instead. Eighteen plus eight point eight. Push them both hard to the right and the working gives you ten point six. The real answer is twenty six point eight. That is not a slip in one digit. It is less than half the size, and it is smaller than one of the two numbers you started with.

4:45Now a bigger one, written out in full, because the full version says something the compact one hides. Seventy five point three four five, plus eighty six point six nine one. Write every place out on its own line. Seven tens with eight tens. Five units with six units. Three tenths with six tenths. Four hundredths with nine hundredths. Five thousandths with one thousandth. Now add each column, starting from the right, and let each carry land in the next one along.

5:21Five thousandths and one thousandth is six thousandths. Nothing to trade there. Four hundredths and nine hundredths is thirteen hundredths, so ten of those go up as one tenth. And on it goes, leftwards, one column at a time. Here is the finished row of column totals. Sixteen tens, twelve units, ten tenths, thirteen hundredths, six thousandths. And here is something worth being careful about. Seven tens and eight tens is fifteen tens, not sixteen.

5:57The sixteenth ten is the one arriving from the units column next door. Every number in that row already has its neighbour's carry sitting inside it. Now trade them. Twelve units is one ten and two units. Ten tenths is one unit and no tenths. Thirteen hundredths is one tenth and three hundredths, and sixteen tens is one hundred and six tens. One hundred and sixty two point nought three six, and every carry in it is a ten for one trade you can put your finger on.

6:35One more, because this one catches almost everybody. Nine point nine, take away nine point nought nine. The tempting answer is nought point nine, and the reasoning sounds perfectly sensible. Nine take away nine is nothing, and nine take away nought is nine. But look at which columns those digits are actually sitting in. The first number has nine tenths and nothing in the hundredths. The second has no tenths and nine hundredths.

7:07So the hundredths column reads nothing take away nine hundredths, and there is nothing there to take it from. Borrow a tenth, break it open into ten hundredths, and ten take away nine leaves one hundredth. Nought point eight one. The gap between that and the tempting answer is nine hundredths. A different use of the same arithmetic. Here is a list of numbers. Four point four, four point eight, five point two, five point six, six point nought.

7:42Each one comes from the one before by adding the same amount, and finding that amount is a subtraction. Four point eight take away four point four is nought point four. So is every other gap along the list. So the step is four tenths, and now you can run it forwards as far as you like. Six point four, six point eight, seven point two. Steps do not have to be small, either.

8:10One list runs thirteen point five, sixteen, eighteen point five. That is a step of two and a half. And they do not have to climb. Five, four point nine five, four point nine, is falling by five hundredths at a time. A falling list is still a list with a step. The step is simply a subtraction instead of an addition. Last of all, a claim that somebody makes, and it is worth taking seriously before you believe it.

8:44Add two decimal numbers, they say, and the total always comes out above the sum of the two whole number parts, and always below that plus two. Try it. Twenty five point nine three six, plus eight point two nought two. The whole parts are twenty five and eight, so the claim is that the answer lands between thirty three and thirty five. It is thirty four point one three eight. Comfortably inside.

9:13And the upper half of that claim can never fail, for a reason you can see. Each fractional part is less than one, so the two of them together can never reach two. But the lower half does fail, and breaking it takes one line. Three plus four. Nothing after either point, and the answer is seven, which is not more than seven. A claim that sounds right is not a true one until somebody has tried to break it.