Adding and subtracting componentwise, and undoing an addition

0:00Here are two of these numbers, and the question is what their sum ought to be. The obvious answer is: add the first parts, and add the second parts. The obvious answer is also the right one. But obvious is not a reason, and a definition is a choice somebody made. So take the choice seriously for a minute. Here are eight rules you could have written down instead. Each one takes the four ordinary numbers and says what the two slots of the answer are.

0:33One adds slot by slot. One crosses the slots over. One keeps the left-hand second part and throws the other away. One pours everything into the first slot — which is exactly what you are doing if you ever write two plus three i as five. They cannot all be right, so how would you tell? You might try the usual tests. Does the order matter? Does the grouping matter? Is there a number that changes nothing?

1:04Run those three tests on all eight, and three of the eight pass all three. Three respectable-looking candidates, and only one of them is the one you want. So the usual tests are not what settles this. Here is what does settle it, and it is not a test at all. Look again at what a plus i b actually says. It says: an ordinary number, plus the symbol times another ordinary number. It is already a sum.

1:36So put two of them side by side. a plus i b, plus c plus i d. That is four things being added, and ordinary arithmetic lets you add things in any order you like. Gather the plain ones. Gather the ones carrying the symbol. a plus c. And i, times b plus d. Nothing was defined and nothing was chosen. The rule fell out of rearranging a sum. And that is the test the other seven fail.

2:08Set each of the eight against what plain rearrangement gives, over eighty one pairs of numbers, and exactly one agrees every single time. The one that crosses the slots gets seventy two of the eighty one wrong. So: two plus i three, plus minus six plus i five. First slots, two and minus six make minus four. Second slots, three and five make eight. Minus four plus i eight. The rule ran once in each slot, and that is all it ever does.

2:44Look at that answer for a second longer. Minus four plus i eight is again one ordinary number, plus i times another ordinary number. It has the same shape both inputs had. That property has a name, closure, and it is worth a minute even though it sounds like a formality. It says the system is self-contained. Add two members and you get a member. You never fall out. Six hundred and twenty five pairs were added here, and not one of them left.

3:18And it is not automatic. Closure is a claim about a collection and an operation together, and it can fail. Five collections were tried. Four of them you cannot add your way out of. The fifth is the numbers whose second part is exactly one — and every sum leaves it, because one and one make two. There is a second thing addition never does, which is get stuck. Try dividing slot by slot on those same six hundred and twenty five pairs, and two hundred and twenty five of them have nothing to divide by.

3:56Addition never refuses. Worth noticing, before it stops seeming remarkable. Now the five properties this system is supposed to have. Addition is closed. Order does not matter. Grouping does not matter. There is a zero. Every number has a negative. Those are not five discoveries about complex numbers. Every one of them is a property the ordinary number line already had, showing up twice — once in each slot. You can see that by breaking the line on purpose.

4:33Build the very same two-slot system on a different operation. Take halfway-between instead of adding. Order still does not matter: zero failures out of eighty one pairs, because halfway-between is even-handed. But grouping now matters badly. Six hundred and forty eight of the seven hundred and twenty nine triples come out differently. Try a lopsided operation instead, one that counts the second number twice. Now order fails at seventy two of the eighty one, and grouping at six hundred and forty eight.

5:12Break the line, and the system breaks in exactly the same place. That is what inherited means, and it is the whole reason those laws are true. Which number does nothing? Do not answer yet. Search for it. A hundred and sixty nine candidates were tried, and the test was strict: it has to leave every number alone, and from both sides. Exactly one survived. Its first part is nothing and its second part is nothing. Written out, that is zero plus i zero.

5:50And it is the same number as ordinary zero, which is why it gets written as zero and nothing is lost. Now the point of searching instead of declaring. Having a zero is a property of the rule, not a fact about notation. Of the three operations this system was built on, only one leaves a zero to be found at all. Halfway-between has none. Ask for it and the search comes back empty and says so.

6:20So when this system turns out to have a zero, it has one because the line had one, and for no other reason. Same question, one step harder. Given a number, which one brings it back to zero? Search again, and every one of the nine numbers tried has exactly one. Take two plus i three. The number that brings it back is minus two minus i three. Both signs flip. Both slots.

6:51Be careful here, because there is a very similar-looking number that does not do this job. Two minus i three flips one slot and leaves the other alone. That is a different object with a different purpose, and it turns up later. The negative flips both. And again, having them is not automatic. The rule that multiplies the second slots does have a zero — zero plus i one, oddly enough.

7:22But three of the nine numbers have nothing at all that brings them back, and they are exactly the three with no second part. Because nothing multiplied by anything is still nothing, and nothing is not one. Now subtraction, and here is the good news: there is nothing to define. u minus v means u, plus the number that undoes v. That is the whole definition. It is not a second rule; it is the first rule with one extra step in front of it.

7:57Which tells you two things immediately. Every subtraction is possible, because every number has something that undoes it, and every addition works. And whatever addition does slot by slot, subtraction does too — because subtraction is addition. Take six plus three i, minus two minus i. Undo two minus i, and you get minus two plus i. Now add. First slots, six and minus two make four. Second slots, three and one make four.

8:31Four plus four i. Once in each slot, exactly as before. There was never a second rule to learn. But here is where subtraction and addition part company. Turn that one around. Two minus i, minus six plus three i. Undo six plus three i, and you get minus six minus three i. Add. First slots, two and minus six make minus four. Second slots, minus one and minus three make minus four.

9:05Minus four minus four i. Set the two answers side by side. Four plus four i, and minus four minus four i. Same two numbers, and both slots have changed sign. So order matters for subtraction, though it does not for addition — and that is not a special fact about these numbers either. Six minus two is not two minus six on the line. Counted over eighty one pairs: addition gives the same answer both ways round all eighty one times.

9:40Subtraction manages it nine times, and those nine are exactly the cases where a number was subtracted from itself. Every law in this video was stated with two words in front of it. For all. For all of these numbers, the order does not matter. Not for these two. For all of them. That phrase is doing real work, and here is why. An example can knock a law down. It can never stand one up.

10:11Take the rule that crosses the slots over — the wrong one. Over those eighty one pairs, it gets nine of them right. Nine. So if you had checked one example, and it happened to be one of those nine, you would have walked away believing a false rule. One more, in the same spirit. What does two plus three i collapse to? Twenty six ordinary numbers were tried, and not one of them is that number.

10:40And that settles nothing, because it is a failed search. Here is what settles it. Whichever ordinary number you pick, the difference has minus three sitting in its second slot, every time. It is never nothing. So no ordinary number is ever going to be it. Three of them, worked all the way through. One minus i, take away minus one plus six i. First slots, one minus minus one is two. Second slots, minus one minus six is minus seven.

11:16Two minus seven i. Next. A fifth plus two fifths i, take away four plus five halves i. First slots, a fifth minus four is minus nineteen fifths. Second slots, two fifths minus five halves is minus twenty one tenths. Ugly, and completely fine. Last. Add a third plus seven thirds i to four plus a third i, then take away minus four thirds plus i. First slots, a third plus four plus four thirds is seventeen thirds. Second slots, seven thirds plus a third minus one is five thirds.

11:58Now look at the three answers together. Only one of them has whole numbers in both slots. And all three are still one ordinary number, plus i times another ordinary number. That is closure, doing its quiet job while you were busy with the fractions. One last thing, and it is the reason this had to come before multiplying. Go back over everything in this video and look for where i squared equals minus one got used.

12:30It did not. Not once. Adding these numbers never touches it. And that is checkable. Change what the symbol squares to, and every one of the eighty one sums comes out identical. Change it, and thirty six of the eighty one products change. So addition is the part of this system that costs nothing extra. You get a self-contained collection, a zero, a negative for every number, and subtraction thrown in free.

13:03And all of it is the ordinary number line, running twice. What you do not have yet is any way to multiply two of these together. That is where the one assumption finally gets spent, and it is where the two slots stop being independent of each other. Until then: two slots, one operation, twice over.