Distance when the segment is parallel to an axis

0:00There is a case of measuring distance that gets waved through as obvious, and I want to stop on it. Two points, and you want the length of the segment between them. If that segment happens to run straight across, or straight up, everyone agrees you just subtract. That is correct. But it is worth knowing exactly why it is correct, because in a few minutes the general case is going to be built out of this one, used twice.

0:31A step you take without understanding is a step you cannot check, and this particular step gets taken hundreds of times. Here are two points. Three, zero. And seven, zero. Before doing anything, notice what they have in common. The second numbers are the same. Both are zero. That shared number is not decoration. It is telling you that one of the two measurements is doing no work at all.

1:02Both points are the same distance up. So whatever separates them, it is not height. The entire separation lives in the first slot. Now look at the line those two points sit on. It runs straight across, and every point on it has the same second number. Here is the claim: that line is a number line. Not like one. It is one. The unit was marked equally on both axes, so the ticks on this line and the ticks on the axis below it line up exactly. Same spacing, same steps, same everything.

1:43And you have known how to measure separation on a number line since long before coordinates. You subtract. That is the whole reason the two axes had to be marked with the same unit. Make the steps different sizes and this line would carry a different ruler from the one below it, and the subtraction would answer a different question. So let us do it on something real. This is the front edge of a wardrobe standing against a wall.

2:15Its two ends are at three, zero and seven, zero. Same second number, so cross it out. It is not going to be used. Seven minus three is four. The wardrobe is four feet wide. And here is the check that matters: four feet is exactly the width the plan already gives for that wardrobe. The subtraction did not approximate anything. It recovered a measurement somebody else had made. The back edge of the same wardrobe runs from three, two to seven, two. Same subtraction, same four feet. And the side runs from three, zero to three, two, which is two. Four by two.

3:01Now a trap worth walking into deliberately. Here is the front edge of a bed. Both of its ends sit five feet up. Not on the horizontal line. Five feet above it. Does anything change? No. The two points still agree in their second number, so the second number still does no work, and the length is still the difference of the first two. What makes the rule work is the shared coordinate. Not the location. A segment four feet up behaves exactly like one on the line itself.

3:37And I am going to leave this one unfinished on purpose. The drawing this came from does not put a number on that far corner. It sits between two marks, and I am not going to guess it. The method is fixed. The measurement is not mine to invent. Turn the whole thing on its side and nothing new happens. Here is a doorway in a side wall. Its two ends are at zero, one and a half, and zero, four.

4:08This time the first numbers agree. Both are zero. So it is the first number that does no work, and the separation lives in the second. Four minus one and a half is two and a half. The doorway is two and a half feet wide. Notice something about that answer. It is not a whole number. Which means you could not have got it by counting squares on a grid. Counting is a check. Subtracting is the method.

4:40Now the detail that trips people, and it is a real distinction rather than a fussy one. Go back to the wardrobe. Seven minus three is four. But three minus seven is minus four. Two different answers from the same two points. Both subtractions are legitimate, and they mean different things. Minus four tells you a direction: you moved four to the left. Four tells you a size. A length has no direction. There is no such thing as a wall that is minus four feet long. So when what you want is a length, you take the size of the difference and throw the sign away.

5:22That is what the two upright bars mean when you see them written round a subtraction. Not a decoration. An instruction to discard the sign. And it is why the order you subtract in never has to be decided. Take the numbers whichever way round they come, and let the bars clean up after you. One more case, because it is the one that goes wrong most often. The wall along the bottom of a bathroom runs from minus six, zero to zero, zero. Same second number, so subtract the firsts.

6:01Zero minus minus six. Two minus signs in a row make people hesitate, and there is nothing to hesitate about. It is six. The wall is six feet long. Keep going in the same direction. From minus six, zero all the way to twelve, zero is twelve minus minus six, which is eighteen. And check it: six feet of bathroom plus twelve feet of bedroom is eighteen. The subtraction across zero and the two rooms added up agree, because they are the same measurement.

6:40Now watch what this does to a shape. Here is a room, given only as four corners. Zero zero. Twelve zero. Twelve ten. Zero ten. No dimensions written anywhere. Just four pairs. Take them in order. The first two agree in their second number, so that side is twelve. The next two agree in their first, so that side is ten. Then twelve again, then ten. Every single side of that rectangle came from spotting which coordinate was shared and subtracting the other. Four sides, four subtractions, no ruler.

7:21Perimeter forty four feet. Area one hundred and twenty square feet. And now the diagonal, from zero zero to twelve ten, shares nothing with itself at either end. There is no single subtraction that reaches it. That is exactly where this stops working. Which brings us to what this was for. The general rule for distance takes both shifts, squares them, adds them, and takes the square root. It looks like a different animal.

7:55It is not. Feed it two points that agree in their second number. The second shift is zero, so its square is zero, and you are left with the square root of the first shift squared. Which is the size of the first shift. The bars, back again. The general rule was never a separate rule. It contains this one. And here is where it earns its keep. Take a slanted segment, drop a corner beside it, and you have made two segments that each share a coordinate. Three up. Four across. Both by subtraction. And the slanted one comes out five.

8:36Neither of those two legs needed anything you have not already got. The hard case was solved by turning it into two easy ones. So the whole of this reduces to one habit. Before you reach for a formula, look at the two pairs and ask which number they have in common. If they share one, that measurement is doing nothing, and the answer is the size of the other difference.

9:05If they share neither, you cannot get there in one subtraction. But you can get there in two, and that is the next thing.