Why a graph stops being trustworthy once the answer is not a whole number

0:00Two equations, two lines, one crossing. Draw them carefully and the answer is simply there on the page. x plus three y equals six, together with two x minus three y equals twelve. Two points for each, a ruler, and the lines meet at six and nought. Put six and nought back into both equations and both balance exactly. So the method works, and a drawing is never wrong. The lines go where the equations send them, and the crossing is where it is.

0:35This whole video is about the step that comes after: getting the answer back out of the picture. Look at the answers that come out of drawn pairs. Six and nought. One and nought. Two and two. Twenty and sixteen. Every one a whole number, landing exactly on a corner of the grid. That is convenient, and it is worth being suspicious of. Is that what pairs of equations actually do?

1:04Take every pair you can build with coefficients and constants between minus four and four. Two hundred and forty-three thousand six hundred and forty-eight of them cross at exactly one point. Sixty-one thousand six hundred and sixteen of those crossings are at whole numbers. That is twenty-five per cent. Close to three crossings in four land somewhere the grid has no corner at all. The whole-number answers are the questions being kind to you.

1:39So what happens when the crossing is not kind? Here are three positions a crossing can perfectly well have. Root three and two root seven. Minus one point seven five and three point three. Four thirteenths and one nineteenth. The middle one you can manage: it sits on marks if your squares are small enough. The other two you cannot, and it is worth being exact about why, because the reason is not a blunt pencil.

2:11Root three is one point seven three two nought five nought eight, and it keeps going. Set the page up. One unit to the centimetre, ruled in two-millimetre squares. That is five marks to the unit. A careful eye splits one square and no better, so about one millimetre, a tenth of a unit, is the finest position you can honestly claim. Now: how far is root three from seventeen tenths?

2:40Nought point nought three two of a unit. At this scale, nought point three two of a millimetre. That is less than a third of what the eye was just granted. On that page, root three and one point seven are the same mark. Not close together. The same mark. Then rule the paper finer, you might say. It does not help, and the reason has nothing to do with eyesight.

3:08Ask whether root three lands on a ruling mark at all. Any ruling, however fine. A ruling with q marks to the unit puts root three on a mark exactly when q times root three is a whole number. Square that, and it needs three q squared to be a perfect square. Count the factors of three in three q squared. One from the three itself, and an even number from q squared. An odd total, always.

3:38A perfect square has every prime an even number of times. An odd count forbids it. So there is no ruling at all, not two marks to the unit and not two thousand, on which root three sits on a line. Two root seven is the same argument with sevens. The third coordinate needs no surd to defeat the paper. Four thirteenths of a unit is three point nought eight millimetres from the axis.

4:09One nineteenth is nought point five three millimetres from it. On two-millimetre squares, four thirteenths is past the first square and just inside the second. One nineteenth is barely off the axis. Half a millimetre, under what the eye can place at all. Drawn carelessly they both go in the first square, which puts one of them in the wrong square and the other nowhere in particular. Here is the distinction the whole topic turns on.

4:42The crossing is derived. It comes out of arithmetic, and it is exact. The reading is measured. It comes off a ruled sheet, and it carries that sheet's resolution. Deriving loses nothing at all. Reading rounds to the nearest mark, every single time. And once you have rounded, the answer has stopped being an answer. Put a rounded reading back into both equations and neither of them balances. Watch that happen.

5:15Seven x minus fifteen y equals two, together with x plus two y equals three. These cross at forty-nine twenty-ninths and nineteen twenty-ninths. About one point six nine and nought point six six. Read that off at five marks to the unit and you get eight fifths and three fifths. One point six and nought point six. Substitute. The first equation is out by a fifth. The second is out by a fifth the other way.

5:48Neither balances, and no amount of care with the ruler was going to change that. To land both coordinates on marks you would need twenty-nine marks to the unit. Squares nought point three four millimetres across, nearly six times finer than the paper. There is a second thing a drawing can lose, and it is worse than precision. It can lose which of the three pictures you are looking at.

6:17Two x minus two y minus two equals nought, with four x minus four y minus five equals nought. Both say that x minus y is a constant. One, and one and a quarter. So they are parallel. They never meet. How far apart are they? One point seven seven millimetres. Less than one small square of the paper. Drawn by hand, those two lines are one line, and the verdict you would read off the page is the wrong verdict entirely.

6:52Now compare x minus y equals eight with three x minus three y equals sixteen. Same shape of question, same kind of answer, gap of eighteen point nine millimetres. More than nine small squares. Nobody mistakes that one. The two gaps differ by a factor of exactly thirty-two thirds, and that single number is the whole difference between a picture you can trust and a picture that lies to you.

7:24The mistake runs the other way as well. x minus two y equals nought, beside two hundred and one x minus four hundred y equals thirty. Across a sheet running from minus ten to ten, those two lines are never more than one millimetre apart. Half a millimetre at one edge, a whole one at the other. They look like a single line. They are not parallel. They cross exactly once, at thirty and fifteen. That is twenty units past the right-hand edge, a whole sheet further on.

8:01Only the coefficients can tell you that. The drawing has no way of knowing. None of this makes the drawing worthless. Quite the opposite. The drawing is the reason anybody believes there are only three pictures in the first place. You can see two lines cross, or run alongside, or lie on top of one another, and you can see there is no fourth thing for them to do. It is still the quickest way to understand what a pair of equations means.

8:35What it is not is a way of reporting a number. So keep the picture for the meaning, and get the number some other way. That other way has to be exact: every step arithmetic, with no position measured against anything. Two such methods do the job. One replaces a variable by an expression; the other adds the two equations to cancel a variable out. Both are coming. A word on how the claims here were checked, because a claim about what cannot be read is easy to make and hard to test.

9:14The trap is that a reader which cannot resolve anything confirms it for free. So the reader used here is a single routine, snapping a number to the nearest mark, and it was proved to be a reader before it was shown anything awkward. A value already on a mark comes back unmoved. A value a hair below one snaps up onto it. Everything it hands back is a mark, and never more than half a mark from the truth.

9:45Then it was put to two populations, not one: crossings at whole numbers, where the reading must survive being substituted back, and the twenty-ninths, where it must fail. A reader that returned its input unchanged would pass the first and fail the second. One that returned nought would fail the first. Root three landing on no ruling was checked over two thousand different rulings, by two routes that share no code: whether three q squared is a perfect square, and how many threes it has in it.

10:22The controls sit beside them. Root four and root nine land on every ruling there is, and both routes say so. Every millimetre spoken here was bracketed by exact whole-number arithmetic rather than typed in as a decimal. And the separation formula was not taken on trust either. Two thousand four hundred and two points along the second line were tested, and not one of them was nearer than the formula claims.

10:56A drawing gives you the crossing exactly. It is the reading that costs you. A reading is only as fine as the marks it is read against, and no sharper pencil changes that. Some numbers land on no mark at any fineness at all. A rounded reading, put back into the equations, does not balance. That check is what catches it. And when two lines sit closer together than one square of the paper, the picture can give you the wrong case, not merely a wrong number.

11:29Keep the graph for what a pair of equations means. Take the number from arithmetic, where nothing has to be measured.