Steepness as the tangent of an angle, and the one line that has none

0:00Here is a plane, a line across it, and two languages for saying things about that line. One is the language of shape: it leans, it climbs, it is steeper than that one. The other is the language of number: pairs of coordinates, differences between them, quotients of those differences. The trade this whole subject runs on is that anything you can say in the first language you can arrange to compute in the second.

0:30But there is exactly one line whose steepness the second language cannot say at all, and why is the whole of today. One piece of arithmetic first. The area of a triangle from its three corners is half the size of a bracket built out of the coordinates. On four and four, three and minus two, and minus three and sixteen, that bracket comes to minus fifty-four, so the area is twenty-seven.

1:02Twenty-seven is not nothing, so those three corners are NOT on one line. But when the area does come out as nothing, there was never a triangle there at all — the three points lie on a single line. That is a collinearity test hiding inside an area formula. Take a line that is not flat and follow it to where it crosses the horizontal axis. Something awkward happens there: the line makes two angles with the axis, one on each side, together filling a straight angle.

1:39If the line rises, one is sharp and the other blunt. So we must say which one we mean, in a way that leaves no doubt. The rule: start along the positive horizontal direction and sweep anticlockwise until you meet the line. One starting ray, one direction of sweep, one answer. For a line that rises, that sweep hands you the sharp angle. For a line that falls, the same sweep goes past the sharp one to the blunt one — not a flaw in the rule, but the rule working.

2:15That angle has a name: the inclination of the line. A line lying flat along the horizontal has inclination nothing — you sweep, and you are already there. Tip the line up and the inclination grows: through the sharp angles, past the right angle, into the blunt ones. Come round towards a straight angle and the line lies flat again — but a flat line already had inclination nothing, so a straight angle is never reached.

2:47Closed at the bottom, open at the top, and every value between belongs to some line. Now the definition, and it is not the one most people carry around. The slope of a line is the tangent of its inclination. Notice what is NOT in that sentence. No two points, no coordinates, no differences, no quotient. A slope is a fact about one angle, and a line has that angle whether or not anybody ever picks two points on it.

3:21So everything you know about the tangent becomes a fact about slopes. It is above nothing on sharp angles, so lines that rise have slopes above nothing; below nothing on blunt ones, so lines that fall have slopes below nothing. And at exactly one angle inside the range, it has no value at all. That angle is the right angle, and the line that has it is the upright one.

3:51Set the two axes side by side: two boundary cases, and not the same kind of boundary. The horizontal axis has inclination nothing, the tangent of nothing is nothing, so its slope is nothing. That is a number. You can compare it, add it, put it in a formula. The upright axis has inclination a right angle, and the tangent of a right angle does not exist. So it has no slope. Not a slope of nothing — no slope.

4:24One is an answer and the other is the absence of one, and sliding between them is the commonest way to get this topic wrong. It is tempting to call the upright line's slope infinite. Resist that. Watch what happens as a line tips up towards upright. One step across and a hundred up gives a slope of a hundred. It takes a hundred and one steps up to pass a hundred, a thousand and one to pass a thousand, a million and one to pass a million.

5:00Name any bound and there is a line steeper than it. Now come down to upright from the other side: one step LEFT and a hundred and one up, and the slope is minus a hundred and one. Those lines are edging towards upright too, and their slopes run off the other way entirely. Two approaches to one line, and the numbers head for opposite ends. No single value is sitting there to be assigned. That is why the honest answer is no slope, and why calling it infinite invites a comparison that means nothing.

5:39So slope is the tangent of an angle. Where does rise over run come from? It is not the definition. It is a theorem, and here is the proof. Take a line that rises and mark two points on it, P and Q, with Q up and to the right. Drop a vertical from Q towards the axis and a horizontal from P across to meet it: they meet at a corner, giving a right triangle.

6:08The flat leg is the difference of the two horizontal coordinates. The upright leg is the difference of the two vertical ones. And here is the step that makes it work: the angle at P inside that triangle is the SAME angle as the inclination down at the axis. It has to be, because the leg at P runs parallel to that axis and the line cuts across both. So the tangent of the inclination is the tangent of the angle at P, which in a right triangle is the far leg over the near one.

6:46Upright difference over flat difference. Rise over run. That argument leaned on the line rising, twice: once when Q went up and to the right, once when the angle came out sharp. So it proves nothing about lines that fall, and those are half of all lines. Redraw it. Same construction, but now Q sits up and to the LEFT. The corner moves to the other side, and the angle at P is no longer the inclination — it is that OTHER angle at the crossing, the one the sweep went past.

7:24The two fill a straight angle, so the tangent of one is minus the tangent of the other. There is the minus sign. It did not come from a convention; it came from a supplementary angle. Now the legs. The upright one is still the difference of the vertical coordinates, but the flat one runs the OTHER way round, because Q is now to the left. So the quotient in front of you is minus the tangent, and it is the upright difference over the NEGATIVE of the flat one.

7:59One minus sign on each side, and they cancel. What is left is the upright difference over the flat difference, taken exactly the same way round as before. Two arguments, two pictures, one expression at the end of both — and why both were needed is worth measuring rather than asserting. Suppose you had done only the rising case and assumed it covered everything. You would have a rule that reads both legs as lengths, which is exactly what that first picture invites.

8:35Run that rule against the truth on ninety-two lines. It gets forty-eight right and forty-four wrong. Every one it gets right has a sharp inclination. Every one it gets wrong has a blunt one. And on those it hands you the true slope with its sign thrown away: a falling line reported as though it climbed. That is the price of skipping the second case — not a gap in a proof, but a rule confidently wrong about half the lines there are.

9:07One more thing: the expression has two points in it, and nothing said which to call the first. Swap them and the top turns over, the bottom turns over, and a quotient with both parts turned over is the one you started with. Ninety-two lines, every pair of points on each, counted both ways round — one thousand one hundred and four of them. Pairs where the quotient came apart from the tangent: none. Pairs where the order changed the answer: none either.

9:42Now put it to work: four lines, all through three and minus two. The first also passes through minus one and four. The upright difference is six, the flat difference is minus four, so the slope is minus three over two. Below nothing, so the inclination is blunt and the line falls as you read it left to right. The sign told you the shape. The second passes through seven and minus two. Both points sit at the same height, so the upright difference is nothing: nothing over four is nothing, and the line lies flat.

10:22The third passes through three and four. Now both sit at the same horizontal place, so it is the FLAT difference that is nothing. Six over nothing is not a large number. It is not a number. Hold those two against each other: one has nothing on top, the other nothing underneath, and only one is an answer. The fourth line is given not by a second point but by its inclination: sixty degrees. You need no points — slope was never defined by points.

10:59The tangent of sixty degrees is the root of three, so that is the slope — and it sits between three halves and seven quarters, which you can pin down without working the root out. Now run the machine backwards. A line through three and minus one and through four and minus two: the differences are minus one and one, so the slope is minus one. Below nothing, so the inclination is blunt — and the blunt angle whose tangent is minus one is a hundred and thirty-five degrees, which makes the sharp angle on the other side forty-five.

11:39One more, and it catches people. A line makes thirty degrees with the UPRIGHT direction, swept anticlockwise. The inclination is not thirty. The sweep starts along the horizontal, so it is a right angle plus thirty: a hundred and twenty degrees, and the slope is minus the root of three. An inclination is measured from one named ray — not from whichever axis happens to be nearer. So what does a slope tell you? Read it in two passes.

12:14First the sign. Above nothing means the inclination is sharp and the line rises to the right; below nothing means blunt, and the line falls. One hundred and eighty-three different slopes were asked for, and a line was found for every one; the number where the sign disagreed with the kind of angle is none. Second the size: bigger means steeper, on whichever side of nothing the sign has landed.

12:43And now the part that gets forgotten: a slope says nothing about WHERE the line is. Among those ninety-two lines, four have a slope of exactly minus one, and the number of points those four share is none. Four lines, one slope, not a single point in common. A slope fixes a direction and leaves the position open — which is why one number is never enough to name a line.

13:13So: what was bought. Not a formula for steepness. Rise over run is a theorem, proved in two pictures — and if you can only say what a slope is by drawing two points, you have the by-product and not the thing. A slope is the tangent of the inclination. One angle, one number. And putting the angle first is what makes the exception make sense. The upright line has an inclination of a right angle; the tangent of a right angle does not exist; so that line has no slope.

13:51Meanwhile, in the language of coordinates, two points on an upright line share their horizontal place, so the difference underneath is nothing and the quotient refuses. The same lines turned away twice, for what look like two different reasons. They are not two. They are one fact, told once in angles and once in algebra — and the trade between those languages is what this whole subject is.