An angle as an amount of turning, and what its sign records

0:00The name of this subject means, literally, the measuring of triangles. Two old roots: one for the three-cornered figure, one for measure. Now look at where it actually gets used. Reading earthquakes. Designing electrical circuits. Describing the states of an atom. Predicting the height of the tide. Analysing the shape of a musical note. There is not a triangle among them. The subject kept its name and outgrew the thing the name refers to.

0:34And the very first step in outgrowing it is a change to what an angle is. That change is the whole of this video. Here is the angle you arrive with. Two rays meeting at a point, and the angle is the corner between them. It is a shape. You can draw it, and you can measure how open it is. But a shape carries no direction, and it has a ceiling built into it.

1:02You cannot draw a corner wider than a complete turn, because after a complete turn there is no more room. And you certainly cannot draw a corner that is less than nothing. So this definition reaches from nought up to one full turn, and stops at both ends. That is not a fact about geometry. It is a fact about the definition. Change the definition and both ends open. The replacement is a rotation.

1:33Take one ray. Pin its endpoint. Turn it. The angle is not the corner you end up with — it is the record of the turning. How much, and which way. A running total, not a shape. Turn a quarter, then another quarter, then another, then another. The record reads three hundred and sixty degrees. Walk through it and you find one complete revolution with nothing left over. Turn a quarter and then a quarter back the other way, and the record reads nought.

2:07Nine quarter turns: eight hundred and ten degrees. Two complete revolutions, and a right angle left over. Nothing stopped it at one turn, because nothing in the definition says stop. Three names, because the rest of the subject uses them constantly. The fixed endpoint the ray turns about is the vertex. The ray in its starting position is the initial side. The ray in its finishing position is the terminal side.

2:40Those two sides are the same ray, photographed twice. Now notice what those three names do not include. A vertex, a starting ray and a finishing ray tell you where the turn began and where it ended. They do not tell you how much turning happened in between. Hold on to that. It comes back at the end and it is the point. Now the sign, which is where people go wrong first.

3:08Turn the ray anticlockwise and the record is positive. Turn it clockwise and the record is negative. The minus sign is not saying the angle is somehow smaller than nothing. It is a direction flag. Nothing else. Minus thirty and thirty are the same amount of turning: thirty, either way round. And they are not the same angle, because they do not finish in the same place. One ends thirty above the initial side. The other ends thirty below it, which is three hundred and thirty measured anticlockwise.

3:46Those two finishing positions are three hundred apart. Reverse a turn twice and you get it back, every time. Reversing never changes the amount of turning — only where it finishes. Take six sample turns and reverse each one: four finish somewhere new, and two do not. The two that stay put are the ones that finish on the line itself — at nought, and at a hundred and eighty.

4:16What unit should a record of turning use? The definition suggests one immediately: the complete turn itself. One revolution. That sounds childish until you meet something that spins. A wheel turning fifteen times a second. In revolutions that is easy to say. In degrees it is five thousand four hundred a second. In one sixtieth of a second, that wheel turns ninety degrees. A right angle, in a sixtieth of a second.

4:48After one whole second the record reads five thousand four hundred, and the walk finds fifteen complete revolutions with nothing left over. After an eighth of a second it reads six hundred and seventy-five: one complete revolution, and three hundred and fifteen left. After a minute the record reads three hundred and twenty-four thousand degrees. Nine hundred turns. You choose a unit to fit the size of the thing you are measuring.

5:22For small angles, revolutions are useless. So cut one into equal pieces. Three hundred and sixty of them. Each piece is one degree. A right angle is ninety of them, and four right angles make the turn. A straight angle is a hundred and eighty, and two of those make the turn. Why that number? Because of what divides it. Twenty-four whole numbers divide three hundred and sixty exactly. Take away one and the number itself and twenty-two are left.

5:58Two, three, four, five, six, eight, nine, ten, twelve, fifteen, eighteen, twenty, twenty-four, thirty, thirty-six, forty, forty-five, sixty, seventy-two, ninety, a hundred and twenty, and a hundred and eighty. Nine of the first ten whole numbers divide it. The only one that does not is seven. That is the usual explanation, and on its own it is not quite honest. Take every whole number from three hundred to four hundred and twenty and count its divisors.

6:34The most any of them has is twenty-four — and two of them reach it. Three hundred and sixty. And four hundred and twenty. So the divisor count alone does not pick our number out. What separates them is how far you can keep cutting. Halve three hundred and sixty and you get a hundred and eighty, then ninety, then forty-five. Three halvings before you leave the whole numbers. Halve four hundred and twenty and you get two hundred and ten, then a hundred and five. Two.

7:09In thirds it is the same story: two for one of them, one for the other. And four hundred and twenty misses two of the first ten — eight and nine. For contrast, look at the neighbours. Three hundred and fifty has twelve divisors. Four hundred has fifteen. Three hundred and sixty-one has three. And three hundred and fifty-nine has two. Nothing divides it but one and itself. Below one degree, the counting is not in tenths.

7:42One degree is sixty minutes. One minute is sixty seconds. So one degree is three thousand six hundred seconds. This is where a very plausible mistake lives. Write forty degrees, twenty minutes. It is tempting to read that little mark as a decimal point and call it forty point two. It is not forty point two. Twenty minutes is twenty sixtieths of a degree, which is a third. Forty degrees twenty minutes is forty and a third degrees, exactly.

8:16Convert forty point two honestly and you get forty degrees, twelve minutes. The gap between the right answer and the plausible one is eight minutes. Four hundred and eighty seconds. Two fifteenths of a degree. Six sample readings taken into degrees and back came out unchanged, all six. Six recorded turns, and what the walk finds inside each one. Three hundred and sixty: one complete revolution, nothing left over. The terminal side is lying on the initial side.

8:54A hundred and eighty: no complete revolution, a hundred and eighty left. Two hundred and seventy: none, two hundred and seventy left. Four hundred and twenty: one complete revolution, and sixty left. Minus thirty: no complete revolution, turned clockwise, finishing thirty below the initial side — three hundred and thirty, measured the other way. Minus four hundred and twenty: one complete revolution clockwise, and sixty more. It finishes sixty below the line, at three hundred.

9:32Three of those six are outside what the old definition could reach. Two of them are negative. Three carry at least one complete loop. And every one of them is a perfectly ordinary angle now. Here is the thing the old definition could not say at all. Draw a terminal side sitting sixty above the initial side. Which angle is that? Sixty works. So does four hundred and twenty. So does seven hundred and eighty.

10:05So does minus three hundred, and minus six hundred and sixty. Within five complete turns either way there are eleven of them. Every single one finishes in exactly that place, and no two of them are the same angle. The walk found something different inside every one: four clockwise loops at one end of the list, five anticlockwise at the other. Eleven records. One finishing position. Do the same at twelve different finishing positions and you have a hundred and thirty-two recorded turns, twelve finishing positions, and a hundred and thirty-two different records.

10:47The drawing is not wrong. It is incomplete. What it is missing is the loops. So what did the change actually buy? The old angle was a shape: bounded above by one turn, and with no way to say which direction. The new one is a record of a rotation. An amount, and a direction. No ceiling and no floor. Three hundred and sixty and nought leave the ray in exactly the same place, and they are not the same angle.

11:20Four hundred and twenty and sixty, the same. That is not a flaw in the definition. It is the reason for it. Everything that comes next depends on a measure that keeps running in both directions without ever repeating itself. A drawing of the finished ray stops at one turn. The record does not.