The Language of Equations

Here is a confession the mathematics classroom rarely makes: an equation is not a puzzle set by an enemy. The moment a line of physics grows letters instead of words, a lot of otherwise curious people quietly decide the room was not meant for them. This chapter is a rescue. Take the very first equation the courses will lean on, d = v × t, and read it as the plain sentence it is: the distance you cover is your speed multiplied by the time you travel. You have known that since your first long car journey. Sixty miles an hour for two hours is a hundred and twenty miles, and you did not need algebra to feel it.

So the letters were never the difficulty. The difficulty was only ever the disguise. This chapter takes the disguise off: what the letters stand for, why the two halves of an equation must balance, and the single rearranging move that turns one remembered fact into every question you might ask of it.

A symbol in physics is just an abbreviation, chosen so that a fact fits on one line instead of sprawling across a paragraph. Written out longhand, the equation d = v × t reads: the distance travelled equals the speed multiplied by the time spent travelling. Physicists shorten distance to d, speed to v (for velocity, the fuller word chapter 4 will earn), and time to t, for the same reason a recipe writes “tbsp” instead of “tablespoon” – not to hide anything, but because you are going to write it a hundred times and life is short.

So a letter is a noun wearing small clothes. It stands for a measured thing – a number with a unit attached, exactly the kind chapter 1 insisted on – and the moment you say the noun back in full, the fear tends to drain out of the line. This course will make a habit of it: every time a new equation walks on, it gets read aloud as an ordinary English sentence before anyone is asked to do a thing with it. Try it now with d = v × t and notice there is nothing left to be afraid of.

The busiest symbol in all of physics is the one people skate over: the equals sign. It is not an arrow that means “and now the answer”. It is a promise, and the promise is balance – whatever sits on the left is the very same amount as whatever sits on the right. d = v × t swears that the distance and the “speed times time” are two names for one quantity, the way 50 and “half of a hundred” are two names for one number.

That single idea is the entire engine of rearranging. If the two sides are truly equal, then anything you do to one side you may also do to the other and the balance holds: divide both by the same amount, or add the same amount to both, and a true sentence stays true. The figure below makes it something you can feel rather than take on trust. Choose which quantity to find and watch the same weighed picture answer for it – that, and nothing more elaborate, is what “solving” has always meant.

Here is the payoff, and it is worth going slowly for, because it is the one algebraic move the whole catalogue leans on. Start from the fact you already believe, d = v × t. Suppose you know the distance and the speed and want the time. The equals sign lets you divide both sides by the speed v; on the right, dividing “v times t” by v leaves just t, and you are holding t = d ÷ v – time is distance divided by speed. Nothing was invented. You asked the same true sentence a different question.

Do it once more for the speed and you get v = d ÷ t – speed is distance divided by time. So one remembered fact has quietly become three recipes: distance from speed and time, time from distance and speed, speed from distance and time. The figure lets you pick whichever of the three you are missing and watch the same picture answer for it. And here is the honest headline: that move – balance a sentence, do the same thing to both sides – is very nearly all the algebra the seven courses ahead of you will ever ask you to perform.

There is one last way to write an equation down, and it is the one physics loves best: draw it. A graph is an equation with the numbers poured back in. Give it two axes – a horizontal one for time, say, and a vertical one for position – and every dot on the paper is a matched pair, a “where” printed directly above its “when”. A single dot is one instant of the story; the trail of dots is the whole story taken in at a glance.

Reading a graph is a skill worth having ready before chapter 4 asks you to lean on it, so learn the vocabulary here, where nothing is moving yet. The two axes are the two quantities; a point pairs one value from each; and the shape of the line is the personality of the motion – flat means nothing is changing, a steady climb means steady change, a bend means something just got interesting. Chapter 4 is the real classroom, where a dot drives along a track and draws its own graph in real time. For now it is enough to know that a line on a grid is not decoration. It is a sentence you can take in with your eyes.