Here's an attempt at an answer to the question "How would anyone think of algebra?" (a genre of question that I very much like).
Suppose I'm a certain number of miles away, and I come back at a certain number of miles per hour. 1/
tiktok.com/@gracie.ham/vi…
Suppose I'm a certain number of miles away, and I come back at a certain number of miles per hour. 1/
tiktok.com/@gracie.ham/vi…
How many minutes will it take me? To work it out, I take the number of miles I was away, divide it by the number of miles per hour, and multiply by 60.
That was a bit of a mouthful, but there's a nice way to say it more succinctly. 2/
That was a bit of a mouthful, but there's a nice way to say it more succinctly. 2/
Let's call the first number D (for "distance") and the second number S (for "speed"). Then the number of minutes I'll take is 60 x D / S (or, as we usually write it, 60D/S). That's a lot quicker to write than "60 times the number of miles away divided by the number of mph". 3/
More generally, every time you have something you want to do to numbers (or, as we like to say, a function), describing that action is much more convenient if we can refer to a general number in some succinct way, and using a single letter is about as succinct as you can get. 4/
In the next tweet is how the Indian mathematician Brahmagupta described the solution of a quadratic equation at a time when it was not standard practice to use letters to stand for numbers. 5/
To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. 6/
Compare that with the modern
( -b ± √(b^2 - 4ac) ) / 2a
It's important to stress that using letters for numbers involves a trade-off. A page full of letters and formulae is less immediately easy to understand than a page full of words. 7/
( -b ± √(b^2 - 4ac) ) / 2a
It's important to stress that using letters for numbers involves a trade-off. A page full of letters and formulae is less immediately easy to understand than a page full of words. 7/
But with practice, it becomes easier, and if you are doing the same thing over and over again, then the investment repays itself many times over.
It's a bit like making a machine to do something. If you use the machine once, it probably wasn't worth it, but if ... 8/
It's a bit like making a machine to do something. If you use the machine once, it probably wasn't worth it, but if ... 8/
it's for a task that you needed to do repeatedly, then it was.
And since there are many mathematical tasks that have to be done repeatedly, but with small variations, the invention of algebra as a time-saving device was pretty well inevitable. 9/
And since there are many mathematical tasks that have to be done repeatedly, but with small variations, the invention of algebra as a time-saving device was pretty well inevitable. 9/
In fact, I'm surprised it took as long as it did.
The kicker is that once it's invented, it suddenly makes a whole lot of things possible to think about that would previously have required too much brain space. 10/
The kicker is that once it's invented, it suddenly makes a whole lot of things possible to think about that would previously have required too much brain space. 10/
And some of those get pretty complicated and need their own clever short cuts and time-saving devices, which lead to yet deeper thoughts becoming possible, etc. etc. And pretty soon there's a whole hierarchy of depth of thought, that we call higher mathematics. 11/
The trade-off I described earlier applies here too. Should I make the effort to understand some bit of maths better? If I'm going to use it enough, then definitely yes. If not, then perhaps not, but perhaps it would take me in unexpectedly fruitful new directions. 12/12
