It’s possible to be fairly proficient in solving equations yet have little or no feel for what numbers mean. In fact, it seems to be fairly common. Not as common as having neither the proficiency nor the feel, but still pretty common. Like Plato musing on virtue, I’m not sure having a feel for what numbers mean is something that can be taught. It certainly can be developed, if it exists, but I suspect it’s a bit like color-blindness: one simply can’t explain the difference between green and red to someone who can’t see it. But, fools rush in:
(Disclaimer: I am so slight a math guy that I’d probably flunk anything fancier than a high-school algebra test if I had to take it right now, and I belong nowhere near people with a truly well-developed feel for numbers. But I have a little, and it has served me well, so here goes.)
Say I make manhole covers: circular chunks of metal heavy enough to drive a truck over.
To keep the numbers super simple, let’s say I get an order for a huge cover 1 meter across. Its diameter is 1 meter, in other words. Let’s say I want to know how big around – the circumference – my manhole is going to be, once I make it. (1)
So, this being me, I first look up the formula, just to make sure: C = πd. The circumference equals the diameter times π. All that’s left is to plug in some numbers.
d = 1
π = 3.14159….
So, here’s where the feel for numbers comes in: Since π is an irrational number with an infinite number of decimal places, and, assuming I’m doing this by hand, I’m going to need to decide how many decimal places to use.
In other words, for my purposes, are 5 decimal places (.14159) enough? Too many? Off the cuff, I’d probably go ahead and use 3.14159, knowing that that’s way overkill for my purposes. I strongly suspect 3.142 would be plenty close. I much more than suspect it, since the answer to my question is obvious upon inspection: the circumference is 3.14159… meters.
Let’s break it down:
- The ‘3’ gives me 3 meters. Whatever error I’m introducing by leaving off all the decimal places is less than a meter – just looking at it, it’s 0.14159 meters, in fact. But that would still be about a 150 centimeters shortfall, which is kind of a lot….
- So I could go with 3.1. That gives me a 3.1 meter circumference, off by a little more than 4 centimeters (4.159 centimeters = a little over 1.6 inches). OK, so maybe a little more accurate than that?
- Using 3.14, I get 3.14 meters, which is off by 1.59… millimeters – I’ll be short about one 1/16th of an inch, if I round to 2 decimal places. That’s plenty close for a freaking manhole cover.
- If, for some crazy reason, I’m milling this manhole cover on modern milling equipment, then I might use 3.1416 (rounding up) and thus be off by 0.0004 inches – 4/10,000th/in. Modern milling equipment can easily do that. Why they’d want to in this case is unclear.
My instincts – my feel for the numbers or, more essentially, for the problem I’m trying to answer – were that maybe 3 decimal places would be more than plenty. Then, just doing it, I find 2 decimal places are more than enough for all practical purposes. I’m making a manhole cover, for heaven’s sake!
Note: All this math is not what I actually do. None of this is conscious. I just look at the problem I’m trying to address, look at the formulas involved, and it’s usually pretty clear how exact I’ll need to be. Can I be wrong? OF COURSE! My instincts have been embarrassingly wrong once or twice. But contrast that with the vast amount of time saved by having a feel for what the answer should looks like. Besides, errors in feel tend to reveal themselves almost instantly once you start working on the problem.
Super-trivial example. A big part of getting this example is recognizing that the math is simple, with no place for the numbers to ‘blow up’. The circumference gets bigger in a direct (linear) way as the the diameter gets bigger. That’s all. So, those trailing decimal places aren’t going to cause something unexpected to happen. 3.14159 is going to be plenty accurate for just about all real-world needs, and way overkill for almost all of them.
Working with financial models, one does sometimes run into cases where numbers out 4 or 5 or more decimal places really do matter, as well as cases where tiny changes cause the model to blow up – where discontinuities arise. As you change one input by a tiny amount, the outputs likewise change by a tiny amount, until, suddenly, they don’t anymore. In practice, what I sometimes saw was some tax or accounting threshold got tripped, different rules suddenly applied, and so the results changed dramatically. There are also gotchas in the math itself, where tiny changes will trigger a bifurcation in possible results, where more than one answer ‘solves’ the problem. With practice, one can get a good feel for even these sorts of issues, but more important, a feel for when you’ve passed into No Man’s Land, and your intuitions are no longer trustworthy.
Those are extreme cases, in this context. Just as how the problems people have understanding science rarely involve highly technical issues but rather basic failures in expectations and logic, most errors in assessing math seem to be much, much simpler even than this manhole cover example. If you add a number in the hundreds with a number in the thousands, your answer cannot be in the billions. Something may happen a million times, but if there are trillions of occasions when it *might* happen, it could still be an unlikely event in any one case. And so on.
This, unfortunately, needs expansion. As time permits.
- In the real world, assuming this is a custom manhole cover and not a mere run of the mill standard one, what I’d really do: make a 1/2 meter jig for my plasma torch, lay out the material, select a center point, and cut it out. Then, grab a grinder and clean it up. Or, better, slap the material on a CNC cutter, and push ‘go’. IOWs, I’m unlikely to care about the circumference, as it doesn’t figure into the process of making the cover. Sorry, just nerding out here…