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Comment by leecarraher | Hacker News Reader

"However, the exponential effects of aerodynamic drag"

quadratic? otherwise we're in trouble

People in informal language constantly use "exponential" to mean "more than linear." Which drives me completely insane, but I think it's time to just accept it.

Well, it's big-O of exponential!

People often use big-O (asymptotically bounded above) when they really want big-Θ (theta, asymptotically bounded above and below). Strictly by the definition, for instance, binary search is O(e^x), as well as O(x^12), O(x), and O(lg x). However, binary search is Θ(lg x), and not Θ(any of those other functions).

I think that your parent was making precisely that point as a joke.

With big theta you strictly have to talk about best and worst cases, because for e.g. insertion sort, what's true of worst case is not true of best case (and vice versa). For big-O, what's true of worst case is also true of best case, so talking about worst case suffices. Big-O is less precise and therefore more useful, because average case usually = worst case anyway.

Why can't you use big theta to talk about average case?

You can, but strictly speaking you have to say, "big theta of the average case"!

Maybe you can explain it for the ignorant masses in terms we would understand? :D

It's the difference between x^2 (quadratic) and 2^x (exponential). Plug in some increasingly large values for x and you'll see how different they are.

Linear: n * x^1

Quadratic: n * x^2

Exponential: x^n

In all cases, x is assumed to be a variable, and n should be assumed to be a constant > 1.

If you pick the same value for n and put these functions into El goog, the answer should become fairly clear.

That's not right at all. Exponential is constant^x.

Also you can put "n *" in front of all of them if you want, but multiplying by a constant is usually omitted because it doesn't affect the shape of the graph.

That's not correct, exponential should, in your example, be n^x, not x^n.

Hell I'd be happy if it were used to describe sequences with more than just two values. I hear it constantly used in comparisons of just two quantities where the intended meaning is just "a lot more".

Yes, this gets to me too. You can fit any sequence of n points with a curve described by a polynomial of n-1 degree. therefore, you need at least 4 points to know that the curve is not cubic, and an infinite number to know that it is exponential (though you can usually surmise as much sooner, because many natural phenomena are exponential or logarithmic).

People in informal language constantly use "insane" to mean "frustrated."

What frustrates me is people saying "exponentially more" when comparing two things, like the Bolt has exponentially more range than the Leaf. I mean, you could fit an exponential curve to those data points, but also logarithmic, so why bother?

Yes, people who use “exponential” to describe the difference between two numbers are almost certainly illiterate in one way or another.

There's no hope of curing this in the popular press, but I think technical people should still be careful with the term. I think it's great that someone's called it out on HN.

>I think it's time to just accept it

Well, I'll continue to downvote comments here that do it. Mathematical concepts are an important part of our culture and of the public conversation.

Yes. To be precise, power, which is energy per unit time, is cubic in velocity: it takes 8 times as much power to go twice as fast (because of air resistance). But for a given distance, the energy required to traverse it is quadratic in velocity: it takes only 4 times as much energy to cover the distance at twice the speed, because it takes you half as long to get there.

ETA: There are other sources of drag, of course -- tire drag, and mechanical friction in the drivetrain -- but AFAIK the power required to overcome those is linear in speed, and so the energy is constant for a given distance. Anyway, at highway speeds, air resistance dominates.

It's quadratic. Polynomial would also have been acceptable. But what they meant to say is "superlinear". Like aetherson, this drives me nuts.

I'm guilty as well. Even though I know, mathematically, the difference colloquially it just roles off the tongue easier.

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