A few quick examples:

1) Negative exponents are just powers of the denominator of a fraction. I'm sure that there are numerous "real world" examples of that; for example, if you halve repeatedly (say, n times), what's a convenient way to represent that? 1/(2^n), or 2^(-n). (Pardon the clumsiness, but this stuff is not readily typed.)

2) Zero exponents are what you get when the numerator and denominator are the same - it means that you have a whole whatever it is that you've got.

3) Absolute value - think of this as the "distance" between two things, since we generally do not concern ourselves with negative distances (although there are non-contrived applications of this as well): one usually says "We live five miles from town," without regard to direction.

Hope this helps.

Wordsmith, yes, I understand what they are and how they work, I just don't know what real use they have. I'm sure there is a real use for all of them, probably involving nuclear physics, aerospace technology, or something else way over my head.

I can't offer much in the way of "real world use." But if pragmatism is the measure of why we learn stuff, then much of theology would fall short, would it not? That is, as one person once said, "Why bother discuss Romans 5 and original sin? What's it got to do with me?"

My response to that question, and to your son's, is simply, while we may not perceive an immediate benefit or practical use for it; there should be a joy in discovering the glorious design of God. Whether that design is reflected in mathematical relationships, or in the nature of Christ vs. Adam, etc. There is a beauty to all of creation, including mathematics (maybe especially), and we have an opportunity to be in wonder at the amazing mind of God as He created the universe.

That may not help the "short term." But I'd hope this would be a foundational principle in formal schooling in these disciplines.

I took calculus in both high school and college, and I can't offer specific instances where I used anything particular from it. In fact, I can't remember most of what was taught in both sequences...

However, what I was told in high school was not that we'd necessarily use this stuff every day, but that we were (supposed to be) learning how to think. It's not that you'll use absolute value or negative exponents or integrals or whatever, but that you take the concepts, synthesize them, and apply them. If you can take an abstract concept like negative exponents and apply it within the realm of mathematics, then you can use the same thinking skills in any environment.

David - I enjoyed puting together some answers last night - to such an extent that I am going to make it a generic post at the armoury later this evening. But I'll say here that my wife was a math major and I - a physics major. Our conversations can become quite interesting over these kinds of questions. For her, she is satisfied with the theoretical aspect of math alone - for myself, I want to see the use and application of the theory. Point: some of us are better aided by seeing how theory is applied in the real world. More than that, some of the physical examples that came to mind offer some encouraging disclosures of God's wisdom and glory in creation, after all, mathematics is the mechanical foundation of all that He made.

Speaking about absolutes, I have a simple real-life example. It's not sexy, but maybe it will help, as you need to understand absolutes.

When I'm trying to figure out the concession on something I'm selling, I'll subtract the sales price from the net price. The difference is the concession.

However, I don't always start with the larger of the two prices, for a variety of reasons. But since I'm just trying to find the difference in the two number, it doesn't matter if it is positive or negative. Saves lots of time when you understand the math behind what it is you're trying to do.

Oh, and speaking of understanding the math, negative exponents are used in the formula for Compound Interest. Check out this link, scroll down to "Present Value Basics," and you'll see the formula. Note: in most places, it is quoted as FV / (1+i)^n. My math understanding is obviously inadequate, as I didn't realize you could restate it as a negative exponent (prolly 'cause we never worked it that way in any of my umpteen finance classes).

Hope this helps, though I will surmise that someone looking for "real-life examples" might just as easily dismiss these as being "not applicable." Good luck.

Matt, that's an excellent example. Thanks.

Mike, I look forward to reading your applications.

I have three examples at the armoury -

1. The 3rd or Zeroth Law of Thermodynamics.

2. The Universe's ratio of omega (rho/rho critical).

3. The common use of stellar distances.

As Einstein would say - better nate than lever.

Funny your friend is fielding such questions from his son. I also have an *ahem* friend who's homeschooled daughter was reading outloud from her science book today and with a most exasperrated 16 yr old tone said to her mother:

"This is retarded, why do I need to know why 2 astronauts travelling at such a high rate of speed toward each other, would never even see each other coming before they collided!?"

My friend answered her daughter and said "eww" (homeschooling moms are brilliant at elaborating).

This is a frustrating question. I asked it in school myself (and to my knowledge have never used or needed algebra or most of the scientific information they forced us to learn), and now my 16 yr old asks the same question.

Sigh...