4 posts tagged “math”
So, my beloved figured out something kind of cool about numbers... not that he was the first to discover it or that it's some kind of awesome mathematical breakthrough, but I mean that he figured it out on his own whilst playing around with perfect squares, 'cos, ya know, that's what math geeks do... for fun. Yes, it's that nerdy around these parts.
I hope I can explain this properly.
Adjacent perfect squares (that is, a number squared once - for instance 4 squared is 16, so 16 is a perfect square - oh heck, let's let Wikipedia explain it:)
An integer which is the square of an integer, i.e. can be written in the form n2 for some integer n (and because of this a square is always nonnegative). Thus a perfect square always has a square root that has no decimal expansion.
Examples: 0, 1, 4, 9, 16, 25, 36, 49, 64 of which the square roots are 0, 1, 2, 3, 4, 5, 6, 7, & 8 respectively.
Okay... the first nifty thing is that adjacent perfect squares increase in an orderly odd number sequence. Notice in the example above that 0 to 1 is 1, 1 to 4 is 3, 4 to 9 is 5, and so on - 1, 3, 5, 7, 9, and on and on up. The next nifty thing is that the difference from subtracting two adjacent perfect squares is equal to the sum of adding the bases of the squares. For example, 16 and 25 are adjacent perfect squares. Their bases are, respectively, 4 and 5. So, 25 minus 16 is 9 and 4 plus 5 is also 9. As I understand it, these two things are true on up the line...forever, I suppose.
Maybe not the most useful pieces of math niftiness, but kind of cool, all the same.
Yikes! My very first Physics test will be this coming Monday morning. This weekend will be all about the Physics reviewing. So far the first four chapters are just the "easy" stuff, easy being very relative between, say, my Math major beloved and my linguaphile self. However, I will say that being married to a Math Brain is really handy when one needs a tutor. Not that I don't return the favor, mind you - he does have a research paper to write this semester...
I jokingly asked my teacher if I could just write an essay.
He did not agree.
Go figure...
A vector would appear to be like any other old line, except that, like other things in Physics, it is not what one thinks it is by way of general knowledge. Kind of like that difference between speed and velocity, which I mentioned in my last Physics post. In fact, it is exactly like that difference between speed and velocity, and is the very definition of that difference.
Here we go... A line has length, some number that we could call a magnitude and it happens to be either positive or negative, and that's about it. A vector, however, also has direction: North, South, East, West... 45 degrees, 22 degrees... whatever. And whaddya know, speed and velocity have the same relationship. Speed is just a number, a magnitude - the magnitude of velocity, as it turns out. But velocity, besides having a magnitude (say 55 miles per hour), also has a direction. With velocity, you aren't just going 55 mph, you actually must be going 55 mph in a specific direction (preferably in the correct lane).
Things that just have magnitude, but no specific direction (other than positive or negative) are called scalars. Speed is a scalar, but that's difficult to think about, as we are used to thinking of speed occurring in a given direction, even though, apparently, it doesn't. Here's an easier one: temperature. Temperature has a magnitude, a particularly high one here in Florida, but it doesn't particularly go anywhere. It can be positive or, so I've heard, negative, but it doesn't have a direction.
The fun thing to do with vectors is, of course, to get lost in the woods. Okay, maybe not so much the getting lost part, as the finding your way back part. Say you walk 8 miles North in the forest, then run into a river and travel another 5 miles along it, going Northeast. Say that you somehow knew that your Northeast travel was at a 60 degree angle - well, come on, you knew you'd walked exactly 8 miles and 5 miles, why not exactly 60 degrees, too? After 5 miles you decide you're really too tired to care about getting across the damn river anyway, and you'd much rather go back to camp and roast some marshmallows. You could walk 5 miles back along the river and another 8 miles back to camp but, really, you're pretty tired of that river now and all that running water makes you really, really need to find a bathroom. Could you get back to camp by a more direct route?
Well, if you knew some Physics (and had a calculator in your backpack) you could quickly (or at least within about half an hour) figure out that if you took the first vector of 8 miles North and added it to the second vector of 5 miles Northeast that you would have yourself a new vector of 11 miles going Southwest at about 20 degrees which would get you back to the camp, the marshmallows, and the porta-potty in two miles less than it took you to get out here...which you could have just walked in the half hour it took you to figure this out.
Our first lesson in Physics is all about the essentials: measurement. After all, you can't make any certain statements about a thing until you can measure it properly. The first thing to keep in mind is to always, always, always state what units you are using. Whatever kind of math you are doing, get into the habit as soon as possible of jotting down the unit of measurement along with your numbers. This could prevent, say, sending a Mars probe sailing past its target if you happened to be measuring in inches and your coworker in centimeters.
The next thing to keep in mind is that, even if you are perfectly comfortable with American measurements and completely revile the metric system, you WILL have to learn to also use the International System (or SI) - yes, the dreaded metric system. It comes up. You might as well get the hang of it, even if it doesn't seem all that useful to you right now. The sooner you are comfortable with it the less problems you'll have later.
And here's the point... you might actually need to use that math you suffered through in high school! I hated math in high school. In fact, it is still a very difficult subject for me, and I certainly don't love it. At the time I intended to become an English teacher and I just didn't see the point of taking math. But even an English teacher needs to know some math. I had to take Statistics for my education degree, and I would have had to calculate grades and be able to weight different assignments at different percentages. Of all of the other students in my Physics class, none of them are planning on becoming scientists, and only one is even a math major. They want to be dentists, pharmacists, and doctors. Some want to design houses, build bridges, or plan cities. Some are just into computers or want to be programmers. I just want to stargaze! But all of us are there because we still need the higher math and the heavy science.
So, if you are still in some type of school, pay attention to the math, go ahead and get the hang of metrics, don't let it frighten or intimidate you. Believe me, it comes up! It'll be easier to keep up with it now than to learn it all over again later.
