in a time before calculators

19/05/2012 § 2 Comments

The current unit we are studying in class is this fabulous thing called logarithms.

log•a•rithm – (n.) a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

In simpler words, the log of a number is the power (x) that makes the base number (y) equal n. Since that actually wasn’t any simpler than the dictionary definition, let’s put that in actual equations.

(Way more happening under the cut.)

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Round and round, here we go again.

29/03/2012 § 1 Comment

It honestly isn’t something one thinks about on a daily basis but if one does bother to put some thought into it, they might realise that trig cycles exist almost everywhere in life, in little bits and pieces. To quickly define a cycle, really a cycle could be anything that occurs or happens in life as long as it starts from somewhere, gradually gets to the end and starts over again. For example, a bicycle wheel is a cycle because, say, if you make a red mark on the wheel, it will constantly turn again and again as the bike moves. That is a kind of cycle that is needed in order to move the machinery.

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a mathematical glance at the semester

24/01/2012 § 1 Comment

First things first: How do I feel about my progress so far? I feel that I’ve done a lot more with math than I ever had in any other year in math because it’s been more of a challenge this year. I think I’ve learned how to understand math with more depth than I have before, really looking at why something does something else, how this and this affects that and that, etc. At the same time, I think I’ve been able to keep up with the class and learn the many units quite well so far. Of course there are far more skilled geniuses in the same room but I’m proud of my progress because I’ve been keeping up and doing well in class.
I like that math class is very smooth and we usually have something very educational to do, which is what I expected from math. What has worked well for me as a learner are the Scrapbook files that we make for each lesson. Those really help when I have to go back to check on something.
I think the class could improve if we did less in-your-seat work and did more interactive and physical activities. For example, to learn about inverse variations, Grace and I were given the opportunity to go outside and measure things outside of class and move a little bit. That was helpful and enjoyable.
I’ve already started but I’d like to study a lot earlier for quizzes and tests and to actually study a lot each time. This means going through the Scrapbook notes, my handwritten notes, and going to look for help a few days to a week before the test. I know this will help me because I used to do it before and I’ve kind of lost the amount of time needed to do it all. By achieving this, I obviously need to plan my time well and stop using the Internet for things I don’t need to use it for. I need to narrow my focus to one thing at a time and instead of living along the thought process of “I need to finish this to get some sleep,” I should start thinking of something along the lines of, “If I finish this now, I’ll have more time to myself later and maybe time to look over this and improve it.” I’ve also done this before and I know it’ll help. I have no major goal except to continue working hard – which won’t ever actually stop – and organize myself better.

My favorite animals live ‘this’ long

21/01/2012 § 5 Comments

Sometimes we should all stop and wonder how wonderful of a life we have. Other times, we should stop and wonder how wonderful of a life all the animals around us have. Humans are lucky. We all have the ability to live ‘til around 70-years-old while most animals usually live up to 15 – 20 years old tops. The lifespan of an animal – mammals, in this case – can be determined by their heart beats. Typically, the slower the heart rate, the longer the mammal will live. This relationship is, in completely mathematical terms, an inverse variation relationship. Less of something means more of something else.

The animals used in this little experiment were a monkey, horse, cow, pig, rabbit, elephant, giraffe, and whale. All of these animals are some of my favourites. (From: http://slowvelder.files.wordpress.com/2010/02/giraffehead.jpg)

The graphing process was not frustrating at all, contrary to what I’d expected. I set up sliders as seen in the below image at the top of the image. The lines with a, h and k above each line helped me move around my graph in order to fit the points I inserted with the data I gathered from the heart-rate website with all the information.

From the makers of GeoGebra.

Of course, it was not easy to find a perfect position for the graph where the line passed each dot but I did my best to find the right trend. Also, we need to take into account that I am actually missing half of the information. The person sitting next to me, Grace, shared the set of data with me and we took half of the information each. Thus, I am missing some data that could help shape the graph better. Despite this though, I think the trend I have presented in my graph is still legitimate and very close to accurate.

The final equation of this graph was y = 1596.6 ÷ x.

All my favorite animals live THIS long.

It’s domain: x > 0 and it’s range: y > 0. We can find the domain and range because these values are directly linked to the asymptotes. In this graph, the vertical asymptote is 0 on the x-axis, and the horizontal asymptote is 0 on the y-axis. Because it is not possible to have a negative lifespan and negative heart rate, it is clear that both values must be positive therefore the graph can only be in Quarter I (positive x, positive y). The asymptotes are the barriers for the graphs and now we know that even if the graph will extend upwards forward and to the right forever, the line will never touch 0. Because of this, any value of y = 1596.6/x will always be greater but never 0.

This makes sense in mathematical and realistic logic. Realistically, the fact that any value of this equation can’t equal zero makes sense because you can’t have a heart rate of 0 and still live (when x = 0). At the same time, you can’t have a lifespan of 0 and still have a heart rate (when y = 0). When there is one, there also must be the other.

Whale, whale, whale – what have we here? (From http://carinbondar.com/wp-content/uploads/2010/10/whale6.jpg)

In fact, although the graph will never touch either x- or y-axis, the graph will extend as far as it can without touching the asymptote, in this case the axes, the numbers will still grow, even if it’s only by a little. As the graph gets closer to the vertical asymptote, for example, this means that the life span grows by more while the heart rate decreases by only a little. This could make sense in real life. Notice point F on the graph is just to the right of point H but H is much higher compared to point F. Think of it like the rise-and-run (aka think of the slope of the line that connects points H and F). The slope is steeper, meaning the rise is greater than the run. If the rise is greater, it means the change in life span is more than the change in heart rate. It is just vice versa with the other values; the more the change in heart rate, the lower the life span. This also makes sense because the faster a mammal’s heart rate is (see the slope of points A and E), the shorter amount of time they’ll live. This makes sense if especially if you think about exercise. The faster one’s heart rate is (according to Mr. Rabb), the more chance he or she might have heart disease. This applies greatly to this inverse variation relationship.

Other examples of inverse variation relationships are really basic ones – the greater the diameter of a cylinder, the lower the height of the same amount of water in another cylinder. Pressure and temperature are also inversely related to each other. In economics, the lower the agriculture percentage in a country, the greater it’s GDP is. There are surely other inverse variation relationships; examples you will see in other peoples’ blogs and assignments.

That was a joke.

It can’t get more imaginary than this.

19/09/2011 § 1 Comment

At first, I was a bit upset when we started learning about complex solutions because it meant that we could really no longer just write a simple “No Solution” as an answer for certain problems. Admittedly, I’m still a bit upset that there’s a bit more math to the work I’ll have to do during certain problems that involve finding the square root of negative numbers. The good part is that the concept of using i isn’t too hard; it simply takes a bit of time for me to find out what i-to-the-power-of-something will equal but usually I get it quickly.

One very good reason to have complex solutions is because the fact that it’s used commonly nowadays is sure sign of the potential of a human’s mental capacity. According to BetterExplained.com, “it is a testament to our mental potential that today’s children are expected to understand ideas that once confounded ancient mathematicians.” Also, although complex numbers are theoretical numbers that humans invent, they are useful and they fill in the blanks of numbers and ideas we don’t know how to explain completely. A little like negatives, there was a concept that was confusing and mathematicians in the past invented negatives or complex numbers to explain that concept as best as possible.

Also, complex numbers are useless when calculating headings and orientations (North, East, South, West). Again derived from the information from BetterExplained.com, instead of using cosine or sine, one can use complex numbers to find directions easily. (See the article for the actual explanation.)

I would think that using complex solutions in a quadratic simplifies a quadratic more. Instead of leaving the quadratic in its form with a negative in the discriminant, we could use i to simply it a little more and end the equation with two answers using that contain i. It’s more work but in my opinion, I think it cleans up the equation instead of leaving lots of numbers in there.

References:

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