I was a math major as an undergrad, but by the end of it I was very close to the limits of my abilities. In fact, I never passed differential equations, although I did very well in more "advanced" classes dealing with linear programming, algorithmic anlaysis, and algebraic coding.
So it's exciting to hear about new math that I actually understand. This is called rational trigonometry.
Instead of being based on the length of sides and angles, it's based on concepts called quadrance and spread.
Quadrance and is basically distance squared, although the text says, "The relationship between the two notions is perhaps more accurately described by the statement that distance is the square root of quadrance."
Spread is a way of quantifying the separation of two lines. Imagine two lines intersect. Draw a segment perpendicular to one line to the other line. The ratio of the quadrances of the portions of the line between the intersection of the lines and the segment is the spread. This is much clearer when you look at the diagram on page 6 of the link. The spread is always a rational number in the coordinate system of the lines.
What's really cool about all this is that, apparantly, it lets you do all the side/angle calculations you do in basic trigonometry using polynomial math.
I don't know how interesting this is to anyone else, but I think I'll check this book out and play around with it.
Posted by pooka (Member # 5003) on :
huh, I'm not sure I quite understand the benefit of that over regular trigonometry. I guess fewer solutions that are irrational numbers? I don't quite remember what an irrational number is. One that cannot be espressed as a ratio?
Posted by Dagonee (Member # 5818) on :
A rational number can be expressed as the ratio of two integers. All others are irrational. This includes certain ratios, certain special values, and certain roots.
One benefit is that calculations can be done by hand. Right now, most people can easily use the sine to find the length of one side when an angle and another side are known, but what people forget is that we basically never calculate the values of of sin, cos, and tan. We use a computer or a calculator or a table. While practically this works fine, the book seems to suggest that there are relationships obscured in the calculations of the values of trignometric functions.
Perhaps some of these relationships will be made accessible to more people this way.