What follows is excerpts of a couple of messages I wrote to a math-hungry email buddy who's struggling to figure out how to make up for missing out on math in high school. It well illustrates how I can sometimes let my enthusiasm for basic math carry me away, even though I'm not licensed to teach math. I was inspired that morning by a couple of educational videos from Ontario that I saw on TV. The first one was about the quadratic formula and imaginary numbers. The second was on conic sections, which reminded me of where I got my favorite use of the word "degenerate".
I don't know if you have any of the concepts in your head about relating geometry to equations, so I'll start by noting that each geometric figure, when placed on graph paper, can be described by some equation. It turns out that there are two major sources for the equations we deal with on a frequent basis: triangles and conic sections. Let me describe the latter for you in a 3D image. Imagine a pair of cones lined up, pointed at each other and connected at their tips. Conic sections are what happens when you cut across those cones with a plane (picture a flat sheet of paper or gigantic ax) at various angles and places. If the plane is a clean, perpendicular slice across one of the cones, the intersection gives you a circle. If the plane is tilted somewhat, you get an ellipse. Tilt it in beyond the edge of the cone, and one end of the ellipse opens to give you a parabola. Take it further, and you start hitting the other cone too (avoid the tips for the moment), giving a hyperbola. If you lay the plane through the tips along the cones' edge, you get a line. Keep the plane attached to the tips, but separate from the cone sides, and the line degenerates to a point. Thus mathematicians refer to the case where all complexities drop away as the "degenerate" case.
When I first learned arithmetic, the problems were often presented as an equation with a box or underline to fill in. For example,
4 + 23 = _______ 16 - 7 = _______ 27 + 9 = _______
This works great when you are presented the problems on a sheet where you can write your answers, and the objective is to consider each equation on its own. But when you are working on more than one equation, and they are related, sometimes it's useful to uniquely label the boxes as variables so you can tie the information together to solve a larger problem:
4 + 23 = X 16 - 7 = Y X + Y = _______
In this problem, you can substitute the numeric values for the X and Y into the latter equation to get the value of the answer:
(4 + 23) + (16 - 7) = ______ 27 + 9 = 36
What I just did sounded pretty simple, but I understand it may have been a huge leap for you. Let me know if that's so.
In other problems, you may need to do more complicated things than numeric substitution to get to a point where arithmetic gives you the answer. Those things you do to equations to work toward answers are called Algebra. The beauty of algebra is that it is a set of rules you can use on both numbers and variables. However, you can't easily apply algebra to empty boxes and blank underlines, which is why we use variables.
Just as we learn addition by counting things and asking what we have if we add this bunch to that bunch, we can lay things out on paper to convey these basics on a line drawn like a ruler. Imagine or draw a line horizontally across a page, and starting in the middle of the page, copy toward the right side of the page the marks and numbers from a ruler onto your line. Where the ruler marks start, add a zero. What you should see is an unmarked side of the line to the left of center, a zero mark at the center, and numbers 1, 2, and so forth increasing toward the right. Add similarly spaced marks to the left side of your line. Now label the left side of the line to mirror the right side, except use negative numbers, -1, -2, and so forth, toward the left. What you have drawn is a number line.
I believe you will agree that if you wanted to add 2 plus 3, you could start on the number line at 2, and count three marks to the right to get the answer 5. Similarly, if you wanted to subtract 9 minus 5, you could start at the number 9 and count five marks to the left to get 4, assuming you were working with a large enough number line. You can even see that you can subtract past zero, yielding negative numbers if you subtract a larger number than you started with. For example, picture 3 minus 5. Start at 3, and count five marks to the left, which brings you to -2.
Just as you can have a number line across the page, so can you have a vertical number line on the page that shares the same zero point. Now when we add or subtract numbers, we need to keep track of which dimension we're talking about, vertical or horizontal. In a graph, we call each of these number lines an axis. By convention, we often call the horizontal number line the "X axis", and the vertical number line the "Y axis". A point on the graph will have a horizontal distance from the axis, which we call the point's X value. It will similarly have a vertical distance from the axis, called the point's Y value. We write these values using parentheses and commas, starting with the X value, and finishing with the Y value. For instance, a point with an X value of 4 and a Y value of 7 would be written (4,7), which we call the point's coordinates. If we wanted to talk in three dimensions, we would add a Z axis, pointing from the center of your numberlines out of the page toward your nose. A point would then have X, Y and Z values, such as (4,7,3). For the rest of this conversation, let's stick to two dimensions, so everything stays on the flat page.
So if you're with me now, you've got horizontal (Y=something) and vertical (X=something) lines. (If not, try to give me a clue where you got lost.) What happens when equations include both X and Y? Then you get a blend of horizontal and vertical in your direction. Let's try a simple diagonal, which includes the points (0,0), (1,1), (2,2), and so forth. Notice the pattern? The Y value is the same as the X value. How would you write that as an equation? How about Y=X? If you connect the dots, you will include such points as (-1/2, -1/2) and (54, 54) and every other place where Y=X. Where, you're wondering is (-1/2, -1/2)? It's half a block into Greenwich Village below the numbered streets! You've visited New York? What do you think the opposite diagonal line's equation will be? Can you draw these on graph paper? How about Y=2X? How does Y=2X+1 compare?
You can make each of those equations more complicated and yet still have them say the same thing. For instance, all the points for the equation X=4 also work for the equation 2X=8 or the equation X=10-6, because in each case, X works out to be 4. Any equation that is equivalent to X=4 will contain the same answers, err, points, err line. Would you agree that (4,7) is also at the intersection of 3X=12 and Y+3=10? How about 25X=100 and 2Y=15?
Generally, the first algebra class is about relating simple lines on graph paper to equations, such as 3x-2y=20. You learn to manipulate equations like that, doing simplifying things, while keeping both sides of the equation in balance:
3x-2y=20 (the equation given as a starting point)
3x-3x-2y=20-3x (subtracted 3x from both sides)
-2y=20-3x (number minus itself is zero, which disappears)
-1(-2y)=-1(20-3x) (multiplied both sides by -1)
-1(-2y)=-1(20)+(-1(-3x)) (associated the -1 into the RHS parentheses)
(RHS stands for right hand side)
2y=(-20)+3x (neg times neg is pos; pos times neg is neg)
2y=3x+(-20) (commuted the RHS about the plus sign)
2y=3x-20 (adding a negative is the same as subtracting a positive)
y=(3x-20)/2 (divided both sides by two)
y=(3/2)x-(20/2) (associated the /2 inside the RHS parentheses)
y=(3/2)x-10 (20/2 is 10)
Were you able to follow that? Why is the latter an improvement? This form is called slope-intercept, from which you can directly read what the graph looks like. The general description of slope-intercept form is Y=mX+b, where b represents the value of Y when X is zero (which is where the line intersects the Y axis), while m represents the slope. Slope can be thought of as rise over run, so in the above equation, the line rises 3 squares for every 2 squares it crosses to the right. It crosses the Y axis at -10, which means that if you plug in the value zero for x, then you get y=-10. To draw the line on graph paper, you'd draw your axes, and start with the point (0,-10). You'd count three squares up and two squares right and draw another point at (2,-7) and perhaps another at (4,-4). Connect the points with a line, and the (x,y) value of every point along that line will fit into the equation. To test this, look where the line crosses the X axis (y=0), and see if the values fit the equation.
As I mentioned above, the other source of common equations is a triangle. The relationships between the lengths of the sides and the angles opposite those sides are central to the study of trigonometry. Swing the triangle around, pinning one corner in place, and you start playing with the portion of trig known as circular functions. These generate the wavy patterns we commonly refer to as sine waves. If you're still with me in this, I can follow up in a later message. That business of quadratic formula has to do with finding the intercepts of parabolas across your axes. Imaginary numbers are what you get when you look for the intercepts but the parabolas don't cross your axes. How much of this message has been spew for you? For me, it's more fun than playing dreydel, so thanks for bringing it upon yourself and helping with my happy Hanukah.
$Date: 1998/12/18 00:46:42 $