On Patience in Mathematics
One of the greatest lessons of an education in algebra is the virtue of patience, especially in the face of ambiguity. We may not know everything about a problem at its outset, but if we trust our steps and tread carefully, we can find what we need in the end.
A classic example is the word problem.
Next year, I will be one-third as old as my father. Three years ago, I was one-fourth as old as he was. How old am I?
This problem is loaded with missing information, most notably the age of the speaker. But there's also the father's age. And we don't know what those ages will be next year, or what they were three years ago.
It is easy to get overwhelmed with all these unknowns, but algebra teaches us another way. We start by naming the unknowns, say by calling my current age m and my father's current age f. There are other unknowns, but we don't need new variables for them. To wit, our ages next year are m+1 and f+1, and our ages three years ago were m-3 and f-3. Once we have expressions for all the ages, we can translate the sentences into equations:
m+1 = (1/3)*(f+1)
m-3 = (1/4)*(f-3)
At this point, the hard work is done, and we just need to manipulate symbols a bit to get our answer. To avoid fractions, we multiply the first equation through by 3 and the second by 4:
3m+3 = f+1
4m-12 = f-3
Therefore, f is simultaneously equal to 3m+2 and 4m-9, and so
3m+2 = 4m-9 .
Subtracting 3m from and adding 9 to both sides gives the answer, m=11.
The hardest part of solving that problem was pressing on in the face of ambiguity. And notice that we didn't resolve all of it. We never bothered to figure out the father's age, or the ages next year or in the past. It's possible for us to figure out those values now that we know what m is, but why? We have the answer we need, and anything else would just be extra work to feed our curiosity. Algebra teaches us how to focus on what we need to know, and to tolerate not knowing the rest.
It's sometimes hard to apply that lesson to new problems, even mathematical ones. For example, consider this problem I stumbled upon recently.
In what proportion does the line between (-2,8) and (0,4) cut the segment from (-2,1) to (0,7)?
The problem is easier to understand if we sketch it out. Start with the segment from (-2,1) to (0,7), and then draw the line from (-2,8) to (0,4) on top.
NOTE: It could get confusing here with a line and a segment in the same problem. We will consistently use the word "segment" in referring to the piece between (-2,1) and (0,7), drawn in black above. When we say "line", we're talking about the gray line stretching across (-2,8) and (0,4) and beyond.
The line intersects the segment, cutting it into two pieces, and it's the ratio of those pieces' lengths we're interested in. In the language of the diagram below, we want to know this/that.
Like the last one, this problem is rife with unknowns. We don't know where the intersection is, or how long the segment pieces are. I'll admit that my first instinct was to eliminate those unknowns - compute the point of intersection, and then calculate the lengths to determine their ratio. But I realized that this method was an impatient one. I was trying to get all the information right away. In the process, I was going to have a lot of work to do:
solving a system of equations to find the intersection point,
applying the distance formula twice, potentially involving some tricky square roots
find the ratio, quite possibly a fraction with square roots in it.
Maybe it would be easier if I could find some more patience with these unknowns.
As an exercise in patience, let's try to solve the same problem, but without finding the point of intersection. We can still find the equation of the line in question. The slope is (4-8)/(0-(-2))=-2, and using the y-intercept, we find it has an equation of y=-2x+4.
With that equation we can find any point on the line. Using some foresight, let's find the points on the line that are horizontally across from the endpoints of the segment. Those endpoints are at y=1 and y=7, which plugged in to the line's equation give x=3/2 and x=-3/2. Those two new points, (3/2,1) and (-3/2,7), form triangles with the endpoints of the segment and the still-unknown point of intersection.
Those aren't just any triangles, they're similar triangles: all three of their angles match. It is a known fact that similar triangles have side lengths proportional to each other. In the language of the diagram below, all three ratios
LONG SIDE : long side
MIDDLE SIDE : middle side
SHORT SIDE : short side
Luckily for us, it's the proportion between the middle sides of these similar triangles that we were interested in from the beginning. That ratio is equal to the ratio of the horizontal sides, 7/2 : 3/2. We can simplify that ratio to get our final answer of 7:3.
Looking back, this approach required less computation, and the computations that were done were relatively painless. Our patience was rewarded with ease of calculation.
It is possible to take this lesson further. We could exercise extreme patience, not computing any new points. We know the line that cuts the segment has equation y=-2x+4. The lines y=-2x-3 and y=-2x+7 are parallel to that line and pass through the segment's endpoints. The differences in the constants added to -2x on the right show us that the distances between the lines are in a 7:3 ratio, which, if you think about it, was exactly what we wanted to know.