Algebra—Why and When
AlgebraThe branch of mathematics that deals with operations on sets of numbers and relationships between them. can seem abstract and remote, utterly disconnected from daily life. We all know we need arithmetic to pay our bills and make sure there are enough pizza slices for everyone at our parties. But quadratic functionsA function of the form `y = ax^2 + bx + c` where `a` is not equal to zero. ? Square rootsAny number `x` multiplied by itself a specific number of times to produce another number, such that in `x^n = y`, `x` is the `n`th root of `y` - for example, because `2^3 = 8`, `2` is the `3`rd (or cube) root of `8`.? Nonlinear equations? Not likely those are ever going to come up in conversation. So who needs it?
Turns out, you do. All of us do. Arithmetic is great for organizing and arranging. But algebra is for dreaming and exploring. It's not just for mathematicians—it underlies art and music, science and medicine, engineering and athletics.
Algebra turns up wherever there is uncertainty, whenever a number isn't known or a value is variable. Let's look at some examples of algebra out in the "real world."
Turn up your favorite song on the radio—can you hear the math? Music and algebra may not seem to have much overlap, but in fact they're closely tied together. A song is a sequence of notes, arranged together into pleasing patterns. And algebra is the study of patterns, like the ones in music. Take a look at this series of numbers:
`0,1,1,2,3,5,8,13,21,34,55,89`,...
It's called the Fibonacci sequence, and it's made up of numbers that are each the sum of the previous two numbers. It's a sequence that artists and musicians (and mathematicians!) have been studying for hundreds of years. Many believe that art and music that incorporate the proportions and ratios of numbers in the Fibonacci sequence are naturally more pleasing and enjoyable than those that do not. That’s a matter of taste, but there is definitely a link between this sequence and music.
For example, on a piano keyboard, it takes `13` keys to travel from one `C` note to the next, `8` white and `5` black. And the black keys are in groups of `3` and `2`. So `13,8,5,3,2`—yes, that's part of the Fibonacci sequence. You'll find out how to recognize and analyze mathematical patterns like this when we study functionsA kind of relation in which one variable uniquely determines the value of another variable. and patterns.
When biologists want to figure out how an ecosystem works and how to preserve it, they have to consider a bewildering array of factors. Species diversity, population changes, resource availability, climate cycles, reproductive patterns, interactions between populations, ... whew! There's an awful lot going on. The only way to figure it out is with algebraic equationsA statement that describes the equality of two expressions by connecting them with an equals sign., equations that combine several variablesA symbol that represents an unknown value. in order to see how changing one part of the ecosystem affects the other parts. Here's an example:
|
Example |
|
Isle Royale is a small isolated island in Lake Superior. It's got moose, and it's got wolves. Scientists have been studying the relationship between these species for many years, wondering if this ecosystem can possibly survive. If the moose population gets too big, it will overgraze the vegetation on the small island and the ecosystem will crash. If wolf numbers grow too large, they'll eat all the moose and then with no moose to eat, they'll starve. The equations below describe the predator-prey relations between wolves and moose (the Greek letters represent aspects of the wolf-moose interaction): |
|
`text(#moose)/("time")="moose"(alpha-beta*"wolves")` |
|
`text(#wolves)/("time")=-"wolves"(y-delta*"moose")` |
These equations have a lot of variables, and if you drew them on a graph, the lines they make would not be straight. This sort of thing may not make much sense to you now, but you'll be learning how they work when you study nonlinear equations. (As for the wolves, their population is dwindling due to inbreeding and disease. These equations predict that the moose population will spike as the wolves disappear, and then plummet in turn.)
So, now let’s look at how algebra applies to the design of a bicycle. There are many things to consider. The frame needs to be strong, because it will take a lot of wear and tear out on the road. But it also needs to be light, because lighter bikes are faster and take less energy to pedal. How should a design engineer balance strength vs. weight? By comparing equations! Our bike maker can draw up a series of graphs for frames made with different designs or out of different materials. Graphs like these:
These graphs show the trade-offs between weight and durability among three different materials. Now the designer has a basis for choosing which option is the best fit (I'd go with carbon fiber—for the same strength it is lighter than steel and aluminum.) You'll learn how to make and interpret graphs of equations like this in the unit on analyzing and graphing linear equationsAn equation that describes a straight line..
We may not always see it or appreciate it, but algebra is part of our daily lives. For a mathematician, its beauty alone may make studying the field worthwhile. For everyone else, algebra is important because it gives us a way to make sense of a complicated world and find creative and effective ways to describe and explore our place in it.