The Algebra Playbook
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In the set of activities that we call games, we find a subset that we call puzzles, problems that we contrive for ourselves solely for the pleasure of solving them. At its most basic, a puzzle consists of an array of objects (and we include disarray, as in the case of the initial state of a jigsaw puzzle, as an array), some description of the array that represents the solution of the puzzle, and a set of rules for solving the puzzle.
Consider, as an example, the Towers of Hanoi. The puzzle consists of three spindles jutting up from a board and a small set of discs set one on top of another on one of the spindles by holes drilled through their centers. The discs all have different diameters and we so place them on the spindle that no disc lies on top of a smaller one. We say that we have solved the puzzle when we have moved the discs onto another spindle in the same order. Three simple rules constrain what we can do to solve the problem, making it more challenging (and, therefore, interesting) and shaping the play that we must find: 1) a disc can only come to rest on one of the spindles, 2) we may move only one disc at a time, and 3) no disc may come to rest on top of a smaller disc. The set of moves, following those rules, that solve the problem constitute a play.
So we include algebra (along with other parts of higher mathematics beyond simple arithmetic) as an element of the set of puzzles. The objects (or game pieces) of algebra consist of algebraic formulae, abstract recipes for doing arithmetic with letters that represent both constant numbers and variable numbers. We set up the puzzle in one of its endless forms by taking a pair of formulae and putting an equality sign between them. Usually we count the puzzle solved when we have one of the variables by itself on one side of the equality sign and only on that side of the equality sign and all of the other letters and numbers organized into a formula on the other side of the equality sign. And we have only one rule for obtaining that solutionB we can apply any of the operations of arithmetic to the equation so long as we do to the formula on one side of the equality sign exactly what we do to the formula on the other side.
Consider, for example, the famous (or infamous if you wish) quadratic equation,
We want to solve that equation for x; that is, we want to manipulate the equation into a form that leaves x by itself on one side of the equality sign and leaves some formula involving a, b, and c on the other side. We can= t see immediately how we might obtain such a solution: it looks well nigh insoluble. But we do have a solvent for that equation, a play called completing the square:
1. subtract c;
2. divide by a;
The formula on the left looks like part of the square of some formula. We want to restore the missing term, so
3. square A+B;
4. by comparison between that equation and Equation 3, let A=x and B=b/2a, so add b2/4a2 to Equation 3;
5. give the right side a common denominator (this is more a matter of"housekeeping", keeping the formulae as simple and as neat as possible, and is not an actual move in solving the equation);
6. extract the square root;
7. subtract b/2a;
in which the plus-or-minus sign represents the fact that the square root can give us either a positive number or a negative number. We thus have two possible solutions of the equation. Q.E.I. (which stands for quod erat inveniendum, Latin for"what was to be found").
Like the quadratic equation, many equations will appear insoluble at first, until some clever soul gains an insight that leads to a solution. Since few of us are so gifted that we can do such things consistently with every difficult equation that we encounter, we really should have a playbook, somewhat like the Table of Integrals, recording some of the better plays solving the most difficult equations. In these essays I will try to create the beginning of such a playbook. So now on to the more interesting plays.
Appendix: Euclid's Common Notions
In his famous book on plane geometry, The Elements, the Greek mathematician Euclid presented to his readers five Common Notions that they might use in the solving of problems in geometry. But we can also see that they apply equally well to algebra. Expressed in English, rather than Euclid's Greek, we have them as:
1. Things equal to the same thing are also equal to one another.
2. If equals are added to equals the wholes are equal.
3. If equals are subtracted from equals the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
We can actually extend that list with the other four basic operations of arithmetic:
6. If equals are multiplied by equals the products are equal.
7. If equals are divided by equals the quotients are equal.
8. If equals are raised to the same power the ponents are equal.
9. If the same roots are extracted from equals the radicals are equal.
It all boils down to the same basic rule of algebra: if we perform the same operations on equals, the results are equal.
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