Chapter 16
Special Manipulations
Asked to add every whole number from one to a hundred, you can start at the left and work across, and it will take a while. Or you can look at the list first and notice how it is built. Algebra is full of problems like that. The methods you have spent this course learning work on all of them, patiently and blindly. A problem with a shape of its own rewards anyone who notices the shape first, and this chapter is about what that noticing is worth, and what it costs.
What You'll Explore
3 lessons.
- Raising Equations to Powers
Adding the same number to both sides of an equation never changes which values solve it. Multiplying both sides does not either. Raising both sides to a power looks like one more move of the same family; you will test whether it behaves like one, and what that means for a variable trapped under a root sign.
- Self-Similar Expressions
A square root with another square root underneath, and another under that, going on forever: can something endless like that have an exact value? Repeating decimals raise the same worry. This lesson asks what lets such an expression be pinned to a single number, and when that reasoning quietly fails.
- Using Symmetry
Swap the two letters in a problem and, sometimes, nothing about it changes at all. That looks like a curiosity rather than a tool. This lesson takes it seriously and asks what a problem that cannot tell its two unknowns apart is quietly telling you, and how far that single observation can carry you.