Choosing Good Notation

In one sentence

The notation you choose at the start of a solution often decides whether the rest goes smoothly or gets stuck at every step. Good notation does three things: it reduces the number of unknowns, it keeps the symmetry, and it matches the operations to the structure: rotation becomes multiplication, iteration becomes addition, choosing elements becomes a string of $0$ s and $1$ s.

Example: x + y = 5, xy = 3; find x³ + y³

Step 1 / 4

Notation 1: solve for x, y
$x, y$ are the two roots of $t^{2} - 5 t + 3 = 0$ , namely $\frac{5 \pm 13}{2}$ . Then cube them and add, dragging the square roots along.

$x, y = \frac{5 \pm 13}{2}$
Notation 2: only s and p
The recurrence
The power of notation

Predict first

$x + y = 5$ and $x y = 3$ . What is $x^{4} + y^{4}$ ?

Choose an answer first; then you can see the result.

Stop 1 · Elements and Sets (Vol. 1)

To count pairs of subsets, change the notation: do not record the subsets, record the state of each element.

One problem, several solutions

How many ordered pairs of sets $(A, B)$ satisfy $A \subseteq B \subseteq {1, 2, \dots, n}$ ?

Notation: choose B first, then A

There are $(j n)$ sets $B$ with $∣ B ∣ = j$ , and each such $B$ has $2^{j}$ subsets $A$ .

The total is $j = 0 \sum n (j n) 2^{j} = (1 + 2)^{n} = 3^{n}$ .

CostNatural, but you need the binomial theorem to finish.

Notation: the state of each element

Each element $x$ has only three possible states: $x \in / B$ ; $x \in B$ but $x \in / A$ ; $x \in A$ .

The pairs $(A, B)$ correspond one-to-one to “three-state strings” of length $n$ , and there are $3^{n}$ of them.

CostOne sentence: change the notation and the sum disappears.

Stop 2 · Iteration of a Function (Vol. 2)

Iterating $f (x) = \frac{x}{1 + x}$ gets messier with every step. Switch to the notation of reciprocals, and iteration is just “add one”.

Example: f(x) = x/(1 + x); find f⁽ⁿ⁾(x)

Step 1 / 4

Compute twice directly
$f^{(2)} (x) = \frac{x}{1 + 2 x}$ , $f^{(3)} (x) = \frac{x}{1 + 3 x}$ … You can guess the pattern, but each time you must simplify a compound fraction.

$f^{(2)} (x) = \frac{\frac{x}{1 + x}}{1 + \frac{x}{1 + x}}$
Change notation: take reciprocals
Iteration becomes addition
Flip back

Stop 3 · Complex numbers (Vol. 4)

Write points of the plane as complex numbers, and “rotate $9 0^{\circ}$ counterclockwise about $A$ ” becomes “multiply by $i$ ”.

Example: square ABCD has its vertices in counterclockwise order, A(1, 1), B(4, 2); find C and D

Step 1 / 4

Change notation
Write the points as complex numbers: $A = 1 + i$ , $B = 4 + 2 i$ , and $A B$ corresponds to $B - A = 3 + i$ .

$B - A = 3 + i$
Rotation = multiply by i
Write down both points
Conclusion

Stop 4 · Choosing a Good Coordinate System (Vol. 7)

A coordinate system is notation that we put on the figure ourselves. Put it in a symmetric position, and the formulas come out symmetric.

Example: P is any point in the plane of rectangle ABCD; prove that PA² + PC² = PB² + PD²

Step 1 / 4

Choose the coordinates
Put $A$ at the origin and two sides along the axes: $A (0, 0)$ , $B (a, 0)$ , $C (a, b)$ , $D (0, b)$ , $P (x, y)$ .

$A (0, 0), B (a, 0), C (a, b), D (0, b)$
Left side
Right side
Exactly the same

Practice

Exercise 1Basic

How many ordered pairs of sets $(A, B)$ satisfy $A \subseteq B \subseteq {1, 2, 3, 4, 5}$ ?

Exercise 2Basic

Let $f (x) = \frac{x}{1 + 2 x}$ . Find $f^{(10)} (1)$ ( $f$ iterated $10$ times).

Exercise 3Advanced

$x + y = 3$ and $x y = 1$ . Find $x^{4} + y^{4}$ .

Exercise 4Advanced

Square $A B C D$ has its vertices in counterclockwise order, with $A (2, 0)$ and $B (5, 1)$ . Find the coordinates of $D$ (write them as $(x, y)$ ).

In one sentence

Example: x + y = 5, xy = 3; find x³ + y³

Stop 1 · Elements and Sets (Vol. 1)

Notation: choose B first, then A

Notation: the state of each element

Stop 2 · Iteration of a Function (Vol. 2)

Example: f(x) = x/(1 + x); find f⁽ⁿ⁾(x)

Stop 3 · Complex numbers (Vol. 4)

Example: square ABCD has its vertices in counterclockwise order, A(1, 1), B(4, 2); find C and D

Stop 4 · Choosing a Good Coordinate System (Vol. 7)

Example: P is any point in the plane of rectangle ABCD; prove that PA² + PC² = PB² + PD²

Practice

Mistake Museum · Exhibits from this stop

Thinking that x² + y² = (x + y)²

Reading f⁽ⁿ⁾(x) as (f(x))ⁿ

Stops on this line