PG2.6 Stage review: choosing a technique

Simulate

Check the claims with numbers

This review has four claims, and each is easiest to prove with a different technique from this stage. First, test them all. The first claim compares $a^2 + b^2$ with $2ab$ , for 10,000 random pairs of exact decimals from $-10$ to $10$ .

import random from fractions import Fraction random.seed(4) gaps = [] for _ in range(10_000): a = Fraction(random.randint(-1000, 1000), 100) b = Fraction(random.randint(-1000, 1000), 100) gaps.append(a**2 + b**2 - 2 * a * b) print("smallest value of a^2 + b^2 - 2ab:", min(gaps)) print("pairs where it is 0:", sum(g == 0 for g in gaps))

⌘/Ctrl + Enter runs the cell

The gap $a^2 + b^2 - 2ab$ is never negative, and its smallest value, 0, turns up only twice. The second claim is about integers $m$ and $n$ whose product is even.

ns = range(-30, 31) even_products = [(m, n) for m in ns for n in ns if (m * n) % 2 == 0] fails = [(m, n) for m, n in even_products if m % 2 == 1 and n % 2 == 1] print("pairs with an even product:", len(even_products), " with both numbers odd:", len(fails))

⌘/Ctrl + Enter runs the cell

None of the 2,821 pairs with an even product has both numbers odd. The third claim adds two distances on the number line, $|x - 1| + |x - 3|$ , for 801 exact points from $-2$ to $6$ .

from fractions import Fraction xs = [Fraction(k, 100) for k in range(-200, 601)] print("points:", len(xs)) values = [abs(x - 1) + abs(x - 3) for x in xs] smallest = min(values) where = [x for x, v in zip(xs, values) if v == smallest] print("smallest value:", smallest, " reached from x =", min(where), "to x =", max(where))

⌘/Ctrl + Enter runs the cell

The sum is never below 2, and it equals 2 for every $x$ from 1 to 3. The last claim is about $\log_2 3$ , the power of 2 that gives 3. If it were a fraction $\frac{p}{q}$ , then $2^p = 3^q$ (the proof shows why), so the cell searches for that.

import math print("log2(3) =", math.log2(3)) print("p, q up to 60 with 2^p = 3^q:", [(p, q) for p in range(1, 61) for q in range(1, 61) if 2**p == 3**q])

⌘/Ctrl + Enter runs the cell

No pair works. Decimals alone can never show that a number is not a fraction, so this claim needs a proof too.

Conjecture

a

Every test passes, but each claim is about infinitely many numbers. Prove all four.

Choosing a technique

The shape of a claim suggests a technique. Here are the four of this stage, with the lessons that taught them.

If the claim	try	lesson
follows from a known fact by algebra	a direct proof	PG2.1
is "if $P$ , then $Q$ ", and "not $Q$ " is easier to use than $P$	the contrapositive	PG2.4
is about every number, and the numbers split into kinds	cases	PG2.4
says something does not exist, or is irrational	contradiction	PG2.5

Each model proof's checklist names the technique it uses. Another technique can work too, as long as every step is justified.

Format 4

Write it

Prove that $a^2 + b^2 \ge 2ab$ for all real numbers $a$ and $b$ . Say which technique you use, and why.

Your proof

Saved in this browser as you type.

Write your own first: a proof you have tried to write teaches far more than one you only read.

Format 4

Write it

Prove that for all integers $m$ and $n$ , if $mn$ is even, then $m$ is even or $n$ is even. Say which technique you use, and why.

Your proof

Saved in this browser as you type.

Write your own first: a proof you have tried to write teaches far more than one you only read.

Format 4

Write it

Prove that $|x - 1| + |x - 3| \ge 2$ for every real number $x$ . Say which technique you use, and why.

Your proof

Saved in this browser as you type.

Write your own first: a proof you have tried to write teaches far more than one you only read.

Format 4

Write it

Prove that $\log_2 3$ is irrational. Say which technique you use, and why.

Your proof

Saved in this browser as you type.

Write your own first: a proof you have tried to write teaches far more than one you only read.

(a - b)^2 \ge 0

Stage review: choosing a technique

Check the claims with numbers

Choosing a technique

Write it

Write it

Write it

Write it

Summary card

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