What can be proved about whole numbers and their ratios, and where do they stop being enough?
Level 1
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6 ideas. Read the hook, pick an answer, see the picture.
The Seven Liberal Arts, Hands-On
The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium
Euler's polynomial, n squared plus n plus 41, gives a prime for every n from 0 to 39. At n = 40 it gives 1681, which is 41 squared. Testing refutes a rule quickly; it cannot prove it for every n.
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You want a new square with exactly twice the area of an old one. What should its side be?
Doubling the side quadruples the area; one and a half times the side gives 2.25 times. Cut the old square along its diagonal: the square on that diagonal holds four such halves, the old square two. Both wrong options are the slave boy's guesses in Plato's Meno (82e, 83e); Socrates leads him to the diagonal (85a-b).
Proof and Precise Reasoning: From Arguments to Theorems
Euclid's proof that the primes never run out is often presented as a proof by contradiction. His own version is a recipe. Give it any finite list of primes, and it produces a prime not on the list.
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Take 'If it rained, the road is wet'. Its truth table has four rows: rain or not, wet or dry. In how many rows is it false?
Only rain with a dry road makes it false. Without rain it promised nothing, so it counts as true, wet road or dry.
What Counts as Proof
Example
Shows a claim holds once
Can fail at the next case
Computation
Checks particular cases
Refutes; proves only if every case is checked
Diagram or simulation
Suggests a truth
Can hide assumptions
Evidence about a model only
Proof
Each step follows from definitions
Covers every case in the domain
Only a valid proof establishes a universal claim (checking every case of a finite domain is one); the rest suggest or refute.
Discrete Mathematics and Graphs: Counting, Relations and Networks
Milgram's 1967 small-world article reports: 'Of 160 chains that started in Nebraska, 44 were completed and 126 dropped out.' Add them up before you read on.
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You join 100 computers with cables, each running between two machines. Every machine can reach every other, and the cables form no closed loop. How many cables are there?
Connected with no cycles is a tree. Every tree on n vertices has exactly n - 1 edges, so 100 computers take 99 cables.
Discrete mathematics topic order
1Logic, setsUnion, intersection, complement, power sets.
2RelationsRelations and functions, including equivalence relations.
3CountingCount without overcounting; foundation of probability.
4InductionProve for all n; recurrences like Fibonacci.
Square both sides of sqrt(x + 2) = x and you get two answers: 2 and -1. Only 2 survives the check, since sqrt(1) is 1, not -1. Squaring smuggled in an impostor.
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Cancelling turns (x^2 - 1)/(x - 1) into x + 1. For which values of x do the two actually give the same number?
At x = 1 the fraction divides by zero and has no value, but x + 1 gives 2. Simplifying does not erase that restriction.
Algebra's dependency groups
7Quadratics, growthFactoring, completing the square, the quadratic formula, exponential models; leans on groups 4, 5 and 6.
6FactoringCommon factors, difference of squares, trinomials; leans on groups 1 and 5.
5Exponents, radicalsLaws of exponents, rational exponents, simplifying roots; leans on group 1 only.
4FunctionsNotation f(x), domain, range, graph reading, composition; leans on groups 2 and 3.
3Lines, systemsSlope, intercepts, graphing, elimination and substitution; leans on groups 1 and 2.
2EquationsSolving, rearranging formulas, flipping an inequality when multiplying by a negative; leans on group 1.
1ExpressionsVariables, like terms, distribution, evaluating; rests on arithmetic.
↑ higher level
Seven groups, numbered as in the table on the full page. Each leans on earlier ones, so repair the group where the error began.
A Map of Mathematics: Twenty-Five Areas and What Depends on What
Euler proved no walk crosses Konigsberg's seven bridges once each, and graph theory began. Routers now run graph algorithms. Shannon showed Boole's algebra of logic describes switching circuits. Some regions of mathematics must be crossed before others.
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Which of these can you start with algebra and careful reading, before any calculus?
Introductory linear algebra needs algebra and proof familiarity, not calculus. Real analysis, differential equations and mathematical statistics all need calculus first.
Mathematics in Six Regions
FoundationsAlgebra, geometry, trigonometry, then precalculus and calculus.
TrunkProof, discrete mathematics, algorithms and probability.
DataStatistics, linear algebra and data modelling.
NetworksGraph theory, optimization, information theory, queueing and time series.
ChangeNumerical methods, differential equations, control and dynamics.
JudgmentBayesian decisions, causal inference and game theory.
Six regions, then three advanced bridges: computability, cryptography and spectral methods. Dependencies matter more than numbers; many areas can be studied in parallel.
Drop from 100 chips to 50, and a 50 percent gain only gets you to 75. Getting back to 100 takes a 100 percent gain. Losses and gains are not mirror images.
Pick one
You put a 10 percent chance on a faulty router. Then you see a symptom three times likelier with a faulty router than with a good one. What chance should you put on it now?
As odds, 10 percent is 1 to 9. The symptom triples the faulty side: 3 to 9, which is 3 in 12, or 25 percent. Triple the odds, not the 10 percent. Answering 75 percent ignores where you started.
Six Tiers of Decision Math
6SystemsConstraints, bottlenecks and feedback loops with delays.
5StatisticsConfounders, survivorship bias, relative versus absolute risk.
4RiskAverages hide spread; survive before you optimise.
3ProbabilityBase rates, conditional probability and expected value.
2CompoundingRepeated growth looks slow, then overwhelming.
1NumeracyPercentages, ratios, estimation; notice when a claim is unreasonable.
↑ higher level
Order study by the cost of ignorance, not by topic coverage.
8 questions from the Arithmetic curriculum, Round 1.
Question 1 of 8
Which statement is closest to what Euclid actually states in Elements IX.20?
Euclid refuses to speak of a completed infinity and claims only that any assigned multitude of primes is exceeded. The construction does not always give a prime, and it does not find the next one.
Question 2 of 8
In the sense of Euclid and the Academy, what is a ratio?
The Academy defines a ratio as a relation between two magnitudes of the same kind, a comparison and not a number. Euclid's Book V handles ratios of incommensurable magnitudes precisely because they cannot be written as fractions.
Question 3 of 8
A classmate checks a formula for n = 1, 2, 3 and 4, finds that it works each time, and says it holds for every positive integer n. What is the flaw, and what would fix it?
A computation checks particular cases. It can refute a universal claim but cannot establish one unless the domain is finite and every case is checked. Induction proves a statement for all n from a base case and a step.
Question 4 of 8
Apply the construction to the list 3, 5, 7. The product plus 1 is N = 106. Which statement is correct?
105 = 3 x 5 x 7, so 106 leaves remainder 1 when divided by each of 3, 5 and 7. 106 = 2 x 53, and neither 2 nor 53 is on the list. The proof only needs a prime divisor that is missing from the list; N itself need not be prime.
Question 5 of 8
What does it mean to say that a square's diagonal and side are incommensurable?
Incommensurable means having no common measure. It does not mean unmeasurable or infinite: the diagonal is exactly as long as it is. What fails is the assumption that every pair of lengths is a ratio of whole-number counts.
Question 6 of 8
In the induction step for 1 + 2 + ... + n = n(n + 1)/2, which expression equals n(n + 1)/2 + (n + 1)?
n(n + 1)/2 + (n + 1) = (n squared + n + 2n + 2)/2 = (n squared + 3n + 2)/2 = (n + 1)(n + 2)/2, which is the formula with n + 1 in place of n.
Question 7 of 8
A formula gives a prime for 40 values of n in a row, then fails at the 41st. What does this show?
A computation checks particular cases. It can refute a universal claim with a single counterexample, but it cannot establish one over an unbounded domain. Euler's example shows how persuasive a long run of successes can be.
Question 8 of 8
A board is 0.75 m long and a pipe is 1.25 m long. A rod 0.25 m long measures both exactly. What is the ratio board : pipe in whole numbers, in lowest terms?
0.75 = 3 x 0.25 and 1.25 = 5 x 0.25, so the ratio is 3 : 5. The ratio 5 : 3 reverses it, and 75 : 125 is not in lowest terms because both numbers are multiples of 25.