Ratio and common measure: Arithmetic, Round 1

1 Learn

You will be able to

Define a ratio and a common measure, and write a ratio of two lengths in lowest terms.
Find the greatest common measure of two whole-number lengths by repeated subtraction and check it.
Reconstruct the proof that a square's diagonal and side have no common measure.
Say what incommensurable means and what it does not mean.

LabThe AcademyA ratio is a relation between two magnitudes of the same kind, not a number.
LabThe AcademyNo common measure however small, and why that is not the same as unmeasurable.
Lab√2 is irrationalThe proof that the diagonal and side have no common measure, with its lowest-terms step.
LabThe golden ratio is irrationalCutting a line in extreme and mean ratio gives phi squared = phi + 1; phi is irrational, and its continued fraction is [1; 1, 1, 1, ...].
LibraryGeometry, Euclid and Trigonometry · A track for Euclid and ApolloniusEuclid's Book V handles incommensurable lengths without assuming every ratio is a fraction.2 min glance · 5 min

2 Practise

Answer each card from memory, then grade yourself honestly. The cards join your review deck and come back just before you would forget them.

recall
What does it mean for two magnitudes to be incommensurable?

Show answer
They have no common measure, however small: no unit divides both a whole number of times. The diagonal and side of a square are the classic example.
recall
Which equal-tempered musical interval has a frequency ratio of exactly √2, and what does the proof that a square's diagonal and side are incommensurable say about it?

Show answer
The tritone, exactly half an octave: two tritones make the octave's ratio of 2, so each is √2. The proof shows √2 is not a ratio of whole numbers, so this interval can never be written as one.
recall
The proof that a square's diagonal d has no common measure with its side s supposes d : s = p : q, with whole numbers p and q in lowest terms. Which theorem then gives p² = 2q², and how?

Show answer
The Pythagorean theorem. The diagonal is the hypotenuse of a right triangle whose legs are two sides, so d² = s² + s² = 2s². Since d : s = p : q, this becomes p² = 2q².
explain
In the proof that a square's diagonal and side are incommensurable, how does p² = 2q², with p : q in lowest terms, lead to a contradiction?

Show answer
p² is even, and the square of an odd number is odd, so p is even: p = 2m. Then 4m² = 2q², so q² = 2m², and q is even by the same step. Both even contradicts lowest terms, so no common measure exists.
recall
Cutting a line in extreme and mean ratio (whole : greater = greater : less) defines the golden ratio φ. What equation does φ satisfy, and what is its value?

Show answer
Setting the lesser part to 1 and the greater to φ gives (φ + 1) : φ = φ : 1, so φ² = φ + 1. Keeping the positive root, φ = (1 + √5)/2 ≈ 1.6180339887.
explain
Why is the golden ratio φ = (1 + √5)/2 irrational?

Show answer
If φ were rational, then √5 = 2φ − 1 would be rational too. But 5 is not a perfect square, so √5 is irrational, and therefore φ cannot equal any fraction.
recall
What is the continued fraction of the golden ratio φ, and in what precise sense is φ called the "most irrational" number?

Show answer
φ = [1; 1, 1, 1, …], with every term as small as possible. That makes φ the slowest of all numbers to be approximated by fractions, the sense made precise by Hurwitz (1891).

3 Prove it: the mastery check

5 questions drawn from a pool of 12. The pass mark is 80%. There is no time limit, and you can retake it with new questions; your best result counts.

4 Prove it: the performance task

Measure one length by another

Part A. Use Euclid's repeated subtraction (Elements VII.2) on three pairs of lengths in centimetres: 84 and 36, 65 and 40, and 144 and 89. At each step subtract the shorter length from the longer as many times as it fits, then continue with the shorter length and the remainder, until the remainder is zero. Record every step in a table with the columns longer, shorter, times it fits, remainder. Name the last non-zero length as the common measure, check that it divides both original lengths exactly, and write the pair as a ratio in lowest terms. For 144 and 89, add one sentence on what you notice in the 'times it fits' column. Part B. Write the proof that a square's diagonal and side have no common measure as five numbered lines, with a reason beside each. Part C. In three or four sentences, explain to someone who has not seen the proof why the diagonal is exactly as long as it is and yet is not a ratio of whole numbers to the side.

What to hand in: One page: three step tables with the common measure and the ratio in lowest terms for each pair, the five-line proof with reasons, and the short paragraph.

Check your work against the rubric Each step table follows from the one before: every remainder equals the longer length minus the shorter length times the number of times it fits, and the next row starts from the shorter length and that remainder. For each pair, the last non-zero length is named as the common measure, both original lengths are shown to be whole multiples of it, and the pair is written as a ratio in lowest terms. The proof has five numbered lines, each with a reason beside it, such as 'Pythagorean theorem', 'the square of an odd number is odd' and 'contradicts lowest terms'. The paragraph says the diagonal is exactly as long as it is, says that what fails is the assumption that every pair of lengths is a ratio of whole numbers, and does not call the diagonal unmeasurable or infinite. The sentence on 144 and 89 says that the 'times it fits' entry is 1 in every row except the last, which is 2, and links this to the golden ratio's continued fraction [1; 1, 1, 1, ...].

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5 Discuss: the seminar

Read the text, then think, write or talk through the question with someone. There is no answer key: the aim is a better question.

Arithmetic · Round 1

Nicomachus of Gerasa, Introduction to Arithmetic, translated by Martin Luther D'Ooge (Macmillan, 1926), Book I, chapters 7 to 13Chapter 7 divides number into even and odd. Chapters 8 to 10 divide the even. Chapters 11 to 13 divide the odd into the prime and incomposite, the secondary and composite, and the kind that is composite in itself but prime and incomposite relative to another. Chapter 13 also describes the sieve of Eratosthenes and a test by repeated subtraction for whether two odd numbers have a common measure. D'Ooge's translation was published in 1926, so it entered the public domain in the United States on 1 January 2022. This Zenodo record holds a full PDF scan of the 1926 volume. Cite by book and chapter, not by page.
Euclid, Elements, Book VII (David Joyce's online edition, Clark University), Definitions 1 to 16; Propositions 1 and 2Definitions 11 to 14 define prime, relatively prime, composite and relatively composite numbers by what measures them. Propositions 1 and 2 are linked from the same page. Proposition 2 finds the greatest common measure of two numbers by repeated subtraction; Elements X.2 applies the same idea to magnitudes.

Euclid calls a prime a number 'measured by a unit alone' (Book VII, Definition 11), while Nicomachus calls it 'prime and incomposite' and files it among the odd numbers (I.11). What does each way of defining a prime make you notice, and what does each hide?

On Euclid's Definition 11, does the number 2 count as prime? Where would Nicomachus put it, given that he sorts primes among the odd numbers?
Nicomachus describes a sieve (I.13) that separates primes from composites by a procedure. Does knowing how to find primes tell you what a prime is?
In VII.1 and VII.2 Euclid subtracts the smaller number from the larger again and again. What does he assume about what it means for one number to 'measure' another?

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Ratio and common measure