Lagrange number

Key takeaways

Lagrange number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lagrange number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lagrange number from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Lagrange numbers (A382098 and A382099 in the OEIS) are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.

Definition Hurwitz improved Peter Gustav Lejeune Dirichlet's criterion on irrationality to the statement that a real number α {\displaystyle \alpha } is irrational if and only if there are infinitely many rational numbers p q {\textstyle {\frac {p}{q}}} , written in simplest terms, such that

| α − p q | < 1 5 q 2 {\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}} . This was an improvement on Dirichlet's result which had 1 q 2 {\textstyle {\frac {1}{q^{2}}}} on the right-hand side. The above result is best possible, since the golden ratio φ {\displaystyle \varphi } is irrational. If we replace 5 {\displaystyle {\sqrt {5}}} with any larger number in the above expression, we will only be able to find finitely many rational numbers that satisfy the inequality for α = φ {\displaystyle \alpha =\varphi } . Hurwitz also showed that if we omit φ {\displaystyle \varphi } (and numbers derived therefrom), we can increase the 5 {\displaystyle {\sqrt {5}}} to 2 2 {\displaystyle 2{\sqrt {2}}} . Again this new bound is best possible, this time with 2 {\displaystyle {\sqrt {2}}} being the problem. If we omit 2 {\displaystyle {\sqrt {2}}} , we can further increase the 2 2 {\displaystyle 2{\sqrt {2}}} to 221 5 {\textstyle {\frac {\sqrt {221}}{5}}} . Repeating this process we get the infinite series 5 , 2 2 , 221 5 , 1517 13 , … {\textstyle {\sqrt {5}},\;2{\sqrt {2}},\;{\frac {\sqrt {221}}{5}},\;{\frac {\sqrt {1517}}{13}},\;\ldots } which converges to 3. These are the Lagrange numbers, named after Joseph Louis Lagrange.

Relation to Markov numbers The n {\displaystyle n} th Lagrange number L n {\displaystyle L_{n}} is given by

L n = 9 − 4 M n 2 {\displaystyle L_{n}={\sqrt {9-{\frac {4}{M_{n}^{2}}}}}}

where M n {\displaystyle M_{n}} is the n {\displaystyle n} th Markov number—the n {\displaystyle n} th-smallest integer m {\displaystyle m} such that the equation

m 2 + x 2 + y 2 = 3 m x y {\displaystyle m^{2}+x^{2}+y^{2}=3mxy}

has a solution in positive integers x {\displaystyle x} and y {\displaystyle y} .

References

Cassels, J.W.S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. Zbl 0077.04801. Conway, J.H.; Guy, R.K. (1996). The Book of Numbers. New York: Springer-Verlag. ISBN 0-387-97993-X.

External links Lagrange number. From MathWorld at Wolfram Research. Introduction to Diophantine methods irrationality and transcendence Archived 2012-02-09 at the Wayback Machine - Online lecture notes by Michel Waldschmidt, Lagrange Numbers on pp. 24–26.

Worked examples

Example 1 — a first encounter with Lagrange number

Start with the simplest possible case. Write down what Lagrange number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lagrange number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lagrange number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lagrange number

In research: Lagrange number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lagrange number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lagrange number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Diophantine approximation, so understanding it makes those chapters shorter.
In everyday life: Look for Lagrange number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Lagrange number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lagrange number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lagrange number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lagrange number in simple terms?

In mathematics, the Lagrange numbers (A382098 and A382099 in the OEIS) are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.

Why does Lagrange number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lagrange number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lagrange number.