mathematics

Ostrowski's theorem

Ostrowski's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ostrowski's theorem rather than just read about it. In short: In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. Theorem statement An absolute value on the rational numbers is a function | ⋅ | ∗ : Q → R {\displaystyle |\cdot |_{*}:\mathbb {Q} \to \mathbb {R} } satisfying for…

Key takeaways

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

Reference excerpt

In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value.

Theorem statement An absolute value on the rational numbers is a function | ⋅ | ∗ : Q → R {\displaystyle |\cdot |_{*}:\mathbb {Q} \to \mathbb {R} } satisfying for all x , y ∈ Q {\displaystyle x,y\in \mathbb {Q} }

| x | ∗ ≥ 0 {\displaystyle |x|_{*}\geq 0} , with equality if and only if x = 0 {\displaystyle x=0}

| x y | ∗ = | x | ∗ | y | ∗ {\displaystyle |xy|_{*}=|x|_{*}|y|_{*}}

| x + y | ∗ ≤ | x | ∗ + | y | ∗ {\displaystyle |x+y|_{*}\leq |x|_{*}+|y|_{*}}

Two absolute values | ⋅ | {\displaystyle |\cdot |} and | ⋅ | ∗ {\displaystyle |\cdot |_{*}} on the rationals are defined to be equivalent if they induce the same topology; this can be shown to be equivalent to the existence of a positive real number λ ∈ ( 0 , ∞ ) {\displaystyle \lambda \in (0,\infty )} such that

| x | ∗ = | x | λ {\displaystyle |x|_{*}=|x|^{\lambda }}

for all rational x {\displaystyle x} . (Note: In general, if | x | {\displaystyle |x|} is an absolute value, | x | λ {\displaystyle |x|^{\lambda }} is not necessarily an absolute value anymore; however if two absolute values are equivalent, then each is a positive power of the other.) The trivial absolute value on any field K is defined to be

| x | 0 := { 0 , x = 0 , 1 , x ≠ 0. {\displaystyle |x|_{0}:={\begin{cases}0,&x=0,\\1,&x\neq 0.\end{cases}}}

The real absolute value on the rationals Q {\displaystyle \mathbb {Q} } is the standard absolute value on the reals, defined to be

| x | ∞ := { x , x ≥ 0 , − x , x < 0. {\displaystyle |x|_{\infty }:={\begin{cases}x,&x\geq 0,\\-x,&x<0.\end{cases}}}

This is sometimes written with a subscript 1 instead of infinity. For a prime number p, the p-adic absolute value on Q {\displaystyle \mathbb {Q} } is defined as follows: any non-zero rational x can be written uniquely as x = p n a b {\displaystyle x=p^{n}{\tfrac {a}{b}}} , where a and b are coprime integers not divisible by p, and n is an integer; so we define

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Ostrowski's theorem

Start with the simplest possible case. Write down what Ostrowski's theorem 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 Ostrowski's theorem 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 Ostrowski's theorem 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 Ostrowski's theorem

In research: Ostrowski's theorem 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 Ostrowski's theorem 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: Ostrowski's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in algebraic number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Ostrowski's theorem 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.

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How to study Ostrowski's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ostrowski's theorem 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 Ostrowski's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ostrowski's theorem in simple terms?

Why does Ostrowski's theorem 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 Ostrowski's theorem?

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 Ostrowski's theorem.