mathematics

Non-integer base of numeration

Non-integer base of numeration 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 Non-integer base of numeration rather than just read about it. In short: A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of x = d n … d 2 d 1 d 0 . d − 1 d − 2 … d − m {\displaystyle x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}} is x = β n d n + ⋯ + β 2 d 2 + β d 1 + d 0 + β − 1 d − 1 + β − 2 d − 2 + ⋯ + β − m d − m . {\displaystyle {\begin{aligned}x&=\beta ^{n}d_{n}+\cdots +\be…

Key takeaways

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

Reference excerpt

A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of

x = d n … d 2 d 1 d 0 . d − 1 d − 2 … d − m {\displaystyle x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}}

x = β n d n + ⋯ + β 2 d 2 + β d 1 + d 0 + β − 1 d − 1 + β − 2 d − 2 + ⋯ + β − m d − m . {\displaystyle {\begin{aligned}x&=\beta ^{n}d_{n}+\cdots +\beta ^{2}d_{2}+\beta d_{1}+d_{0}\\&\qquad +\beta ^{-1}d_{-1}+\beta ^{-2}d_{-2}+\cdots +\beta ^{-m}d_{-m}.\end{aligned}}}

The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β−1]. There are applications of β-expansions in coding theory and models of quasicrystals.

Construction β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ2 for β = φ, the golden ratio. A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm, essentially due to Rényi (1957) and formulated as given here by Frougny (1992). Let β > 1 be the base and x a non-negative real number. Denote by ⌊x⌋ the floor function of x (that is, the greatest integer less than or equal to x) and let {x} = x − ⌊x⌋ be the fractional part of x. There exists an integer k such that βk ≤ x < βk+1. Set

d k = ⌊ x / β k ⌋ {\displaystyle d_{k}=\lfloor x/\beta ^{k}\rfloor }

and

r k = { x / β k } . {\displaystyle r_{k}=\{x/\beta ^{k}\}.\,}

For k − 1 ≥ j > −∞, put

d j = ⌊ β r j + 1 ⌋ , r j = { β r j + 1 } . {\displaystyle d_{j}=\lfloor \beta r_{j+1}\rfloor ,\quad r_{j}=\{\beta r_{j+1}\}.}

In other words, the canonical β-expansion of x is defined by choosing the largest dk such that βkdk ≤ x, then choosing the largest dk−1 such that βkdk + βk−1dk−1 ≤ x, and so on. Thus it chooses the lexicographically largest string representing x. With an integer base, this defines the usual radix expansion for the number x. This construction extends the usual algorithm to possibly non-integer values of β.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Non-integer base of numeration

Start with the simplest possible case. Write down what Non-integer base of numeration 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 Non-integer base of numeration 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 Non-integer base of numeration 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 Non-integer base of numeration

In research: Non-integer base of numeration 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 Non-integer base of numeration 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: Non-integer base of numeration is common in secondary-school and first-year university syllabi. It links to neighbouring topics Coding theory, Non-standard positional numeral systems, Number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Non-integer base of numeration 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 Non-integer base of numeration in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Non-integer base of numeration 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 Non-integer base of numeration out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Non-integer base of numeration in simple terms?

A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of x = d n … d 2 d 1 d 0 . d − 1 d − 2 … d − m {\displaystyle x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}} is x = β n d n + ⋯ + β 2 d 2 + β…

Why does Non-integer base of numeration 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 Non-integer base of numeration?

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 Non-integer base of numeration.