Sentence examples for a decimal expansion from inspiring English sources

In this sense, one can construct real numbers (i.e., converging choice sequences) that do not yet have a decimal expansion.

The answer to the question in the title of Brouwer's paper "Does Every Real Number Have a Decimal Expansion?" (1921A) turns out to be no.

1920 Start of the "Grundlagenstreit" (Foundational Debate) with Brouwer's lecture at the "Naturforscherversammlung" in Bad Nauheim, published in 1921, "Does Every Real Number Have a Decimal Expansion?"; the debate was amplified by Weyl's defence of intuitionism in 1921, "On the New Foundational Crisis of Mathematics"; Hilbert responds in 1922, "The New Grounding of Mathematics".

A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.

The series definition above is a simple way to define the real number named by a decimal expansion.

First, Richman defines a nonnegative decimal number to be a literal decimal expansion.

This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.

Each irrational number can be expressed as an infinite decimal expansion with no regularly repeating digit or group of digits.

This is because an infinite decimal expansion such as 3.14159… (the value of the constant π) actually corresponds to the sum of an infinite series 3 + 1/10 + 4/100 + 1/1,000 + 5/10,000 + 9/100,000 +⋯,and the concept of limit is required to give such a sum meaning.

The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999... = 1?" by Fred Richman in "[[Mathematics Magazine]]", which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.

The sets of integers (… -3, -2, -1, 0, 1, 2, 3, and so on), rational numbers (all numbers that can be expressed as the ratio of two integers, like 3/4, -5/18, 2/1, etc)., and real numbers (any number that can be represented as a [possibly infinite] decimal expansion) all have infinite cardinality.