Real analysisLesson 1 of 3 in the course Real numbers and sequences

Real Numbers: Order, Completeness, Decimals, and Sequences

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Real numbers describe continuous quantities and form the number system used throughout much of mathematics. This lesson introduces the real line and the main families of real numbers.

A real number is a number that can measure a continuous one-dimensional quantity, such as length, duration, or temperature. Continuity means that between two distinct values there can be values with arbitrarily small differences. The set of real numbers is denoted by ℝ.

All flashcards

Rational number
A number expressible as p/qp/q, where pp and qq are integers and q≠0q\ne 0.
Transcendental number
A real number that is not a root of any nonzero polynomial with rational coefficients.
Dedekind completeness
Every nonempty real set bounded above has a least upper bound in the real numbers.
Cauchy sequence
A sequence whose terms eventually become arbitrarily close to one another.
Convergent sequence
A sequence whose terms eventually lie within every positive tolerance of a fixed limit.

Quiz questions

  1. When both sides of a strict inequality are multiplied by a negative real number, what happens to the inequality?Answer: Its direction reverses.
  2. Which statement expresses the total-order property of the real numbers for any aa and bb?Answer: Exactly one of a<ba<b, a=ba=b, or b<ab<a holds.
  3. What is the additive identity in the real numbers?Answer: Zero, since adding it leaves every real number unchanged.
  4. What does Dedekind completeness guarantee for a nonempty set of real numbers bounded above?Answer: The set has a least upper bound in the real numbers.
  5. What does completeness of the real numbers guarantee about every Cauchy sequence of real numbers?Answer: It converges to a real number.

Based on the Wikipedia article Real number, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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