Density of Numbers Math Example 4

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Example 4

medium
Are there integers between 33 and 44? Are there rationals? Explain what this says about the density of integers vs. rationals.

Solution

  1. 1
    Integers between 33 and 44: there are none (integers are …,2,3,4,5,…\ldots, 2, 3, 4, 5, \ldots with no values in between).
  2. 2
    Rationals between 33 and 44: infinitely many, e.g., 3.1,3.5,72,134,…3.1, 3.5, \dfrac{7}{2}, \dfrac{13}{4}, \ldots
  3. 3
    Conclusion: integers are NOT dense (there are gaps between consecutive integers), but rationals ARE dense (between any two rationals there are infinitely more).

Answer

No integers, but infinitely many rationals, lie between 33 and 44. Integers are discrete; rationals are dense.
Density distinguishes number systems. The integers are discrete — they have 'next' and 'previous' elements with gaps in between. The rationals are dense — no two distinct rationals have a gap with nothing in between. The reals are even denser, containing all irrationals too.

About Density of Numbers

The property that between any two distinct real numbers, there are infinitely many other real numbers—no two are 'adjacent'.

Learn more about Density of Numbers →

More Density of Numbers Examples

Example 1 medium

Find three rational numbers strictly between [formula] and [formula].

Example 2 hard

Show that there is an irrational number between [formula] and [formula], and find one explicitly.

Example 3 easy

Find a rational number between [formula] and [formula] by averaging, and another by finding a decima

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