Density of Numbers Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Find a rational number between 0.30.3 and 0.40.4 by averaging, and another by finding a decimal with more places.

Solution

  1. 1
    Average: 0.3+0.42=0.72=0.35\dfrac{0.3 + 0.4}{2} = \dfrac{0.7}{2} = 0.35. So 0.350.35 is between 0.30.3 and 0.40.4.
  2. 2
    Decimal method: any three-digit decimal 0.3x0.3x with 1≤x≤91 \leq x \leq 9 works, for example 0.370.37.

Answer

0.350.35 and 0.370.37 are both between 0.30.3 and 0.40.4.
Density of the rationals guarantees that between any two decimals (rational numbers), we can always write another decimal with more decimal places, or simply take the average. There is no 'next' rational number.

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 4 medium

Are there integers between [formula] and [formula]? Are there rationals? Explain what this says abou

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