Real Numbers Math Example 4

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Example 4

medium
Give one example of a real number between 2\sqrt{2} and 3\sqrt{3}.

Solution

  1. 1
    2≈1.414\sqrt{2} \approx 1.414 and 3≈1.732\sqrt{3} \approx 1.732.
  2. 2
    Any number strictly between these works. For example, 1.51.5 (since 1.414<1.5<1.7321.414 < 1.5 < 1.732).
  3. 3
    Alternatively, 2+32\frac{\sqrt{2}+\sqrt{3}}{2} is exact but irrational.

Answer

1.5 (or any value between ≈1.414 and ≈1.732)1.5 \text{ (or any value between } \approx 1.414 \text{ and } \approx 1.732\text{)}
The real line has no gaps. Between any two irrationals, there exist infinitely many reals — both rational (like 1.5) and irrational. Using decimal approximations of the bounds makes it easy to find a rational example.

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More Real Numbers Examples

Example 1 easy

Classify each number as rational or irrational, and state whether it is a real number: [formula], [f

Example 2 medium

Show that between any two distinct real numbers [formula] and [formula] (with [formula]), there exis

Example 3 easy

Which of the following are NOT real numbers? [formula], [formula], [formula], [formula].

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