Real Numbers Math Example 2

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Example 2

medium
Show that between any two distinct real numbers aa and bb (with a<ba < b), there exists another real number.

Solution

  1. 1
    Define c=a+b2c = \frac{a + b}{2}, the average (midpoint) of aa and bb.
  2. 2
    Show a<ca < c: c−a=a+b2−a=b−a2>0c - a = \frac{a+b}{2} - a = \frac{b-a}{2} > 0 since b>ab > a.
  3. 3
    Show c<bc < b: b−c=b−a+b2=b−a2>0b - c = b - \frac{a+b}{2} = \frac{b-a}{2} > 0. So a<c<ba < c < b.

Answer

c=a+b2 satisfies a<c<bc = \frac{a+b}{2} \text{ satisfies } a < c < b
The real numbers are dense: between any two distinct reals, there is always another. The midpoint argument works for any pair. This density is a key property distinguishing the reals from the integers.

About Real Numbers

The complete set of all rational and irrational numbers, filling every point on the continuous number line.

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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 3 easy

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

Example 4 medium

Give one example of a real number between [formula] and [formula].

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