Completeness (Intuition) Math Example 4

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

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State whether

\mathbb{Q}

\mathbb{R}

is a better domain for solving

x^2 = 2

, and explain in terms of completeness.

1
The equation $x^2=2$ has solutions $x=\pm\sqrt{2}$ .
2
$\sqrt{2}$ is irrational, so it is not in $\mathbb{Q}$ . In $\mathbb{Q}$ , the equation has no solution.
3
In $\mathbb{R}$ , $\sqrt{2}$ exists due to completeness (the real line has no gaps). The equation has solutions $x=\pm\sqrt{2}$ .
4
$\mathbb{R}$ is the appropriate domain.

Answer

\mathbb{R}\text{ is needed; } x=\pm\sqrt{2} \notin \mathbb{Q}

Completeness of

\mathbb{R}

guarantees that equations like

x^2=2

have solutions. The rationals lack this property, which is why

\mathbb{R}

is the standard setting for real analysis.

About Completeness (Intuition)

The property of a mathematical system where every true statement that can be expressed in the system can also be proved within it.

The real numbers [formula] are 'complete' while the rationals [formula] are not. Illustrate this by

Check that a proof by induction for [formula] is complete: what cases must be covered? Use [formula]

A student proves a statement for all even integers but forgets odd integers. Is the proof complete?