Irrational and real numbers
Fractions crowd the number line, but do they fill it, and what are the numbers that are left?
Where you are 5 inside
What this area is about
Between any two fractions there is another, so it is tempting to think fractions reach every point on the number line. They do not: the side of a square of area 2 is \displaystyle \sqrt{2}, which is not a ratio of integers. Numbers like it are irrational, and the rational and irrational numbers together are the real numbers, which fill the whole line.
How these fit together
Definition. Irrational numbers shows why \displaystyle \sqrt{2} cannot be a fraction, and that its decimal never ends or repeats.
Skills. Sorting a number into its sets decides whether a number is whole, an integer, rational or irrational, often several at once. Approximating irrational numbers brackets a square root such as \displaystyle \sqrt{29} between the perfect squares on either side.
Results. The real number line matches every point of the line with a real number. Properties of real numbers names the rules for adding and multiplying that hold for every real number, which algebra relies on.
A path through it
Irrational numbers first. Sorting numbers, approximating square roots and the real number line can follow in any order, but sorting and the real line both come before the properties.
Next comes proof by contradiction, where the proof that \displaystyle \sqrt{2} is irrational is the leading example.