Irrational Numbers & the Real Number Line

What you’ll learn

• Every repeating decimal converts to a fraction by shifting one block and subtracting; estimate $\pi^2$ or $2\sqrt{3}$ with two-decimal approximations, and order mixed sets by their decimals.
• Rationals are fractions of integers; irrationals are reals that are not rationals; both still have a fixed place on the number line.
• Terminating or repeating decimals signal rationals; familiar irrationals like $\sqrt{2}$ and $\pi$ need symbols or radicals for exact work.
• Squeeze $\sqrt{n}$ between consecutive perfect squares, then refine by squaring trial decimals when both sides are nonnegative.
• Compare a radical to a decimal by squaring both sides when squaring is valid; like roots compare radicands instead.
• Use tighter intervals to plot irrationals; keep $\pi$ or $\sqrt{2}$ in formulas when you mean the exact value, not a truncated string.