L5 · Classifying and ordering real numbers
“I can sort any real number into its most specific family — natural, whole, integer, rational, or irrational — and I can order a mixed set of numbers, including irrationals, on a number line.”
Give the most specific family for each: −8, 0, 3/4, √5, 12, √81. Step 1: Evaluate first: √81 = 9. Never classify a radical before you know its value. Step 2: Walk the nesting boxes, most specific first. 12 and √81 = 9 are counting numbers -> natural (and therefore also whole, integer, rational, real). Step 3: 0 is whole but not natural. −8 is an integer but not whole. 3/4 is rational but not an integer. √5 is irrational (5 is not a perfect square) — it sits outside the rational box but inside the reals. Say it plainly: the families nest. Every natural is an integer; every integer is rational; rationals and irrationals together make the reals. Classify the VALUE, not the costume it arrives in.
Why is √81 classified as a natural number when it is written with a radical sign?
Most specific family for: −3/1, √36, 0.5, −√4. -> −3/1 = −3: integer. √36 = 6: natural. 0.5: rational. −√4 = −2: integer.
"√49 belongs to the naturals because √49 = ___, and 6/2 belongs to the naturals because 6/2 = ___ — in both cases the ___ decides, not the form." -> 7; 3; value
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.