Grade 10 — Mathematics 10C › C10U04 Factors, Roots & Irrational Numbers

Grade 10 › C10U04 Factors, Roots & Irrational Numbers › L4
GRADE 10 · ALBERTAOUTCOME AB-MAT10C-AN2

L4 · Irrational numbers on the number line

Learning goal

“I can tell whether a number is rational or irrational, approximate an irrational number to the nearest whole number and nearest tenth, and place it on a number line with reasons — using neighbouring perfect squares as benchmarks.”

Rational number — a number that can be written as a ratio of two integers, a/b with b ≠ 0. Its decimal either terminates or repeats: 0.75, −3, 0.333…
Irrational number — a real number that is NOT rational: its decimal never terminates and never repeats. √2, π, 0.1010010001…
Radical — an expression with the root symbol √: the symbol is the radical sign, the number under it is the radicand. √50 is a radical with radicand 50.
Index — the small number in the notch of the radical that names the root: ∛ has index 3 (cube root), √ has index 2 (usually unwritten — a square root).
Worked example

Sort into rational and irrational: √9, √10, 0.333…, 1.1010010001…, −√25, 4/11. Step 1: Evaluate every radical you can: √9 = 3 (an integer -> rational). −√25 = −5 (rational). 4/11 is a ratio of integers (rational), and 0.333… repeats (rational). Step 2: √10: 10 is not a perfect square, so √10 cannot be a whole number — and in fact its decimal (3.162277…) never terminates and never repeats -> irrational. Step 3: 1.1010010001… has a pattern (one more zero each time) but no block ever repeats, and it never ends -> irrational. A pattern is not the same as a repeating block. Say it plainly: rational ⟺ terminates or repeats ⟺ writable as a ratio of integers. A square root is irrational exactly when the radicand is not a perfect square.

Check yourself — explain it

Why is 1.1010010001… irrational even though it clearly has a pattern?

Paired problem — example still visible

Which of these are irrational: √16, √17, 0.7̄ (0.777…), −√8? -> √17 and −√8 (16 is a perfect square so √16 = 4, rational; 0.777… repeats, rational; 17 is not a perfect square, and 8 is not either, so both roots are irrational).

Faded completion — last step is yours

"√17 is irrational because 17 is not a ___ ___, so the decimal of √17 never ___ and never ___." -> perfect square; terminates; repeats

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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