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45 multiple-choice questions and 62 flashcards on Rational and Irrational Numbers, about 30% of the 8th Math: The Number System bank. Every one carries a written rationale.
Rational and Irrational Numbers is one of 2 chapters in CoStudy's 8th Math: The Number System bank, and it holds 45 of the bank's 150 multiple-choice questions — roughly 30% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
10 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
Which is irrational?
Answer: C — √15
C) 15 isn't a perfect square; √15 ≈ 3.87. A) 1/2 rational. B) √16 = 4, rational. D) Integers are rational.
Every integer is also a:
Answer: D — Rational number
D) Any integer n = n/1 — rational. A/B) Not always. C) Integers terminate, not repeat.
The decimal expansion of a RATIONAL number is always:
Answer: A — Terminating or repeating
A) Definition of rational: terminates (e.g., 0.5) or repeats (e.g., 0.333...). B) That's irrational. C) 1/3 doesn't terminate. D) 1/2 doesn't repeat in the looping sense.
Which is RATIONAL?
Answer: C — 0.121212... (12 repeating)
C) Repeating decimal = 12/99 = 4/33, rational. A) Sum of two irrationals here is irrational. B) Half of an irrational is still irrational. D) Non-perfect-square root is irrational.
Which describes π?
Answer: C — An irrational number ≈ 3.14159...
C) Definition. A) Integers don't have decimal parts. B) 22/7 is a rational approximation, not equal to π. D) π has infinite non-repeating decimals.
Is 0.121221222122221... rational?
Answer: D — No, because the pattern grows and never cycles
D) A rational repeating decimal has a fixed-length cycle. Here the gap between 1's grows, so no cycle — irrational. A) Pattern ≠ repetition (key trap). C) False — π and √2 are decimals that are irrational. B) Definitionally answerable.
Which fraction equals 0.625?
Answer: D — 5/8
D) 625/1000 = 5/8. B) Wrong arithmetic. C) Not simplified and missing zero. A) 3/5 = 0.6, not 0.625.
True/False: The product of a rational and an irrational (with rational ≠ 0) is always irrational.
Answer: A — True
A) e.g., 2 · √2 = 2√2 irrational. If it were rational, dividing back by 2 would give √2 rational — contradiction. B) Wrong. C/D) Sign doesn't matter.
Convert 0.444... to a fraction.
Answer: D — 4/9
D) Let x = 0.444...; 10x = 4.444...; 10x − x = 4; 9x = 4; x = 4/9. B) Would give 0.4 exactly, not 0.444... . C) Truncates to two places. A) = 0.4, not 0.444... .
Which set contains ONLY rational numbers?
Answer: B — {0.3, √9, −2}
B) 0.3 = 3/10 rational; √9 = 3 rational; −2 integer. A) π is irrational. C) √2 irrational. D) e is irrational.
4 cards from the 62 in this chapter.
Rational number?
A number that can be written as p/q with integer p, q (q ≠ 0).
Convert 0.75 to a fraction.
3/4.
How can you tell if a fraction terminates as a decimal?
If the denominator's prime factors are only 2's and/or 5's, it terminates. Otherwise, it repeats.
What's the relationship between integers, rationals, and reals?
Integers ⊂ Rationals ⊂ Reals. All integers are rational; all rationals are real.
These are a sample. The full Rational and Irrational Numbers chapter runs 107 items with per-chapter progress tracking, on the web and in the iOS app.
Open 8th Math: The Number System in CoStudy →