Question 1 of 3Grade 8 State TestAI-N.RN.3Algebra I Regents · June 2023 · Q2Non-calculatoremerging
Which expression represents an irrational number?
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Unit 1 · The Number System & Roots
Rational vs. Irrational Numbers
A rational number can be written as a ratio of integers. Irrational numbers have non-terminating, non-repeating decimal expansions.
NY-8.NS.1AI-N.RN.3
Notes
The two teams
Rational numbers include integers, fractions, terminating decimals, and repeating decimals. Irrational numbers include √2, √3, π, and non-repeating non-terminating decimals. Between any two rationals there are irrationals — and vice versa.
Why √2 is irrational (sketch)
If √2 = a/b in lowest terms, then a² = 2b², so a is even, then b is even, which contradicts lowest terms. You do not need a formal proof for the Grade 8 test, but you must know √2, √3, π are irrational and √9 is rational.
Vocabulary
rational number
A number that can be written as a/b where a and b are integers and b ≠ 0.
Example: −4, 3/5, 0.25, and 0.333… are rational.
irrational number
A real number that cannot be written as a ratio of integers.
Example: √2 ≈ 1.414213… never repeats.
Worked examples
Try the prompt first. The steps stay folded until you ask to see them.
Classify
Is √49 rational or irrational? What about √50?
Common errors
Watch for
Thinking every square root is irrational
Perfect-square roots are integers.
Fix: Ask: is the radicand a perfect square?
Instructional videos
External links. We do not host copyrighted exam items or paid video libraries.
- Khan Academy: Intro to rational and irrational numbers — Khan Academy (external, freely licensed instructional video)
Guided practice
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Question 2 of 3Grade 8 State TestNY-8.NS.1Grade 8 Mathematics · Spring 2023 · Q35Non-calculatoremerging
Which statement about the value of √50 is true?
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Question 3 of 3Grade 8 State TestAI-N.RN.3Algebra I Regents · June 2023 · Q2Non-calculatoremerging
Which expression represents an irrational number?
Choose or type an answer first. The key stays hidden until you do.
Intervention
Practice converting repeating decimals to fractions before classifying.
Enrichment
Prove that the sum of a nonzero rational and an irrational is irrational (Algebra I N.RN.3).
Explain your reasoning
Sentence starters · Notice and Wonder · Three Reads · Think-Pair-Share · Compare and Connect · Math Talk · Stronger and Clearer Each Time
- I know this number is rational because…
- This decimal is irrational because it…