Identify Rational and Irrational Numbers

Students learn that numbers like √2, √3, and π cannot be written as a ratio of two integers and have non-terminating, non-repeating decimal expansions. They classify given numbers as rational or irrational and explain their reasoning using decimal representations.

Try three of them

Real questions from this skill, graded here. No account, and nothing is saved.

Grade 8 · The Number System · question 1 of 3

0 of 20

20 rungs. A star at the top.

Which of these numbers is rational: √2, √3, √4 or √5?

√2

√4

√5

√3

How it is taught

0:00 of 3:43

A rational number can be written as a ratio of two integers.

An irrational number cannot, and its decimal never repeats.

Read it: Square roots between whole numbers

√20 is the number that times itself makes 20. 4 × 4 = 16 and 5 × 5 = 25, and 20 is between them, so √20 is between 4 and 5.

Find the squares just below and just above the number.
The root is between the numbers that make those squares.
It sits nearer the whole number whose square is nearer.

A square root undoes squaring: 4 × 4 = 16, so √16 = 4, and 5 × 5 = 25, so √25 = 5.

20 is between 16 and 25, so √20 is between √16 and √25: between 4 and 5.

20 is not a square, so √20 is irrational: its decimal never ends and never repeats.

√n is between the whole numbers whose squares are just below and just above n.

Halving the number: 20 ÷ 2 = 10.

A root times itself makes the number: 4 × 4 = 16 and 5 × 5 = 25, so √20 is between 4 and 5.

√71 is between ___ and 9 — 8. 8 × 8 = 64 and 9 × 9 = 81, so √71 is between 8 and 9.

Classify ½, ⅓, and √2

Today we'll learn how to tell if a number is rational or irrational just by looking at its decimal.

A rational number's decimal either ends, or repeats a pattern forever. If it does neither, the number is irrational.

Let's start with one-half.

As a decimal, one-half is zero point five. It ends right there.

The decimal ends, so one-half is rational.

Next, one-third.

Its decimal is zero point three three three… repeating forever.

A repeating pattern still counts, so one-third is rational too.

Now look at the square root of two.

Its decimal is 1.41421356… and it never ends and never repeats.

No fraction of two integers can give us this, so root two is irrational.

Grade 8 · Read the decimal to decide

Decimal ends or repeats? Rational. Neither? Irrational. That's the whole test!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

More of the library

Every skill has a page

Practise this skill free

14 days of every skill, practice packets, Fluency sheets and up to four students. No card.

Start free