One question decides it: can the number be written as a fraction of two whole numbers? These free printable worksheets ask it of roots, fractions, decimals and π.
A radical sign is not what makes a number irrational, and that is the misreading these sheets are built around. √2 is irrational; √16 is 4, which is as rational as a number gets. A child who has just met the word alongside the symbol will call both of them irrational, so the roots sheet deals perfect squares and their neighbors together — √49 and √48 look alike on the page and have opposite answers.
Why this sheet works
The decimals carry the mirror of the same mistake. 0.75 stops and 0.333… repeats, and both can be written as fractions, so both are rational. Only a decimal that neither stops nor repeats is not — which is why no question here shows a run of digits trailing off: eight places and an ellipsis could be the start of a repeating block nobody has reached, and the question would have no honest answer from the page.
How this one works
One question from this sheet, worked through a step at a time.
π
π
π
its digits never end and never repeat
πisirrational
One question decides it: can this be written as a fraction of two whole numbers? Not "does it look tidy", and not "is there a root sign" — those are the two guesses that go wrong.
Look at what kind of number it is: its digits never end and never repeat.
So it cannot be written as a fraction of two whole numbers. That makes it irrational.
Is each number rational or irrational? Circle one.
1πRational/Irrational
28/11Rational/Irrational
30.742Rational/Irrational
4√441Rational/Irrational
50.505050…Rational/Irrational
6√224Rational/Irrational
Rational or irrational · Every kind of numberAnswer key
1. π: irrational — its digits never end and never repeat2. 8/11: rational — it is already written as one whole number over another3. 0.742: rational — a decimal that stops can be written over a power of ten4. √441: rational — it is exactly 21, and any whole number counts as a fraction over one5. 0.505050…: rational — a decimal that repeats forever can still be written as a fraction6. √224: irrational — 224 is not a perfect square, so its root never ends and never repeats
