Rational and irrational numbers

Mathematics · Number Representation & Place Value · ages 13-14

Name ______________________ Date ____________

A rational number's decimal always either stops or repeats.
Circle one: True False
What is the distance between 1/4 and 3/4 on the number line? Give your answer as a decimal.
Answer: ______________
True or false: the equation x² − 4 = 0 has rational solutions.
Circle one: True False
Which of these numbers is rational?
- 3/4
- π
- √2
A right triangle has two legs that are each 5 units long. What is the length of the hypotenuse, rounded to the nearest hundredth?
Answer: ______________
How many numbers are there between 2 and 3?
- Infinitely many
- None, there are no whole numbers between 2 and 3
- Exactly 10
A right triangle has two legs that are each 4 units long. What kind of number is the length of the hypotenuse?
- A rational number, because it's a whole number
- An irrational number, because its decimal never ends or repeats
- Neither rational nor irrational
- You can't tell without a calculator
Which of these decimals belongs to an irrational number?
- 0.333... (repeats forever)
- 0.75 (stops after two digits)
- 0.3030030003... (pattern never repeats)
What kind of solutions does x² − 5 = 0 have?
- Two rational solutions
- Two irrational solutions
- One rational solution
- No real solutions
Ben says 22/7 is exactly equal to pi, since 22/7 is often used to approximate it. Is Ben correct?
- Yes, 22/7 equals pi exactly
- No, 22/7 is rational but pi is irrational, so they can't be exactly equal
- Yes, because both numbers have decimals

Answer key

For grown-ups. Fold this page away before handing over the rest.

True · That is exactly what makes a number rational. No other kind of decimal counts.
0.5 · 1/4 = 0.25 and 3/4 = 0.75, both terminating decimals. The distance between them is 0.5.
True · Solving gives x = ±2, and 2 is a whole number, so it's rational.
3/4 · 3/4 is a fraction of two integers, so it's rational. Pi and square root of 2 can't be written as exact fractions.
7.07 · The hypotenuse is √50, which is irrational, so we round its endless decimal to 7.07.
Infinitely many · There are no whole numbers between 2 and 3, but there are infinitely many decimals, like 2.1, 2.01, and 2.001.
An irrational number, because its decimal never ends or repeats · The hypotenuse is √32. Since 32 isn't a perfect square, √32 can't be written as a fraction of two integers, so it's irrational.
0.3030030003... (pattern never repeats) · The first decimal never settles into a repeating block, so it's irrational. The other two are rational: one repeats, one stops.
Two irrational solutions · Solving gives x = ±√5. Since 5 isn't a perfect square, √5 is irrational, so both solutions are irrational.
No, 22/7 is rational but pi is irrational, so they can't be exactly equal · 22/7 is a fraction of two integers, so its decimal repeats and it's rational. Pi is irrational, so no fraction can equal it exactly. 22/7 is only a close estimate.