Real Numbers — Practice
Every number you meet this year is either rational — a fraction — or irrational. Knowing which tells you how it behaves.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Integer — A whole number, positive, negative, or zero.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Rational — Any number writable as ab with integers a and b, b ≠ 0.
- Irrational — A real number that cannot be written as a fraction — its decimal never repeats and never ends.
Problem 1
Is 0.75 rational?
- Yes
- No
Worked answer
Yes
0.75 = 34, so yes.
Problem 2
Classify √9.
- Rational
- Irrational
Worked answer
Rational
√9 = 3, a whole number, so rational.
Problem 3
Classify √7.
- Rational
- Irrational
Worked answer
Irrational
7 is not a perfect square
√7 = 2.6457… never repeats, so irrational.
Problem 4
Is -5 an integer?
- Yes
- No
Worked answer
Yes
Integers are the whole numbers and their negatives, so yes.
Problem 5
Write 0.3 as a fraction in lowest terms.
Worked answer
1/3
0.333… = 13
Problem 6
Which of these is irrational?
- 0.25
- 23
- √16
- √11
Worked answer
√11
0.25 and 23 are fractions, so rational
√16 = 4, rational
11 is not a perfect square, so √11 is irrational.
Problem 7
Write 0.27 as a fraction in lowest terms.
Worked answer
3/11
Let x = 0.2727…, so 100x = 27.2727…
99x = 27
x = 2799 = 311
Problem 8
Which is largest: 34, 0.7, or √0.64?
- 34
- 0.7
- √0.64
Worked answer
√0.64
34 = 0.75
0.7
√0.64 = 0.8
0.8 is the largest.
Problem 9
Is 227 exactly equal to π?
- Yes
- No
Worked answer
No
227 = 3.142857… and repeats, so it is rational
π = 3.14159… never repeats
They are close but not equal.
Problem 10
Which describes 0?
- Integer and rational
- Irrational
- Neither
Worked answer
Integer and rational
0 is a whole number and an integer
It is rational because 0 = 01.
Problem 11
Write 0.6 as a fraction in lowest terms.
Worked answer
2/3
10x = 6.666… and x = 0.666…
9x = 6
x = 69 = 23
Problem 12
Which of these lies between 3 and 4?
- √5
- √10
- √17
- √20
Worked answer
√10
32 = 9 and 42 = 16
10 lies between 9 and 16
√10 ≈ 3.16
Problem 13
Write 0.45 as a fraction in lowest terms.
Worked answer
5/11
100x - x = 45
99x = 45
x = 4599 = 511
Problem 14
Which is smallest: π, 227, √10, or 3.1?
- π
- 227
- √10
- 3.1
Worked answer
3.1
3.1
π ≈ 3.1416
227 ≈ 3.1429
√10 ≈ 3.1623
3.1 is smallest.
Problem 15
√150 lies between two consecutive integers. What is the smaller one?
Worked answer
12
144 < 150 < 169
So 12 < √150 < 13
Problem 16
Is the sum of two irrational numbers always irrational?
- Yes, always
- No — it can be rational
Worked answer
No — it can be rational
√2 + (-√2) = 0, which is rational
So the sum is not always irrational
It usually is — but "always" is too strong.