Explanation:
We need to sort each number as either
Rational or
Irrational, based on the definition:
- A
rational number can be written as a fraction of two integers (like 3/4, 0.5 = 1/2, or repeating decimals like 0.333… = 1/3).
- An
irrational number cannot be written as a simple fraction — its decimal goes on forever with no repeating pattern.
Let’s go through each number one by one:
1.
3.14
This is a terminating decimal → it stops after 2 decimal places. So it’s rational.
(Note: π ≈ 3.14159… is irrational, but *3.14* alone is just a decimal approximation and is rational.)
2.
0.444
Terminating decimal → rational.
3.
√121
√121 = 11 → integer → rational.
4.
√1000
√1000 = √(100 × 10) = 10√10. Since √10 is irrational, multiplying by 10 keeps it irrational.
Also, 1000 is not a perfect square (31² = 961, 32² = 1024), so √1000 is irrational.
5.
π
Famous irrational number — non-repeating, non-terminating decimal → irrational.
6.
π/7
π is irrational; dividing an irrational number by a nonzero rational number (7) gives an irrational number. So π/7 is irrational.
7.
0.333…
Repeating decimal → equals 1/3 → rational.
8.
1.12313…
The “…” with no bar or pattern shown suggests it's non-repeating and non-terminating. Since no repeating pattern is indicated, and it’s written as “1.12313…” (not 1.12313̄), we treat it as irrational.
(Important: In math problems like this, if it says “…”, and no bar is over digits, it usually means non-repeating → irrational.)
9.
0.12̄
That bar over “12” means 0.12121212… → repeating decimal → rational.
10.
e
Euler’s number — famous irrational number → irrational.
11.
∛9
Cube root of 9. 9 is not a perfect cube (2³ = 8, 3³ = 27), so ∛9 is irrational.
12.
2.9̄
Bar over 9 means 2.99999… = 3 exactly (since 0.999… = 1), so this is rational.
Alternatively, 2.9̄ = 2 + 0.9̄ = 2 + 1 = 3 → rational.
13.
2e
e is irrational; multiplying by 2 (rational ≠ 0) keeps it irrational → irrational.
14.
1.234̄
Bar over “234” means 1.234234234… → repeating decimal → rational.
15.
∛8
∛8 = 2 → integer → rational.
Now let’s list them clearly:
Rational numbers:
- 3.14
- 0.444
- √121 (=11)
- 0.333…
- 0.12̄
- 2.9̄
- 1.234̄
- ∛8 (=2)
That’s
8 rational numbers.
Irrational numbers:
- √1000
- π
- π/7
- 1.12313…
- e
- ∛9
- 2e
That’s
7 irrational numbers.
Double-check count: 8 + 7 = 15 total numbers — matches the grid.
Final Answer:
Rational: 3.14, 0.444, √121, 0.333…, 0.12̄, 2.9̄, 1.234̄, ∛8
Irrational: √1000, π, π/7, 1.12313…, e, ∛9, 2e
Parent Tip: Review the logic above to help your child master the concept of rational and irrational numbers worksheet.
