To determine if a number is rational or irrational, we use these rules:
-
Rational (R): Can be written as a fraction. This includes whole numbers, integers, terminating decimals (decimals that end), and repeating decimals (decimals with a pattern that repeats forever). Square roots of perfect squares (like $\sqrt{4}$, $\sqrt{9}$) are also rational.
-
Irrational (I): Cannot be written as a simple fraction. These are non-terminating, non-repeating decimals. Square roots of non-perfect squares (like $\sqrt{2}$, $\sqrt{101}$) and special numbers like $\pi$ are irrational.
Here is the step-by-step check for each number:
1)
61π: Since $\pi$ is irrational, multiplying it by 61 keeps it irrational. ->
I
2)
42: This is a whole number (integer). All integers are rational. ->
R
3)
75.082106: This decimal ends (terminates). Terminating decimals are rational. ->
R
4)
$\sqrt{101}$: 101 is not a perfect square ($10^2=100$, $11^2=121$). So, the root is irrational. ->
I
5)
65.4279: This decimal ends. It is rational. ->
R
6)
$\frac{29}{6}$: This is already written as a fraction. It is rational. ->
R
7)
$\pi$: Pi is a famous irrational number. ->
I
8)
$5.6\overline{213}$: The bar over 213 means those digits repeat forever. Repeating decimals are rational. ->
R
9)
$\frac{98}{16}$: This is a fraction. It is rational. ->
R
10)
39: This is a whole number. It is rational. ->
R
11)
89.396668...: The "..." indicates it goes on forever without a repeating pattern shown. Non-repeating, non-terminating decimals are irrational. ->
I
12)
$\sqrt{17}$: 17 is not a perfect square ($4^2=16$, $5^2=25$). So, the root is irrational. ->
I
13)
67.714813...: The "..." indicates it goes on forever without a repeating pattern. It is irrational. ->
I
14)
$\sqrt{64}$: 64 is a perfect square because $8 \times 8 = 64$. So $\sqrt{64} = 8$, which is rational. ->
R
15)
$\frac{1}{4}$: This is a fraction. It is rational. ->
R
16)
$\sqrt{25}$: 25 is a perfect square because $5 \times 5 = 25$. So $\sqrt{25} = 5$, which is rational. ->
R
17)
71.5186: This decimal ends. It is rational. ->
R
18)
$\frac{7}{54}$: This is a fraction. It is rational. ->
R
19)
20.455566...: The "..." indicates it goes on forever without a clear repeating pattern. It is irrational. ->
I
20)
97.33997: This decimal ends. It is rational. ->
R
Final Answer:
1. I
2. R
3. R
4. I
5. R
6. R
7. I
8. R
9. R
10. R
11. I
12. I
13. I
14. R
15. R
16. R
17. R
18. R
19. I
20. R
Parent Tip: Review the logic above to help your child master the concept of identify rational numbers worksheet.
