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A rational number can be written as the ratio of two integers. Conversely, an irrational number cannot be written as the ratio of two integers.
C and E
We will select the cards that show irrational numbers.
With this information, if the decimal expansion of a number is non-repeating and non-terminating, the number cannot be expressed as the ratio of two integers and the number is irrational. Conversely, repeating and terminating decimals are rational numbers. From here we will make a table and consider whether the numbers on the cards can be written as a ratio of two integers.
| Can the Number be Written as a Ratio of Two Integer? | Why? | Rational/Irrational | |
|---|---|---|---|
| 10 | Yes | 101 ⇒ Ratio of two integers | Rational |
| 6/5 | Yes | 65 ⇒ Ratio of two integers | Rational |
| π | No | π=3.141592... ⇓ Non-repeating and non-terminating | Irrational |
| 11/4 | Yes | 114 ⇒ Ratio of two integers | Rational |
| 8.25635... | No | Non-repeating and non-terminating | Irrational |
| -7 | Yes | - 71 ⇒ Ratio of two integers | Rational |
| 6.31 | Yes | 6.31=6.313131 ... ⇓ Repeating decimal | Rational |
From the table we can see that π and 8.25635... are irrational, so the cards labeled C and E are the cards we select.
