To determine which numbers are rational, we need to remember that a rational number is any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \) is not zero.
Let's evaluate each of the provided options:
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1217: This is a whole number, which can be written as \( \frac{1217}{1} \). Therefore, it is a rational number.
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12 over 17 (12/17): This is clearly a fraction of two integers, thus it is a rational number.
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4.52525252…: This is a repeating decimal. Any repeating decimal can be expressed as a fraction, making it a rational number.
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√32 (square root of 32): Since \( \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \) and \( \sqrt{2} \) is an irrational number, \( \sqrt{32} \) is also irrational. Therefore, it is not rational.
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π (pi): This is a well-known irrational number; it cannot be expressed as a fraction of two integers. So, π is not rational.
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√70 (square root of 70): Similarly, \( \sqrt{70} \) is not a perfect square, and since \( \sqrt{70} \) cannot be expressed as a fraction of two integers, it is also irrational.
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16 (16): This is a whole number, which can be expressed as \( \frac{16}{1} \), thus it is rational.
From the above analysis, the rational numbers among the options are:
- 1217
- 12 over 17
- 4.52525252...
So the three numbers that are rational are:
- 1217
- 12 over 17 (12/17)
- 4.52525252...
1 answer