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A number has infinite decimal expansion 0.45445444544445, where the number of fours between the fives continues to increase.
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Is this a rational number?
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The decimal form of rational numbers either terminates, that means they stop, or they repeat.
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First of all, this decimal expansion never terminates.
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We know that is an infinite decimal expansion.
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The only way this could be a rational number is if it is repeating.
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It’s really tempting to look at all these fours and fives and think that this is repeating.
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It’s not repeating four five.
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You could think it was repeating the value four five four.
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Then again, it breaks that pattern.
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This decimal value has no set of repeating values.
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And that means it’s not a rational number.