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Thread: Rational Numbers a/b

  1. #1
    Super Member harpazo's Avatar
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    Rational Numbers a/b

    Textbooks teach that rational numbers of the form a/b, where b CANNOT be zero, are typically fractions. However, it is not clear to me what makes a fraction rational. What do you say?
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    Re: Rational Numbers a/b

    It's simply the definition of what a rational number is - it's any number that can be expressed as a fraction. Note that 0 and all the integers are rational, since the denominator can be equal to 1. It turns out that in decimal form any rational number will either (a) terminate (for example 1/8 = 0.125, which has three digits then ends), or (b) goes into an infinite cycle of a repeating string of digits, for example 1/6 = 0.16666... or 1/7 = 0.142857142857... So, by definition an irrational number expressed in decimal form is a number that neither terminates nor goes into an infinitely repetitious loop. Examples include pi and the square root 2, but you can also dream up a slew of irrational numbers that have a pattern that never repeats, such as 0.1011011101111... Hope this helps.
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    Re: Rational Numbers a/b

    Quote Originally Posted by harpazo View Post
    Textbooks teach that rational numbers of the form a/b, where b CANNOT be zero, are typically fractions. However, it is not clear to me what makes a fraction rational. What do you say?
    A rational number is the ratio of two integers. Now in the history of mathematics when that understanding of rational was developed the ancients did not consider negative numbers much less did they even use the concept of a zero. So there was no reason to exclude zero, which did not exist, in a ratio. There is a famous story, most likely a myth, that the apostle who discovered that hypotenuse of an isosceles right triangle was not rational, that man was banished.
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  4. #4
    Super Member harpazo's Avatar
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    Re: Rational Numbers a/b

    Good.
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