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Example 2 Rational numbers are those which can be expressed as a ratio (fraction) of two integers. We commonly think of them as fractions, and indeed they are, but the stipulation is that both numerator and denominator must be an integer. Integers themselves are rational since each can be written with a denominator of 1. I wrote a little earlier in the introduction to Chapter 6 about the derivation of the word “rational.” In decimal form, all rational numbers either repeat or terminate. If the denominator consists only of factors of 2 and 5, then it will terminate, which means that the decimals will come to an end with nothing but zeros afterwards. For example, the fraction 17/80 terminates. In decimals, 17/80 = 0.2125, exactly. All other rational numbers repeat, which means that a specific sequence of numbers will repeat in the decimal over and over, ad infinitum. Most people are aware that 1/3 = 0.333333333 . . . . The threes just keeep on coming when you divide 1 by 3. The fraction 1/7 = 0.142857142857142857 . . . with the sequence 142857 repeating interminably. You can see why this is so by looking at the long division of 1/7.
The numbers I circled are remainders after each step. Since the remainders can never be zero, there are in this instance only six possible remainders since there are six numbers from 1 to 6, i. e., less than the divisor 7 but greater than 0. This partly explains why the repetend, the part that endlessly repeats in the decimal version, is six digits long. In case you've ever wondered, the length of the repetend of a fraction is a factor of one less than the denominator. For example, the length of the repetend of 1/251 is a factor of 250. In this case, the length is actually 50. So if you divide 1 by 251 the decimal string will repeat every 50 digits. (Calculating this exact length without doing the mindlessly dull long division is beyond the scope of this course, unfortunately.) Since every fraction has a repeating or terminating decimal form, then it logically follows that any number which does not have a repeating or terminating decimal form cannot be rational, or a fraction. These numbers are known as irrational. The earliest known example of an irrational number is
A similar statement applies to other kinds of roots such as cube roots or seventh roots. Logically, what is left to show is that irrational numbers do not repeat or terminate in the sense that rational numbers do. Showing that is a bit advanced, but it is true. You can punch a number like So, bottom line as they say, we get introduced to irrational numbers through radicals. There are several well known numbers— | ||
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. It was proven in ancient Greece that it is impossible to represent this number as a fraction. Since then it has become possible to prove the following.
is not an integer, then 
. When I was in seventh grade, before there were handheld calculators, I was shown how to calculate square roots by hand. In a fit of nerdness which typified my childhood I calculated this number to 23 decimal places one afternoon before the calculation became too large for the paper I was writing on. I wanted to convince myself that there was no pattern to these decimals, and indeed there wasn’t.
is the most famous—that are irrational but not the root of an integer or even the root of a rational number. These are called transcendental numbers. | ||
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Copyright © 2005 by Jon Davidson. Duplication for instructional usage is permitted for students and faculty of Southern State Community College. | ||
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