MHB Repeated and Nonrepeated Decimals

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Repeating decimals are rational numbers because they can be expressed as a fraction, such as 0.154 repeating, which equals 154/999. In contrast, non-repeating decimals are irrational, as they cannot be represented as a ratio of integers; an example is the decimal representation of the square root of 2. All decimals fall into either the repeating or non-repeating category, which means not all decimals are rational. This distinction highlights the existence of irrational numbers in mathematics. Understanding these classifications is essential for grasping the nature of numbers in decimal form.
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My question concerns repeated and nonrepeated decimals. Are both rational numbers? Can you give an example for each?
 
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RTCNTC said:
My question concerns repeated and nonrepeated decimals. Are both rational numbers? Can you give an example for each?

A repeating decimal number is rational because you can always express such a number as the string of repeating digits over an equal number of 9's (one of the tricks my father taught me as a child). For example, we may write:

$$0.\overline{154}=\frac{154}{999}$$

A non-repeating decimal is irrational since it cannot be expressed as the ratio of one integer to another. $\sqrt{2}$ is an example of a non-repeating decimal.
 
You realize, I hope, that all decimals are either "repeating" or "non-repeating". So if it were true that "all repeated and non-repeated decimals are rational numbers" then there would be no irrational numbers!
 
MarkFL said:
A repeating decimal number is rational because you can always express such a number as the string of repeating digits over an equal number of 9's (one of the tricks my father taught me as a child). For example, we may write:

$$0.\overline{154}=\frac{154}{999}$$

A non-repeating decimal is irrational since it cannot be expressed as the ratio of one integer to another. $\sqrt{2}$ is an example of a non-repeating decimal.

More important information.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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