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From Wikipedia

Repeating decimal

A decimal representation of a real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of (spoken as "0.3 repeating", or "0.3 recurring") becomes periodic just after the decimal point, repeating the single-digit sequence "3" infinitely. A somewhat more complicated example is where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" infinitely.

A real number has an ultimately periodic decimal representation if and only if it is a rational number. Rational numbers are numbers that can be expressed in the form a/b where a and b are integers and b is non-zero. This form is known as a vulgar fraction. On the one hand, the decimal representation of a rational number is ultimately periodic because it can be determined by a long division process, which must ultimately become periodic as there are only finitely many different remainders and so eventually it will find a remainder that has occurred before. On the other hand, each repeating decimal number satisfies a linear equation with integral coefficients, and its unique solution is a rational number. To illustrate the latter point, the number above satisfies the equation whose solution is

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000..." one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational numbers of the form k/(2n5m). For example, . A terminating decimal can be written as a decimal fraction: . However, a terminating decimal also has a representation as a repeating decimal, obtained by decreasing the final (nonzero) digit by one and appending an infinitely repeating sequence of nines. and are two examples of this.

A decimal that is neither terminating nor repeating represents an irrational number (which cannot be expressed as a fraction of two integers), such as the square root of 2 or the number π. Conversely, an irrational number always has a non-terminating non-repeating decimal representation.

Background

Notation

One convention to indicate a repeating decimal is to put a horizontal line (known as a vinculum) above the repeated numerals (\tfrac{1}{3}=0.\overline{3}). Another convention is to place dots above the outermost numerals of the repeating digits. Where these methods are impossible, the extension may be represented by an ellipsis (...), although this may introduce uncertainty as to exactly which digits should be repeated. Another notation, used for example in Europe and China, encloses the repeating digits in brackets.

Decimal expansion and recurrence sequence

In order to convert a rational number represented as a fraction into decimal form, one may use long division. For example, consider the rational number 5/74:

0.0675 74 ) 5.00000 4.44 560 518 420 370 500

etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50. When we arrive at 50 as the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the same problem we began with. Therefore the decimal repeats: 0.0675 675 675 ....

Every rational number is either a terminating or repeating decimal

Only finitely many different remainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, ..., 73. If at any point in the division the remainder is 0, the expansion terminates at that point. If 0 never occurs as a remainder, then the division process continues forever, and eventually a remainder must occur that has occurred before. The next step in the division will yield the same new digit in the quotient, and the same new remainder, as the previous time the remainder was the same. Therefore the following division will repeat the same results.

Fractions with prime denominators

A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The period of the repeating decimal of 1/p is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the period is equal to p − 1; if not, the period is a factor of p − 1. This result can be deduced from Fermat's little theorem, which states that 10p−1 = 1 (mod p).

The base-10 repetend (the repeating decimal part) of the reciprocal of any prime number greater than 5 is divisible by 9.

Cyclic numbers

If the period of the repeating decimal of 1/p for prime p is equal to p − 1 then the repeating decimal part is called a cyclic number.

Examples of fractions belonging to this group are:

  • 1/7 = 0.142857 ; 6 repeating digits
  • 1/17 = 0.05882352 94117647 ; 16 repeating digits
  • 1/19 = 0.052631578 947368421 ; 18 repeating digits
  • 1/23 = 0.04347826086 95652173913 ; 22 repeating digits
  • 1/29 = 0.0344827 5862068 9655172 4137931 ; 28 repeating digits
  • 1/97 = 0.01030927 83505154 63917525 77319587 62886597 93814432 98969072 16494845 36082474 22680412 37113402 06185567 ; 96 repeating digits

The list can go on to include the fractions 1/47, 1/59, 1/61, 1/97, 1/109, etc.

Every proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation.

  • 1/7 = 1 &time


From Yahoo Answers

Question:2. Classify each decimal according to the following categories _ terminating decimal or nonterminating decimals _ repeating decimal or non-repeating decimal (a) 123:123 (b) 45:4545 (c) 0:101001000100001 : : : (d) 0:10101010 : : : (e) 0:9999 : : :

Answers:(a) terminating; non repeating (b) non terminating; repeating (c) non terminating; non repeating (d) non terminating; repeating (e) a tough one. It is actually equal to 1, so it is non terminating and non repeating.

Question:I'm studying geometry and im wondering if an angle of any regular polygon can be a repeating or non-terminating decimal

Answers:Formula for internal angle sum: (n-2) * 180deg = total deg sum for n number of sides using this formula a triangle: (3-2) * 180 = 180 degree internal sum. square: 4-2 * 180 = 360 degree internal sum. all angles in a square are 90 and theres 4 angles total so thats 360. if you use this formula you will never get decimal values only whole numbers. if you are looking for single angles inside a regular polygon you can get repeating decimals most likely on large odd sided polygons.

Question:

Answers:A decimal that does not go on forever. A repeating decimal, as I have heard it called, is a decimal that continuously repeats. For example, 1/3. When you put 1/3 into decimal form, you will find that you will arrive at an answer of: 0.3333333.... and the 3's go on to infinity. 1/5, however, is a terminating decimal. When put into decimal form, the answer you arrive at is simply 0.2. It end's at 2. EDIT: (Add): By the way, I found this pretty cool website with some pretty good explanations on decimals; terminating/repeating. It even has a decimal - fraction converter. You might want to check it out. http://argyll.epsb.ca/jreed/math7/strand1/1108.htm

Question:Can you please show me the problems that I got wrong? 1) 7 __ = .875 8 2) 5 __ = .55 REPEATING 9 3) 1 1 _ less than _ 4 3 4) 3 _ = .15 20 5) - 5 _ =-.625 8 6) -3 3 _ = -3.75 4 7) - 2 _ = -.22 Repeating whole number 9 8) 4 _ = .363 Repeating whole number 11 9) 7 _ = .4 15 10) 7 _ = .5833 Repeating 33 12 11) 17 _ =.85 20 _______________________ LESS THAN, EQUAL TO, OR GREATER. 1) 0.3 Less than .25 2) 1 _ Less than 0.5 Repeating 5 3) 7/8 Less than 8/9 This is NOT answers for the test, this is in my math book. I know it is like cheating on my homework, but I really want to make sure I did this right. Thank you for understanding. If possible, could you also give me a site for me to learn more about this? Thank you and Good Day! SORRY ABOUT THE FRACTIONS BEING MESSED UP, I HOPE YOU STILL CAN COMPREHEND IT. THANK YOU!

Answers:On the answers you "missed", I included my interpretation of the problem, just in case I interpreted it wrong. A "/" means division, or a fraction. Example: 1/4 is one fourth, or 1 divided by 4. Repeating numbers are usually written as the numbers that repeat have a line over the top of them. But I'll use your method of simply saying what repeats, since I can't make a line over them using this editor anyway. 1) correct 2) correct 3) 1/4 less than 1/3 ? True 4) correct 5) correct 6) -33 / 4 = -8.25 7) correct 8) 4/11 = .3636 repeating 36 9) 7/15 = .466 repeating 6 10) correct 11) correct 1) 0.3 greater than .25 2) correct 3) correct Sorry. I don't know any other sources. For not being sure of your answers, though, you are doing pretty well! I hope these corrections clear things up for you.

From Youtube

Rational vs Irrational Numbers - YourTeacher.com - Math Help :For a complete lesson on rational vs irrational numbers, go to www.yourteacher.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn that the following number sets represent rational numbers natural numbers, whole numbers, integers, fractions, terminating decimals, and repeating decimals. For example, -2, 7, 3/4, 0.0006, and 0.191919... are all rational numbers. However, a decimal that is both non-terminating and non-repeating is an irrational number. For example, 0.12579835781... and 39.779778776775... are irrational numbers.