What is two thirds as a decimal? | Reference.com What is two thirds as a decimal? A: Quick Answer Two-thirds expressed as a decimal is 0.6, with a line over or under the six, indicating that the six repeats indefinitely, or as 0.666, which is accurate to three decimals. To convert a fraction to a decimal, divide the numerator, the top number, by the denominator, the bottom number. Repeating Decimals Repeating Decimals and Cyclic Numbers Most of you are already familiar with the repeating decimal digits of fractions like one third (1/3) or two thirds (2/3) which have these never ending strings of threes and sixes: 1 / 3 = 0.3333333333... and 2 / 3 = 0.6666666666... At some point (preferably before high school), a Math teacher should have explained the convention of placing a bar over repeating decimal digits, or possibly underlining them or placing brackets around them; on this web page, for example, we'll write the repeating decimals using an underline: 1/6 = 0.1666.

For electronic text files, its acceptable to merely bracket the repeating digits; with parentheses or or literal 'brackets' such as: 1/6 = 0.166(6) or 0.166[6]. Math teachers should never allow you to simply round-off the digits to 0.167; unless you are specifically studying some method of doing so, such as the use of Math in the sciences. Note: In the sciences, where numbers often depend upon the accuracy of some measuring tool -- from a simple meter stick to the most advanced scientific instruments available, it's quite reasonable to agree that none of your results should ever be expressed with more digits than the number obtained from the least accurate tool used. For example, although 6.485 times 1.07 times 4.392 should be equal to 30.4758684 in a math class. In a science class, this would most likely be expressed as just 30.5; using a 'round-up' method, since the accuracy of the 1.07 measurement was only three digits.

In a Mathematics class, however, you should never state that 1/6 = 0.167 (they are not equal). If a person feels it's necessary not to include every digit in some result (due to limitations of space to record the digits, or too little time), they should use some kind of symbol which shows the result is an approximation (and state it will be used as such at the beginning of of their paper; or at least in a footnote where it's first used). When writing, one could replace the parallel bars of an equal sign with tilde marks (~), also called a 'wavy' equal sign (≈); 2248h in Unicode (or "&#8776;" in HTML code), or place a dot above the equal sign (≐); 2250h in Unicode, but not available in most fonts (we used Lucida Sans Unicode here). Or define some other way you will show an approximate value in any of your own papers.

The first non-trivial (more than a single digit) repeating decimal fraction is one seventh: 1/7 = 0.142857142857 Even the calculator program that came with Windows® 95 (with its 12 to 13 digit display) would give most people the idea that the same six digits might repeat an infinite number of times. But what's a person to do if the number of digits that comprise a repeating decimal fraction end up being 20, 30, 60 or even hundreds of digits?! This is why we must have some agreed upon notation for expressing approximate values of numbers that must be written with digits to the right of a decimal point.

(On a web page, one might be able to use the 'Wavy equal sign' Unicode character, for example: 1/49 ≈ 0.0212766 and we read this sign as "approximately equal to," or we'll simply have to state that the value is an approximation.) The Decimal Expansion of All Fractions (1/d) from 1/2 through 1/70 What we'll confirm further below (in detail) is that the digit '9' (often a string of many nines) is a key factor in understanding repeating decimal fractions: Because for many of them, the number of digits which repeat are the same as the smallest string of nines which can be divided evenly by the denominator, and that quotient will equal the repeating digit(s)! Stating this in algebraic terms: For each 1/p (where p is a prime number, but not 2 or 5), then for k = 1,2,3... n, [(10^k)-1]/p = the repeating digits for that fraction, when there is no remainder; k also indicates the number of digits that repeat.

This is quite easy to show by way of example: Starting with 1/3, for k = 1: (10^1)-1=9, and 9/3 = 3. Done. The single repeating digit is 3. For 1/7, it's obvious that 9/7 has a remainder, as does 99/7 (and for k=3, 4 or 5 as well). But when we use (10^6)-1=999999, we arrive at the integer only quotient of 142857 (999999 / 7 = 142857). Unlike 1/3, the fraction 1/7 is a special case where k = p - 1, and we call the digits which repeat a cyclic number. For 1/11, k = 2, since 11 divides evenly into 99. Note: There must be two repeating digits in this case (since k=2), and they are: 09.

Likewise, for 1/13, at k = 6, we find that 999999 / 13 = 76923, but there must be 6 repeating digits; they are: 076923. A Cyclic Number is a k-digit-long integer, that when multiplied by 2, 3, 4 ... up to k will result in the same k digits in a different order. It will also have the characteristic that multiplying those digits by k + 1 will produce a string of k nines ( ). Example: Multiplying 0588235294117647 (16 digits) by 16, produces: 9411764705882352; rotating through 8 digits in this case. But if we multiply this number by k + 1 = 17, the result is a string of 16 nines: 9999999999999999.

Fraction 0.0142857142857142857 ( 1/2 * 1/5 times 1/7) General Rule of Exact Decimal Equivalents Note that all the fractions comprised of a power of 1/2 (1/4, 1/8, 1/16, 1/32, 1/64, etc.) have exact decimal equivalents. (This may seem more like binary math, instead of 'decimal' fractions! But dividing '1' by powers of two, gives us digits similar to successive powers of 5; such as: 0.25, 0.125, 0.0625, 0.03125, etc.) As a matter of fact, every fraction that has an exact decimal equivalent consists only of powers of 1/2, or the product of 1/5 times a power of 1/2 or some multiple of these fractions (such as 3 times 1/4 = 3/4 = 0.75); there is no other way for an exact decimal equivalent to exist.

Also note that 2 and 5 are the only prime factors of 10. More examples: 1/20 = 0.05, 1/50 = 0.02 and 1/100 = 0.01. Observations on Repeating Decimals If a fraction is composed of repeating decimal digits, the number of digits that repeat will never be greater than one subtracted from the fraction's denominator: Either k = d - 1, or k < d - 1 (where k is the periodic length of the repeating decimal digits). A mathematical theorem, attributed to M. E. Midy [1] of Nantes, France in 1836, it effectively states that for any reduced decimal fraction a/p (where p is prime), having an even number of repeating digits, if you split the repeating digits in half and add them together, the sum will result in a string of nines. For example, the 2 repeating digits of 8/11 are 72, and 7 + 2 = 9.

The 6 repeating digits of 5/13 are 384615, and 384 + 615 = 999. And for the 18 repeating digits of 2/19, this gives us 105263157 + 894736842 = 999999999 (a string of 9 nines). If the periodic digits of the decimal expansion of a fraction a/d have a length of d - 1, then its denominator must be a prime number. This is the basis for our definition of a cyclic number: If the periodic length of a fraction 1/p equals p - 1 (that is, k = p - 1) then its repeating decimal digits are called a cyclic number. Various properties of these cyclic numbers are given below with examples. (Note: There are other definitions for the term 'cyclic number' within mathematics.) Note: This is not true for all prime number fractions. Only about half of all the 'prime fractions' listed above have this property (8 out of 17). Would this ratio continue to be the same if all 'prime fractions' were included?

Well, the ratio depends more or less on where we stop counting. If we stop at 1/199, then the ratio drops to less than 39% (17 out of 44). To be more accurate, we'd need to determine some point in the total number of 'prime fractions,' that doesn't favor one type over the other. It has been conjectured that for all prime number fractions, this ratio would be the same as Artin's Constant [2] (about 37.4%).

Here's a table showing which of the first 44 prime fractions (1/3 through 1/199) are cyclical: Cyclical Group: 1/101 = 0.0099 (only 4 digits) 1/137 = 0.00729927 (8 digits) And here's a list of the number of repeating digits (in parentheses) for all the prime fractions which are not part of the "Cyclical Group" in the table above: 1/3 (1), 1/11 (2), 1/13 (6), 1/31 (15), 1/37 (3), 1/41 (5), 1/43 (21), 1/53 (13), 1/67 (33), 1/71 (35), 1/73 (8), 1/79 (13), 1/83 (41), 1/89 (44), 1/101 (4), 1/103 (34), 1/10

7 (53), 1/127 (42), 1/137 (8), 1/139 (46), 1/151 (75), 1/157 (78), 1/163 (81), 1/173 (43), 1/191 (95), 1/197 (98), 1/199 (99).

Here's a graph and table of the number of repeating digits, for each of the first 90 fractions; including 1/1 when n=1. A zero indicates those which do not repeat; so for n = 1 through 10, the data shows: 0, 0, 1, 0, 0, 1, 6, 0, 1, 0. It's easy to pick out the values for the cyclic numbers, since they will always increase: 6, 16, 18, 22, 28, 46, 58, 60. There will always be an even number of digits in a cyclic number, since all primes beyond 2 are odd, then k = p - 1 must be even. Therefore, Midy's Theorem will always apply to cyclic numbers. The fraction 1 / 7 is quite amazing...

almost every product of this fraction with another (1/7 times 1/2, 1/3, [1/2]^2, 1/5, [1/3]^2, etc.) has only six repeating decimal digits; although it may take one, two or even three digits (as in the case of 1/56 or 1/84) before we reach the beginning of the digit string that repeats, it will always be: 142857 . All the cyclic numbers larger than 142857 begin with a leading zero. This continues through to the 96 repeating digits of 1/97. From 1/109 through 1/983, there are two leading zeros, and from 1/1019 through 1/9967 there are three leading zeros! This pattern of n leading zeros for all p > 10^n continues into the infinite number of all prime fractions. So 1/7 is the only fraction resulting in a cyclic number that has no leading zeros! The denominators in our Cyclical Group of fractions are also called Full Reptend Primes .

There is still no general method for finding full reptend primes. We must carry out the long divisions of each 1/p then check for p - 1 repeating digits, or try dividing p into a string of p - 1 digits of nines and check for no remainder; if it divides evenly, the result is a cyclic number. Here's a table of the first eleven cyclic numbers: Prime 3142854 4 + 3 = 7; so pattern is still like 14285(7). Hmm... how far can we extend this?

Well, each time we hit a multiple of 7 (14, 21, 28, etc.), we should get something like '999999', but let's check that multiple + 1 (or 29, 36, 43, etc.) for the same patterns we observed above (in rows x 8, x 15 and x 22): 4142853 [x 29; nothing new here; 3 + 4 = 7; so still like 14285(7).] 5142852 [x 36; nothing new here; 2 + 5 = 7; so still like 14285(7).] 6142851 [x 43; nothing new here; 1 + 6 = 7; so still like 14285(7).] 7142850 [x 50; nothing new here; 0 + 7 = 7; so still like 14285(7).] 8142849 [

x 57; now we must change 9 + 8 = 17 into a 1 and a 7 to give us: 1428(4+1=5)(7).] 9142848 [x 64; again, this requires 8 + 9 = 17 to be a 1 and a 7 to give us: 1428(4+1=5)(7).].] We can also see that a definite pattern has emerged in the multiples themselves: Note the sequentially increasing and decreasing digits at both the beginning and end of each new multiple!

This is true for every 7th multiple you compare. When we multiply 142857 by 71, we get: 10142847 which does resemble our pattern once we steal a 1 from the '7' to arrive at 1428(4+1=5)(6+1+0=7). Is there a multiplier that produces a result so different, that our original cyclic number is completely lost in the digital fuzz? Let's try 7 to the 4th, 5th and 6th powers, plus 1, as multipliers: 343142514 [x (7^4)+1 = 2,402; maybe "142" is part of the original, but it's quite fuzzy.

However, I can still use my intuition(?) and imagine that '5' + '1' + 2 (of the '4') = 8, leaving 2 + '3' = 5 and 4 + 3 = 7 for 142(8)(5)(7).] 2401140456 [x (7^5)+1 = 16,808; here we can only guess "14" is what's left of our original pattern! I'm not even going to bother imagining how to get back to it from here!] 16807126050 [x (7^6)+1 = 117,650; I think that's sufficiently fuzzy! And using (7^7)+1, gives us a result of: 117648882351 in which it's impossible to even guess at the starting '1' for our original pattern.]. Wolfram's MathWorld and Wikipedia .

Since you've made it this far, I'm going to tell you where to find a FREE Windows Calculator that can compute (and copy into the clipboard buffer) as many as 10,000 decimal places!!! It defaults to 500 digits which should be quite sufficient for any repeating decimals you'll ever study! Take this offsite link: #freeware to get a copy of the MIRACL Calc multi-precision calculator by Geoff Wilkins. NOTE: Do not trust at least the last 5 to 10 digits of the number Pi (3.14159265...) from this calculator nor possibly any other value you calculate with it. It's always a good idea to carry out a calculation to many digits beyond what you'll actually use, since the digits at the end may be incorrect! And because it produces 500 to 10,000 digits (depending on how you set it up), you can easily afford to forget about the last 10 or so digits!

Previous Updates: October 6, 2007 (2007.10.06); January 24, 2010 (2010.01.24) Revised: October 8, 2012 (2012.10.08) Updated: October 12, 2012 (2012.10.12) What is two-thirds as a decimal and a percent? | Reference.com What is two-thirds as a decimal and a percent? A: Quick Answer According to Fact Monster, two thirds as a decimal is 0.666 repeating (or 0.667 rounded) and as a percent is 66.6% repeating or 66.7%. To convert the two-thirds fraction to a decimal, simply divide the top number by the bottom number, or divide 2 by 3. Full Answer To convert the decimal to the percentage, multiply the decimal by 100 and add a percentage sign. The easiest way to do this is to move the decimal point 2 places to the right. Doing so converts 0.667 to 66.7%.

Other common fraction to decimal and percentage conversions include one-fourth to 0.25 and 25%, one-third to 0.333 and 33.3% and one-half to 0.50 and 50%. Question Corner -- How To Express A Repeating Decimal Number As A Fraction Question Corner and Discussion Area How To Express A Repeating Decimal Number As A Fraction Asked by Dana Shaddad, student, Qatar International on September 22, 1997: How do you find a rule for expressing any recurring decimal as a fraction and such rule to be tested with examples of three digits, four digits, five digits repeating patterns. There is a rule for converting a repeating decimal number into a fraction. Let's first of all suppose that the repeating pattern starts immediately after the decimal point, with zero before the decimal point. Let's say the repeating part is n digits long, so the number looks like Let A stand for the integer (the n-digit number whose n digits are , , etc.). Then the first part of of our number, , is the same as . The next part of our number is , which is , and so on. This means that our number d is equal to the infinite sum .

This is a geometric series and by a well known formula (as described in another question ) its value is . To see that this formula works, consider the decimal . In this case A=142857, n=6, and . Then this decimal is equal to 142857/99999. This fraction then reduces to 1/7. If you have a decimal where the repeating part doesn't start right after the decimal point, first multiply or divide by 10 enough times so that the repeating part does start right after the decimal point, then undo your multiplication or division at the end. For example, suppose you have the decimal d=3.4575757. . . . Multiply by 10 to get 10 d = 34.575757. . . . Our formula tells you that the 0.575757. . . part equals 57/99, so 10 d = 34 + 57/99, and therefore d = 34/10 + 57/990.

Asked by Peter Collins, student, Mill Hill School on January 2, 1998: How can you calculate what a fraction will be in decimal form? More specifically: How can you tell if a fraction is a recurring decimal or not, and if it is what will be the value of the decimal? (without a calculator). This is in relation to the question posted in September about writing a decimal in fractional form. To calculate a fraction in decimal form, you perform long division. You should be familiar with this procedure from early arithmetic lessons. For example, to calculate 3/11, you say that 11 goes into 3 zero times, with remainder 3. So the answer starts 0. Then you find that 11 goes into 30 two times, with remainder 8. So the answer starts 0.2. Next you find that 11 goes into 80 seven times, with remainder 3. So the answer starts 0.27. Now that you have a remainder 3 which you have seen before, you see that the pattern will repeat: 0.272727. . . . Every fraction produces a recurring decimal.

That is because there are only a finite number of possibilities for the remainder, so eventually you will encounter a remainder you have seen before, and the pattern of digits will continue from that point on. Some fractions will give you a decimal where the recurring part is all zeros, such as 4/5 = 0.80000. . . = 0.8. Such decimals are called terminating decimals. The fractions which give you terminating decimals are those which can be written in a form where the denominator is a power of 10 (for example, 4/5 = 8/10). Which fractions can be written this way? Precisely those fractions for which, when you write them in lowest terms (factoring out any common factors from top and bottom), have the property that the denominator is a product of 2's and/or 5's but no other prime factors. Then you can multiply top and bottom by an appropriate number of 2's and 5's to get a power of 10. For example, 37/40 will give you a terminating decimal, because 40 is 2 times 2 times 2 times 5.

If you multiply top and bottom by two more 5's, i.e. by 25, then the denominator will become so . (This is the same answer you would get if you carried out the long-division procedure above). However, 32/39 will give you a non-terminating decimal because, even though it is in lowest terms, the denominator has prime factors other than 2 and 5, so there is nothing you could multiply it by to get a power of ten. This part of the site maintained by (No Current Maintainers) Last updated: April 19, 1999 Original Web Site Creator / Mathematical Content Developer: Philip Spencer Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu Navigation Panel: TIMES MODULE M18 - Decimals and Percentages Decimals and percentages Number and Algebra : Module 18 Year : 5-8 Fluency with addition, subtraction, multiplication and division of whole numbers and fractions. Familiarity with using a number line for whole numbers and fractions. Understanding place value for whole numbers. Students will have had some experience with decimals. return to top Motivation Decimals are used widely. Our money is based on a decimal system.

Common measurements such as the height of students in a class, the price of petrol or the amount of gas used for heating your home are usually expressed using decimals. We can represent decimals as points on the number line by repeatedly subdividing intervals into tenths. A decimal number gives the ‘address’ of a point on the number line − the digits give successively more accurate information and together locate a point. This is an extension of the concept of place value met earlier. Whole numbers are written as strings of digits that correspond to the ‘places’ ones, ten, hundreds, etc. Positive numbers less than one can be written with the same notation with ‘additional places’ tenths, hundredths, thousandths, etc. Despite their common use today, decimals were not used widely until the early seventeenth century. Now, decimals are used extensively and almost exclusively in commerce and science. In mathematics, decimals can be used to represent both rational and irrational numbers. They are often used where measurements are taken and recorded to a given accuracy. Decimals lend themselves easily to approximations which can be given to any stated accuracy. Approximations will be discussed in the module, Introduction to Measurement . Percentages are a way of writing numbers with denominator 100.

For example = It should be observed that 25.24 can also be written as 25 and read as ‘25 and 24 hundredths’. The decimal point separates the units column from the tenths column. To the right of the decimal point, we read the names of the digits individually. For example, 125.2408 is read as ‘one hundred and twenty-five point two, four, zero, eight’. Here is a chart illustrating place value in the decimal system. In decimals the number of places filled by the digits after (to the right of) the decimal point are called the decimal places. Thus, 345.607 has 3 decimal places and 0.2 has 1 decimal place. Comparing decimals Using the number line to compare decimals On the number line 0.8 and 0.34 are both between 0 and 1. Because 0.8 is to the right of 0.34 we know that 0.8 > 0.34.

To make a decision about which of the numbers 4.2, 4.54 and 4.362 is largest we locate each number on the number line. We first focus on numbers between 4 and 4.6. It is necessary to ‘zoom in’ on a section of the number line to show that 4.362 is between 4.36 and 4.37. So we find that 4.2 < 4.362 < 4.54. Comparing numbers by working through the digits from left to right One method for comparing decimals is to write the numbers one underneath the other, with the decimal points lined up. For example, to compare 3.78 and 3.612, we first write the numbers one beneath the other, aligning the decimal points. 3.78 3.612 Next, we compare the whole-number parts. Both are the same, so we proceed to the next digit. We compare the tenths: 7 tenths is larger than 6 tenths. So 3.78 is larger than 3.612.

Writing fractions in decimal notation and vice versa Writing decimals as fractions and mixed numbers To write 0.005 as a decimal, note that there are no tenths, no hundredths and five thousandths and thus we know 0.005 = = To write 2.36 as a mixed number, we write 2.36 = 2 = 2 Writing fractions and mixed numbers as decimals If we have a fraction where the denominator is a power of 10, it is easy to convert it to a decimal. For example, 2 This can also be done as: 2 The short division algorithm can be adapted. We set out 743 ÷ 8 as shown: We have had to add trailing zeros. The decimal 0.875 is a terminating decimal. The division process terminates. By using a technique similar to that of the example above it can be shown that = 0.33333…. The digit 3 is obtained endlessly as new trailing zeros are added and the division algorithm continued. The decimal 0.3333… is called a recurring decimal. These will be discussed later in this module. There are many fractions whose decimal equivalent is not terminating.

The result of the following exercise determines which fractions are terminating decimals. = 9 = 9.29 However this is inefficient and the decimal notation gives us an easier way to calculate such sums. The decimal point in each number should be lined up one under the other. The vertical addition algorithm is shorthand for adding hundreds to hundreds, tens to tens, ones to ones, tenths to tenths, and so on. It is important to line up the place value columns when lining up the decimal point. Subtracting decimals When subtracting one decimal from another, write the numbers one under the other as with whole number subtraction, making sure the decimal points are aligned. Here are the two standard subtraction algorithms showing the methods used for subtracting 16.532 from 23.84. Sometimes, the number of decimal places is different in each of the numbers. It is helpful when doing subtractions that deal with these ‘ragged’ decimals to place additional zeroes at the end of the top number (in this case 23.84). This does not change the number; it simply says that there are ‘no thousandths’.

Multiplication and division by powers of 10 Multiplying When any number is multiplied by a power of 10, each digit is multiplied by the same power of 10. For example 24.06 × 10 = 20 × 10 + 4 × 10 + 0.06 × 10 = 200 + 40 + 0.6 = 240.6 Multiplying by 10 corresponds to moving the decimal point one place to the right and inserting a zero where necessary. Similarly multiplying by 100 = 102 corresponds to moving the decimal point two places to the right and inserting zeros where necessary. For example: 8.7 × 100 = 870. Multiplying by 1000 = 103 corresponds to moving the decimal point three places to the right and inserting zeros where necessary. For example: 8.7 × 1000 = 8700. Dividing When any number is divided by a power of 10, each digit is divided by the same power of 10.

78.4 ÷ 100 Division of decimals Division of decimals by whole numbers If we had a piece of rope that was 4.8 metres long and wanted to cut it into 4 equal pieces we could cut it in two and then cut each piece in two again to get 4 pieces, each 1.2 metres in length. We can use the division algorithm to show this. The procedure is the same as for whole numbers. The decimal point in the answer is aligned directly above the decimal point in the question. We can relate this to division by fractions (invert and multiply) 4.8 ÷ 4 = = 1.2 If we were to divide a 4.8 metre piece of rope into 5 equal pieces, the procedure is a little more involved. We could divide 4.8 by 10 and double the result to get 0.96. This can be also be found using the division algorithm: So, 4.8 cut into 5 equal lengths is 0.96. Each piece of the rope will be 0.96 metres in length.

Now if we divide a 4.8 metre piece of rope into 7 equal pieces, the result cannot be expressed as a terminating decimal. This can be shown using division: Thus the result of 4.8 ÷ 7 is 0.6 5714 and is a recurring decimal where the block of digits which recurs is indicated by a dot over the first and last digits of the block. We will return to decimals of this form later in the module. It should be noted that in a practical situation you would probably want to know the length of each piece of rope to the nearest millimeter. Each piece of rope is to be 0.686 metres or 686 millimetres in length correct to the nearest millimetre. return to top Dividing decimals by decimals When the numbers are ‘nice’ it is possible to perform some divisions by decimal using common sense. For example 1.8 ÷ 0.3 We can show what 1.8 ÷ 0.3 looks like on a number line using jumps of 0.3 until we get to 1.8. It takes 6 jumps of 0.3 to get to 0 from 1.8.

By multiplying each number by the same multiple of ten the division is easier to deal with. 1.8 ÷ 0.3 = = = 6 This suggests that we multiply both numbers by the same power of 10 to make the divisor a whole number and then use the division algorithm. 1.355 ÷ 0.05 = 135.5 ÷ 5 Alternatively we can relate division of decimals to division of fractions by writing each decimal as a fraction and completing the division. 0.6 ÷ 0.4 return to top Recurring decimals Earlier in this module we showed how to express fractions, whose denominator can be factorised as products of 2’s and 5’s, as a decimal. These decimals are terminating. We now look at fractions whose denominator cannot be factored as a product of 2’s and 5’s. These produce recurring decimals; that is decimals with eventually repeating digits. We have met these previously in our discussion of division. (Obviously 3.0000… is not considered to be a recurring decimal.) Numbers, such as when written as a decimal, result in a block of digits that repeats indefinitely.

= 0.185185185… = 0. 8 This is a recurring decimal and the dots above the 1 and the 5 digit indicate that 185 are the recurring digits. Some people prefer to write . where the bar indicates that 185 are the recurring digits. The fraction is an eventually recurring decimal but decimals with this property are still called recurring decimals. EXAMPLE Theorem We prove that every rational number, can be written as a terminating or recurring decimal. The above examples serve as illustrations of the proof. Proof At each step of the division of a by b there is a remainder. If the remainder is 0 then the division algorithm stops and the decimal is a terminating decimal. If the remainder is not 0, then it must be one of the numbers 1, 2, 3,…, b − 1. (In the above example, b = 7 and the remainders can only be 1, 2, 3, 4, 5 and 6.). Hence the remainder must repeat after at most b − 1 steps. Note that calculators can only handle terminating decimals.

EXAMPLE return to top Percentages We meet and use percentage in many aspects of our lives. The word percent comes from the Latin per centum meaning ‘for each hundred’. Percentages will be central in several other modules. In this module their relationship with fraction and decimal notation is emphasised. The symbol for percent is %. Percentage is another way to write fractions with a denominator of 100. For example, 8% = 99% = 150% = To understand percentages we need go no further than remembering that percent means ‘out of one hundred’. A square that has been cut into 100 smaller squares can be used to model percentages. If we colour three of them, we say that ‘3 out of a hundred’ or ‘3 percent’ are coloured in. We can write We proved the following earlier: Every rational number can be written as a terminating or recurring decimal, and vice versa. The decimal is terminating if and only if the fraction in simplest form has a denominator that is a product of 2’s and 5’s only. If the decimal doesn’t terminate, it eventually repeats with the length of the repeating block less than the denominator. It is also true that every real number has a decimal expansion (representation).

If the real number is not rational then the expansion is infinite and not recurring. The representation is not always unique. For example, the decimal 1.5 can also be expressed as 1.4999999….. We can state this more precisely. Every real number x has a decimal expansion. The decimal expansion is unique (if we ignore ending zeros) unless x is a non-zero terminating decimal, in which case x has exactly two decimal expansions, one terminates and one ending in an infinite tail of nines. Conversely every decimal expansion specifies a real number. EXAMPLE Consider the infinite decimal that starts with 0.101 and continues with two 0’s and a 1, three 0’s and a 1 and so on. 0.101001000100001… This decimal expansion is neither terminating nor recurring. This decimal does not represent a fraction. In other modules we will discuss numbers that are not rational. These are called irrational numbers. Converting a repeating decimal to a fraction A method for writing repeating decimals as fractions is shown in the following example. EXAMPLE Any real number can be approximated to any desired accuracy by a rational number.

This can be seen as follows. Every real number x has a decimal expansion. To approximate x within ±10−n, truncate that expansion at the nth decimal place. The result is a finite decimal, and so represents a fraction, and by subtraction the difference is less than 10−n. The number π = 3.14159……..has neither a terminating or recurring decimal representation. We can give approximations to π using decimal notation. For example π ≈ 3.142, correct to three decimal places. There are many interesting properties of decimals. Some of these will be considered in other modules. For example, if p is prime, and not 2 or 5, the decimal expansion of the fraction has a repeating block of length either equal to or a factor of p − 1. For example can be written as a repeating decimal with repeating block of length 6. return to top Decimals Decimal fractions had been employed for the extraction of square roots by Islamic mathematicians such as Al-Kashi (1380 − 1429) and also by the Chinese at an even earlier time.

The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende (36 pages), published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548-1620), then settled in the Northern Netherlands. No one realized their widespread potential for their daily use before Stevin. He felt that this innovation was so significant, that he declared the universal introduction of decimal coinage, measures and weights to be merely a question of time. Stevin’s notation was, however, rather unwieldy. The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus who used it in his trigonometrical tables (1612). It was also used by John Napier in his logarithmic papers (1614 and 1619). It should be noted that the Babylonians used a sexagesimal system.

That is, instead of the ‘places’ of the decimal system they used ‘places’ , etc., This is recorded for instance on Plimpton 322 which was written in about 1800BC. The equivalent to the decimal point did not occur in Babylonian writings until much later (c 300 BC). return to top Percentages Long before decimals were used, the need to work with tenths, twentieths and hundredths was evident. Roman taxes were calculated on the basis of these types of fractions. For example, Augustus levied a tax of 1% on goods sold at auction and there was a tax of 4% on every slave sold. The Romans did not use this notation but certainly employed the concept that we use today. The concept was used throughout the ages, however the use of the sign, % , is quite recent and its exact history uncertain. return to top APPENDIX Many different ways of multiplying have been used over the centuries. The grid (or lattice or Italian) method was used extensively. Hence we adapt it to multiply decimals. Consider the product 24.3 × 8.96. Follow the procedure outlined below to fill in the grid.

The first few numbers have been entered. 3 × 8 = 24. The 2 is placed in the top position. 4 × 8 = 32. The 3 is placed in the top position. Notice the arrows in the grid extending inwards from the decimal points. The diagonal on which these arrows meet (shown as a bold line) separates numbers greater than or equal to 1 from numbers less than 1. Now fill in the grid and extend the diagonal as shown below. Next, add along the diagonals, starting at the bottom right. The sum for the first diagonal is 8. This 8 represents . Check your answers using another method for long multiplication. The sum for the second diagonal is 7 + 1 + 4 = 12. This 12 represents Can irrational numbers (in the decimal system) become rational in a binary, octal or hexadecimal number system? - Quora Quora Numbers (mathematics) Can irrational numbers (in the decimal system) become rational in a binary, octal or hexadecimal number system? If we say that irrational numbers are those which are non recurring and non terminating, can we convert them to terminating or recurring using other number systems and hence make them rational?

No. A simple argumentative proof : Rational numbers can be expressed as p/q, while irrational numbers cannot be. So if base is same, rational and irrational numbers form a disjoint set. If a number can be expressed in form of p/q in any number system with fixed base b then it can be expressed in form of x/y in any other number system with a different base. (p and both are integers. Conversation from one base to another) Hence a number of the form p/q in any number system will remain in the same form. So if a number which is rational in at least one base will also be rational in all other bases, whether hexa, deci or binary. Hence a number is either rational or irrational in all bases. □ Written Apr 22, 2014 No. It can be shown easily that any number which is terminating or recurring in binary, octal or hexadecimal number system is a rational number (i.e. terminates or recurs in decimal number system). And since rational and irrational are completely disjoint sets, an irrational number cannot have a terminating or recurring notation in binary or any other number system.

Proof (written for binary and can be extended): Case (1): Consider any number which terminates in binary number system. Let us try and convert this number to a decimal number. Then it will be a finite summation of integral powers of two. Since all of the integral powers of two are rational, the summation is rational. Thus every terminating no. in binary is rational. Thus if any irrational no. has a terminating representation in binary,it will become rational and irrational at the same time after conversion to the decimal no. system. Thus an irrational no. cannot have a terminating binary representation. Case (2): Let's say it recurs. Let the recurring string of numbers have length n. Consider only the recurring stream (say 110001 or anything like that, here n=6). It can be converted to decimal representation x (as 1+2^4 +2^5 =49). The original number can be represented in decimal system as: 2^(-n) *(x) +2^(-2n) *(x)+........ Which is an infinite GP with common ratio 2^(-n). Since n>1, this GP converges to a finite sum.

=x*(2^(-n))/(1-2^(-n)) Which is rational as n is integer. (Thus 0.110001110001....... can be represented as 2^(-6) *(49) +2^(-12)*(49)+..... =49*(2^-(6))/(1-2^(-6)) = 0.7777..... =7/9.) Thus every recurring no. in binary is rational. Thus if any irrational no. has a recurring representation in binary, it will be rational and irrational at the same time. Thus any irrational no. cannot have a recurring representation in binary. Hence Proved. Updated Apr 25, 2014 The answer given by Ameya above is great. But allow me to give a simpler and less rigorous answer. If you wish to make an irrational number (say 2^0.5) in a particular number system(say decimal) be a rational number in some other number system, one thing you can do is take this irrational number to be the base of the new number system.

Hence the representation of 2^0.5 (decimal system) in a number system with base=2^0.5 will be simply 10. Now all the irrational numbers like 2*(2^0.5),3*(2^0.5),...,n*(2^0.5) are rational in this representation because they have a terminating representation. But the problem is there are infinitely more types of irrational numbers like 3^0.5 or 5^0.3 and so on which will still have a non terminating, non recurring representation in the system considered above. Hence coming back to your question NO irrational numbers cannot become rational in a binary, octal or hexadecimal number system. Certain kinds of irrational numbers can be made rational in specialized number systems that have an irrational base. There exists no number system that does not have irrational numbers in it because irrespective of the base you choose there are still infinitely many irrational numbers that can not be expressed as a linear combination of the base number you considered.

Question: What is the recurring number when two-thirds is expressed as a decimal?
Answer: 6