REPEATING DECIMALS BY JD Neal 2015 J.D. Neal All Rights ReservedI'm using a big font for the sake of those who have high-resolution screens that squeeze textdown to eye-straining small sizes.

WARNING: This discussion involvesgrade school and early middle school math.It is not a college level math discussion andinvolves far simpler mathematical concepts.It seems many people were never taughtthese concepts, never learned them, or haveforgotten them.

Repeating decimals is a subject evenmath geniuses can fail to understand orexplain, and numerous erroneous theoriesabout how they work have been created totry to explain them. This happens becauseeven people with a high level degree inmathematics forget the basics and throughrote memory, not by actually understandingthe background details that explain them.And that is what anyone wanting tounderstand repeating decimals must startwith: the very basics.

What is math? Math is themanipulation of numbers. What arenumbers? They are the symbolicrepresentation of quantities.

For example, when a person withthree cows and sells one to someone itmeans they have two cows left. One of themany ways of representing this inmathematical notation is "3 cows - 1 cow = 2cows."

Thus, we understand what the terms"one" and "1" represent and the aboveshows one way we use them in thinking.

What is a fraction? A fraction is not anumber - it is the symbolic representation ofthe portioning process. In grade school, weare taught that 1 apple split among threechildren is represented by the fraction 1/3.This means 1 item is portioned among 3people; each gets 1/3 portion of the apple.

Another example is 2/3, whichindicates 2 out of 3 portions of something.Take something and split it into 3 parts; thenuse 2 for the given situation.

Suppose I say I will give you 3/2apples. Doing fraction reduction, you convertthat into 1 1/2 apple.

It is entirely possible to do all yourmath work with fractions. That is why we aretaught things like "A field hand is given anequal share of the apples they pick. Theywork at Farm A with 2 people and they pick35 apples. They work at farm B with 7 peopleand they pick 705 apples total. Thus the fieldhand gets 35/3 = 11 2/3 apple from farm Aand 705/8 = 88 1/8 apple from Farm B. Theyhave a total of 99 + 2/3 + 1/8 apple. Thecommon denominator for 2/3 and 1/8 is 24so multiplying 2/3 x 8/8 we get 16/24 andmultiplying 1/8 x 3/3 we get 3/24 whichmeans the field hand has 99 + 16/24 + 3/24= 99 + 19/24 apples.

So, what is a decimal? A decimal isnot a number: a decimal is the decimalrepresentation of a fraction.

Many people learn their decimals byrote; thus they recognize the notation .3333(repeating) as 1/3. Hence if they see a .3333(repeating) portion of 600 they either pull theresult from memory or quickly do the math:600 x 1/3 = 600/3 = 200. This leaves them tobelieve .333 (repeating) is a magic number,when in fact they are doing the math fromrote memory and pattern recognition.

The term .333 (repeating) is not amagic number because it is a not a numberat all: it is a mathematical notation for aprocess and means nothing more complexthan 1/3 - take one instance of a number anddivide into 3.

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We used (and someone invented) thedecimal format because it can be easier todeal with in some cases than raw fractions.To understand how they work, we go back tothe basics.

Finding the decimal notation of afraction is done by dividing the denominator(bottom number) into the numerator (topnumber) of a fraction. Consider the term 1/2,which most people will recognize as one-halfand those with good memory recognize as .5in decimal notation. The initial layout of theprocess of converting 1/2 to a decimal is asfollows: ___2 |1

Looking at 2 we find it will not go into1, so we place a decimal up top and besidethe 1, following it by a zero: _._2 |1.0

We treat 1.0 as 10 and find 2 goesinto 10 five times and get this:

.52 |1.0

We are not done: we have to check fora remainder by multiplying 2 times 5 anddropping it below the 1.0

.52 |1.0 1.0

Subtracting 1.0 from 1.0 gives 0,meaning no remainder.

.52 |1.0 -1.0 ----- 0.0

Thus we have it: without smoke andmirrors and magic, we have converted themathematical notation 1/2 into its decimalformat. The process does not repeat; we donot have a remainder; we have a finitedecimal.

Now, let us do this with the fraction1/3. The layout looks like this: 1 |3

Looking at 3 we find it will not go into1, so we place a decimal up top and in the 1: . 3 |1.0

We treat 1.0 as 10 and find 3 goesinto 10 three times and get this:

.33 |1.0

We are not done: we have to check fora remainder by multiplying .3 times 3 anddropping it below the 1.0

.33 |1.0 .9

Subtracting .9 from 1 gives .1.

.33 |1.0 -0.9 ----- 0.1

And that is where the cult of theworshipers of the "Magical RepeatingDecimal" goes awry. They forget theremainder. The notation .33 (repeating) doesnot mean the 3 repeat infinitely; it means thatthe process of converting the base fraction to

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a decimal always results in a remainder, nomatter how many times you loop through it.You can repeat the process infinitely; but youdo not end up with a magic number: instead,you create a succinct number that has aremainder.

In the above case, we have one of theinfinite possible representations of 1/3converted to a decimal:

.1.3 with a remainder of ---

3

.1or .3 ---

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Said term can also be representedas .33 .01/3 or .333 and .001/3 or any otherof an infinite number of terms.

Consider the concept of multiplying 6x .333 (repeating). Anyone with math skillsrecognizes this as 6 x 1/3 which is 6/3 = 2.The long way of doing the multiplicationthrough decimals is to choose a specific finiterepresentation: decimals with a remainder.

Thus, calculating 6 x .333 .001/3 weget 1.998 + .006/2 = 1.998 + .002 = 2.000.

Which is odd: I didn't need anymagical, mystical theory to show how thatworks. I just used grade school math.

OTHER DECIMALS

Decimals can be deceiving becausethey tend to be taught by rote, without a carein the world given to explaining them in anypractical fashion. Most students recognize PIas some variation of 3.145 - the actualnotation depends on how precise a personrepresenting it is with decimals. No one hasyet to discover a situation where the decimal

notation repeats, which means PI is magical!No, it isn't. PI is a fraction: it is the

ratio of the circumference of a circle dividedby the diameter. The term 3.145 is just one ofcountless ways of representing it as adecimal. You rarely see it represented as afraction because for some reason almosteveryone in the mathematical field thinkssuch things should are not important.

The reason we have decimals for PI isbecause people with common sense drawcircles of a specific diameter; they use string(or some other device) to measure the lengthof the circle's circumference. Which is whatour ancient ancestors did to discover PI andnumerous other mathematical theories longbefore computer and calculators: measuresomething with string and see what you canmake of it.

Placing the circumference over thediameter, a person then have the fractionthat is PI. No one has found a perfectlyagreed upon fraction (likely due to errors inmeasuring) - but they can calculate thedecimal representation of PI by takingfractions and doing the grade school mathshown above.

The fact that a repeating decimal forPI has never been found is not reallymagical: it merely means the remainderskeep changing every time they do theiteration without any recognized pattern. Theperson doing the math can stop any timethey want and preserve said remainders, anddo math that (while likely flawed in tiny waysdue to imperfect measurements) works wellenough.

Thus, while 3.145 works okay, it is aflawed representation of PI because noremainder is provided for the user who wantsan accurate answer.

COMPUTERS AND REPEATINGDECIMALS

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One of the reasons some theoremsseem to work is due to computers.Computers are not autonomous devices;they only do what their programmers tellthem to. Programmers take the basicconcepts behind a computer and manipulatethem to their own ends. Programmers haveto go through hoops to make them work.

There is no limit to the number acomputer can manipulate, except thememory space available; one need onlyprogram it to do as they wish. But, forpersonal uses, most programmers do definea practical limit to the size of the numbersthat programs they create will readilymanipulate; if the user wants to go beyondthat they have to formulate their ownsolution. They literally have to write code tomake math work.

As it is, the programs used to createother programs provide a stock set of basicparameters, defining numbers (and otherdata types) and how big they are for commonuse.

Many years ago common programsworked on an 8 bit (1 byte) limit; then it wasdoubled to 16 bits (2 bytes); then 32 (4bytes); and current operating systems are 64bit (8 byte). The more bytes allowed, thelarger the number allowed in stock code; thebits that make up the bytes can beinterpreted in different ways, so the exactrange can vary.

What is important is that there is alimit to the size of a number, including thenumber of decimals a decimal representationof said numbers can have. At some point theprogrammer has to decide when to hack offanything that cannot fit the data limits of thecode being used.

Thus, if you pull up a spreadsheet andplay with decimals, you might noteinteresting things, depending on who wrote

the program and even the program they usedto write it and perhaps operating system.Odd behavior might fool the unwary intothinking various repeating decimal theorieswork; in reality, the programmer who wrotesaid program decided at some point theywere going to hard code an exception thatmerely makes them seem to work.

Start entering the decimal .9 and keeprepeating it (.99999...) and hit return (pressthe enter key) so often. At some point it mightmagically flip to 1. This is not proof that as adecimal approaches 1 it turns to 1; it meansthe programmer decided that at some pointthe input has reached the maximum numberof decimal points allowed by the way he setthe program up. He or she decided that saidtype of decimal was to be converted to 1 forthe sake of functionality.

A different programmer might not dothat; they might simply stop letting a userinput 9s and leave it as .9999 (carried out tothe limits of the computer's memory space).

Which explains why some programsirritate the user. The user may have numbersthan look like normal, but their spreadsheetsand databases may work erratically,especially when sorting. If they extend thedecimal format of said number, they mightfind that the computer is not (for example)always treating a whole number like 3 as 3 -but rather the computer treating some 3s and3 and others as 2.9999999999 (stretched tothe computers limit), causing odd artifactswhen sorting or doing other things with them.

Computers can be used to understandvarious mathematical concepts; but due tothe unreliable and erratic way they aresometimes programmed, they can also foolthe unwary!

Which is what happens when youinvolve people who create their own, differentimplementations of a system.

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OTHER DECIMALS