(note:  this write-up has to be closed come comments;  comments around it on various other pages will certainly be deleted!)

UPDATE!!:  The saga continues at this post.

You are watching: .9 repeating as a fraction



Every year I get a few kids in my classes that argue through me on this.  and also there room arguers everywhere the web.  and I simply know I'm going come get controversial "but it just can't be true" whiners in mine comments.  yet I feeling obliged to step right into this fray.

.9 repeating amounts to one.  In various other words, .9999999... Is the same number as 1.  They're 2 different ways of composing the same number.  kind of favor 1.5, 1 1/2, 3/2, and also 99/66.  every the same.  I understand some of girlfriend still don't think me, so let me to speak it loudly:


Do you think it yet?  Well, i do have actually a pair of arguments besides only size.  Let's look at some factors why it's true.  then we'll look in ~ some reasons why it's no false, which is something different entirely.  The typical algebra evidence (which, if you change it a little, functions to convert any repeating decimal into a fraction) operation something choose this.  permit x = .9999999..., and also then multiply both political parties by 10, for this reason you obtain 10x = 9.9999999... Due to the fact that multiplying through 10 simply moves the decimal allude to the right.  climate stack those two equations and also subtract castle (this is a legal move since you're individually the same quantity from the left side, where it's dubbed x, together from the right, wherein it's referred to as .9999999..., however they're the same since they're equal.  We said so, remember?):


Surely if 9x = 9, climate x = 1.  But due to the fact that x also equals .9999999... We acquire that .9999999... = 1.  The algebra is impeccable.

But I understand that this is unconvincing to numerous people.  therefore here's another argument.  Most people who have actually trouble v this fact oddly don't have trouble through the reality that 1/3 = .3333333... .  Well, consider the following addition of equations then:


This appears simplistic, but it's very, very convincing, isn't it?  Or try it with some other denominator:


Which functions out very nicely.  Or even:


It will work for any kind of two fractions that have a repeating decimal representation and that add up to 1.

Those space my very first two demonstrations the our fact is true (the last one is in ~ the end).  but then the whiners begin in around all the reasons they think it's false.  therefore here's why it's no false:

".9 repeating doesn't equal 1, it it s okay closer and also closer to 1."

May i remind you that .9 repeating is a number.  That way it has it's location on the number line somewhere.  Which way that it's not "getting" anywhere.  it doesn't move.  it either equates to 1 or the doesn't (it does of course), yet it doesn't "get" closer come 1.

".9 repeating is obviously much less than 1."

Hmmmm...it can be noticeable to you, however it's not evident to me.  Is it really much less than 1?  just how much much less than 1?  No, seriously...tell me how much less?  What is 1 minus .99999999...?


Really???? Infinitely many zeros and also then ~ the infinite list that never ends, there's a 1????  for sure that's stranger 보다 the possibility that .9 repeating just does same 1.  Or for something also stranger, take into consideration this:  if .9 repeating is much less than 1, then us ought to have the ability to do other very basic with those 2 numbers:  discover their average.  What's the number directly in between the two?  Or for that matter, surname any number between the two.  allow me guess:  the average is .99999...05?  for this reason after this infinite list of 9s, there's the opportunity of starting up multiple-digit extensions?  Doesn't that simply raise the apparent question:  What around .9999999...9999999...?  Namely, infinitely many 9s, and also then after that boundless list, there's another limitless list that 9s?  How, specifically is that various from the original infinite list of 9s?  If you observed it written out, wherein would the break between the perform be?

I'm afraid that if you use the "huh??" test of strangeness, you acquire a much higher strangeness variable if you say the .9999999... Is not 1 than you do if you say it is 1.

"Uhhhhh, I'm sorry, however I still don't think you.  .99999... Simply can't equal 1."

Well, let's look a little an ext carefully in ~ what us really median by .999999...:


This equation isn't yes, really up because that debate, right?  It's simply the definition of our location value device made explicit.  That point on the best hand next is dubbed an unlimited geometic series.  They have been studied broadly in math.  the word "geometric" means that each term that the series is the identical multiple (in this instance 1/10) the the previous term. The definition the the amount of an infinite geometric collection (and other series, too, yet we won't obtain into those) walk something this:

Start do a perform of partial sums:  the sum of the first one number, then the amount of the an initial two numbers, climate the amount of the first three, etc.Examine your list closely.  In this situation the perform is: .9, .99, .999, .9999, .... (Note that the actual number .99999.... Is no on the list, because every number ~ above the list has finitely plenty of 9s.)Find some numbers that room bigger than every solitary number on your list.  prefer 53, 3.14, and a million.Of every the numbers that are bigger 보다 every number on her list, uncover the smallest possible such number.  ns think we have the right to all agree the this the smallest number is 1.That the smallest number that can't be exceeded by noþeles on the list is the definition of the amount of the geometric series.

Notice that ns keep placing the indigenous definition in interlocutor face.  (See, ns did it again!)  That's because it's a definition, which isn't yes, really up for debate.  that is the nature of a mathematical meaning that once you acccept it, you have to agree to its consequences.  In various other words, .99999... = 1 by the definition of the sum of a geometric series.  It's also true if you usage the popular formula

a/(1 - r) v a = 9/10, and r = 1/10.

We're left through this:  just saying ".99999... Doesn't same 1" admits the truth that this number .99999... Exists.  and also if it exists, it amounts to 1 by definition.  The only means out because that you now, if girlfriend still don't think it, is to have a different working meaning of the sum of one infinite series (go speak to part math professors, and also see how far you get) or to deny the really existence the the number .9999....  I have actually seen a many of civilization doubt that the number amounts to 1, however very couple of of them are willing to refuse the an extremely existence of that number.  If you desire to pat "there's no such point as infinitely long decimal representations," I'm afraid you won't get very far, due to the fact that there's always the number pi to problem about, too, you know.

See more: Travel Distance From New York To England,Uk) And New York City

Okay, for this reason there's my rant.  .9 repeating equals one.  No, I'm sorry, that does.