## July 05, 2006

The webpage above is a fairly simple article in the Guardian about why 0.9 recurring is equal to 1. The good bit is the comments below.

A few samples:

OK if .99 is equal to one, give her teacher 99 pounds (GBP) and get her to give you 100, not an infinite number of times, but a million will do nicely.
You need to find a better maths teacher. "Another way of proving this, suggested by my daughter's maths teacher, is this. Think of two numbers, x=0.9 recurring and y=1 Can you think of a number that is higher than x but less than y? No, you can't." Yes, I can. x + (y–x) Teachers eh?
All a bit of bollocks really – as usual, when you get what looks like a paradox, it's because your starting conditions are incorrect – in this case it's because infinitely recurring numbers don't actually exist.
I for one will take Mr PikeBishop’s common sense mathmatics over any amount of pointy headed clatrap from some four–eyes who’s probably working on a thesis about “lesbian algebra” (grant aided no doubt).
Isn't maths great?! I like how this discussion stayed on topic & relatively rhetoric free!
Rhetoric free, until you remember how the genocidal and racist policies of Israel are denying all those innocent Palestinian children the chance to learn anything about maths at all!
Tallisker, Wikipedia Schmickipedia, I’m sorry but I simply don’t approve of any of this “fannying about” with our British numbers. In the world of business one plus one equals two and we seem to get along well enough. Now maybe I don’t have a degree–level course in formal logic and set theory, but I know how many beans make five!
The industrial revolution was based on sleeves up gumption, not the latest fashionable nonsense. Oh and by the way I think you'll find that the calculus wasn't invented till 1972, by an Englishman as it happens, although I suppose we're not meant to take pride in our country's achievements these days.
Dude, calculus is not the "latest fashionable nonsense", it is there since the ancient greeks…
Mathematics is a discipline which is centrally concerned with the pure logical examination and definition of concepts. The mere fact that it is mathematically sensible to separately describe 0.9 recurring and 1 prove that they are different.

And most surreal of all…

So I will have to just accept that we have agreed to differ – you at ground level with only a coloring book to your name, peering into crevices, and I, in a box on the shoulders of Robert Hooke with a vast array of luminescent and florescent crayons but (I think you implied) not knowing where I put my spectacles. I can only claim that it might appear so from way down there but . . .

### 37 comments by 1 or more people

1. "infinitely recurring numbers don't actually exist"

Gotta laugh at that one. Bye bye all rational numbers!

05 Jul 2006, 10:28

2. The mere fact that it is mathematically sensible to separately describe 0.9 recurring and 1 prove that they are different.

I don't think this one is worth less than the infinitely recurring numbers. It would make maths even more interesting if it turned out square root of 4, 10log100, and 2 were all actually different.

05 Jul 2006, 10:57

3. I read a blog post on this about a month ago: No, I'm sorry, it does in which a maths teacher explains the premise, walks through several different proofs to demonstrate that it's true, anticipates possible objections and answers them, leaving the reader in no doubt whatsoever that 0.9|9 actually is 1…

…and ends up with the longest comments thread of all time, with people desperately trying to prove that he's wrong. When they can't, they come up with all kinds of "well the assumptions must be wrong" theories, including, yes, "infinitely recurring numbers don't actually exist", "0.3 recurring isn't actually equal to a third, it's just an approximation", "you can't do arithmetic with recurring numbers, so your proofs are wrong", "it's just trickery with numbers", "it's just a peculiar notational outcome of our decimal system (so there must be something wrong with it)" and, oh yes, "maths must be broken if you can use it to prove that 0.9 recurring equals 1."

It's really astonishing the lengths to which people will go to avoid opening their minds to a new concept.

05 Jul 2006, 10:58

4. The comment about x + (y – x). Using the x and y given there, the answer is 1. Which is not strictly less than 1.

05 Jul 2006, 11:51

5. Unfortunately this is something that crops up an awful lot, especially at a level where 'recurring' isn't really defined in a strictly mathematical sense. For people having taken Analysis I, the fact that 0.999… = 1 is somewhat of a trivial deduction from the definition of decimal representations, but a lot of people who perhaps haven't had formal training often can't grasp the concept of limits and infinite summations.

I moderate an online mathematics forum, and this particular topic crops up an awful lot. In fact, it crops up so much that most of the trained mathematicians ignore those threads like the plague. Intuition, and indeed the (mis)interpretation of what we actually mean by 'recurring', seem to play a fundamental role. As I'm sure the mathematicians on here will know all too well, some concepts just aren't intuitive at all.

For whatever reasons, the debate always gets very heated and out of hand rather quickly. I don't really understand why this is so, but nine times out of ten, I'll end up closing the thread. This is a pity in some respects, since sometimes some really interesting conversation can arise.

05 Jul 2006, 20:25

6. Argh those comments made me angry… Not simply because the responses were rude and ignorant but, as Catherine pointed out, the way in which people will happily stick their fingers in their ears and scream "lalalallalal" instead of reading the proofs and thinking about them.

Interesting that so many people who don't understand the proofs (and they are that… proof) of the problem resort to peppering their own versions of why the equation just isn't so with lots of that there f–word.

06 Jul 2006, 20:32

7. You're a complete fucking idiot if you think that 0.99999999999… doesnt = 1

This quote wins it for me. :)

06 Jul 2006, 22:38

8. Catherine, that thread was a classic too, but not as good as the Guardian one – where was the lesbian algebra for example?

06 Jul 2006, 23:48

9. Haha, that linked thread was genius. My own particular laugh–out–loud favourite's were

'This is just the kind of logic that slams a spacecraft into the surface of a planet instead of landing it safely.'

and

'This is simply a mathematical trick, and I'm surprised that someone teaching mathematics could be taken so easily. Here's an example using his trick above that shows you that 9 = 10.
x = 9
10x = 99
– x = 9
—————
9x = 90
x = 10'

Hmmmm….

07 Jul 2006, 00:57

10. Oh no, someone please sort out that apostrophe disaster in the second sentence of my previous comment! It hurts my eyes!

I'm blaming it on it being late….

07 Jul 2006, 00:59

11. #### Michael Jones

Some absolute classics there. Take the x + (y – x) one; applying the axioms leads to the conclusion that this equals y. In order to substantiate his claim that this number is strictly between x and y, would the author care to tell us which axiom he rejects?

07 Jul 2006, 01:19

12. Mr PikeBishop on that board is hilariously stupid.

07 Jul 2006, 01:51

13. He's great. He's always one of the first people to write a comment, and it's always something incredibly reflexively right and stupid. I was pleased to find out he's the same with maths…

07 Jul 2006, 02:01

14. #### chris

Clearly 0.9 recurring does not equal 1.
First of all 1 – 0.9 recurring will give 0.1 recurring. Therefore there is always a difference. The suggestion that the two numbers equal each other is merely lasyness and disrespect for such a small number. The difference is considered insignificant and thus ignored, but it still exists no matter how much you try to pretend it dosnt.
If you are not prepared to except the infenticable small difference, then you should not be prepared to except 0.9 recurring as a real number. You can have it both ways!

07 Jul 2006, 03:17

15. #### Alastair

Woot for chris!

07 Jul 2006, 10:14

16. Thanks for this – I always get a bit depressed when I read stupid articles about politics, news, etc. its good to know that even when people do write articles about mathematics they screw it up.

"1 – 0.9 recurring will give 0.1 recurring" – oh no it won't! 1 – 0.9 = 0.1, and 0.9 recurring > 0.9, so the rhs must be less than 0.1, which 0.1 recurring isn't. I believe the answer to be more like 0.0 recurring, with a 1 at the end. I can't express that very well in just text – 0.01, with the dot above the 2nd 0.

07 Jul 2006, 10:51

17. I do hope #14 is a joke, because clearly:
1 – 0.9 = 0.1
1 – 0.99 = 0.01
1 – 0.999 = 0.001

continuing we get our result, which is not 0.1 recurring.

The problem with this whole thing is that infinity is a purely mathematical construct, and a result of the set theoretic axioms. These axioms are not the axioms of the real world, because no infinities exist in the real world. wrt #14 yes, if you stop the sequence at any point, and consider the difference between 0.99…9 and 1, there will be one, but the point is that we are considering the difference at infinity. We want a to find a non–negative number (which will be the difference) that will be smaller than the differences at each point in the sequence, but still as large a possible. Clearly as the differences in the sequence get arbitrarily close to zero, this number is also zero.

If you have trouble grasping the concept of the decimal number system, you're really going to love what the axiom of choice says!

07 Jul 2006, 11:02

18. Actually, I just read the article, which was entitled "Can two different numbers be the same?" Thats a horrific quetion to ask! Completely ambiguous meanings of different and same. The way it initially seems to be interpreted in the question is for the difference to be one of representation, and the same be of value – but this is surely cheating? As someone above observed, "It would make maths even more interesting if it turned out square root of 4, 10log100, and 2 were all actually different." Ie, its not very controversial to claim that two numbers with equivalent values can have a different representation.

What would be controversial (and wrong by definition) would be to claim that different values are actually the same value – which is what he has provided an argument for. I don't believe him, and my blog is called true contradictions!

07 Jul 2006, 11:08

19. #### James Black

‘First of all 1 – 0.9 recurring will give 0.1 recurring’

No, 1–0.9=0.1

1–0.99=0.01

1–0.999=0.001

Etc.

But you are right. In this world new world we need to respect all numbers, regardless of their size, race or sexual orientation, and its just plain laziness to ignore them. Its as Marx said, ‘The Mathematicians have only interpreted the world, in various ways; the point, however, is to change it’. By this he meant that professional Mathematicians always only speak bollocks aimed at sustaining the status quo. We the common people, with our superior skills of intuitive common sense, we know the truth. To try to understand ourselves will only lead to us becoming indoctrinated, docile and alienated from the class–consciousness. The result will be that these contradictions would survive. The only answer is revolution by us the people. Popular democratic common sense is born of the people and hence is always better then elitist opinion born of a small group, as their opinions are always biased (they want to make maths complicated on purpose so to sustain their almost mystical status). Only after a popular revolution can the fundamental assumptions of mathematics change, only after popular revolution will they be grounded in democratic common sense and not elitist complicated mysticism.

Amateur Mathematicians of the world have nothing to lose but their chains, they have the world to gain. AMATUER MATHEMATICIANS OF ALL COUNTRIES, UNITE!

07 Jul 2006, 11:43

20. Uhoh, they've realised what we're up to.

My fellow members of the Military–Industrial–Mathematical complex… RUN!

07 Jul 2006, 13:34

IT'S A LIVING BREATHING MONSTER!

07 Jul 2006, 16:08

22. LOL

I looooove maths! It's like being in a secret club.. shhhh

07 Jul 2006, 17:55

23. Oop, they're off.

08 Jul 2006, 12:04

24. "Can two different numbers be the same?"

Hmm. No. Can one number be represented in more than one way? Yes.

Oh dear, I'm being drawn into the argument …

08 Jul 2006, 12:05

25. "1 – 0.9 recurring will give 0.1 recurring"

That deserves to be looked at in more detail:

0.9 recurring + 0.1 recurring = 1.1 recurring (probably need proof by induction to prove this step rigorously, but writing it out by hand makes it obvious)

Taking 0.1 recurring from both sides:

0.9 recurring + 0.1 recurring – 0.1 recurring = 1.1 recurring – 0.1 recurring
Hence:

Looking at the left hand side:

0.9 recurring + 0.1 recurring – 0.1 recurring = 0.9 recurring

and the right hand side:

1.1 recurring – 0.1 recurring = 1.0 recurring = 1

Hence:

0.9 recurring = 1

So the initial mistake leads to quite a neat proof.

08 Jul 2006, 13:16

26. Cross–application of my previous comment reveals that "chis" is indeed a complete fucking idiot.

10 Jul 2006, 12:45

27. #### Gareth Herbert

"Math, my dear boy, is nothing more than the lesbian sister of biology"

But seriously, you want to know why maths will never reveal a higher truth to you – because it's really boring.

11 Jul 2006, 00:28

28. Bah Humbug! Maths shows the governing dynamics. Indeed, Sir, there could be a mathematical explanation of how bad your comment is. Anyhow, I must be off, I cannot waste anymore time memorising the weak comments of lesser Mortals!

(P.S. If anyone doesn’t understand this, watch ‘A Beautiful Mind’).

11 Jul 2006, 12:59

29. #### Michael Jones

I read this somewhere. It's very easy to find a counterexample to, since I'm fairly bad at physics and much worse at chemistry and biology, but I'll share it with you anyway:

Anyone who can do maths can do physics.
Anyone who can do physics can do chemistry.
Anyone who can do chemistry can do biology.

11 Jul 2006, 20:07

30. Haha Gaz! You're just jealous I can talk about history and politics and other real–world nonsense with you and (appear) to do it with a vague level of competence. Try talking about maths and suddenly everything becomes Greek to you – perhaps one day you'll permit me to enlighten you on the rudiments of binary operations of the group of integers?

Can only agree with the person who said maths is a like being in a secret club! Half the point seems to be those in the know trying to confuse those who don't – and if they can manage that they get bonus points from everyone else in the know. On the other hand some mathematicians actually are totally mental as anyone who's done Algebra II will know:

"So a martian comes down to earth and asks you 'what is this map?'. You try to explain in terms of some space and he says 'what is this space? I only have tentacles'."

mmm well at least we aren't physicists – or heaven forbid – engineers!

11 Jul 2006, 23:46

31. Ahem, I'm a mathematicain, and I'm on my way to becoming an engineer. Don't assume the club to be so exclusive Thomas!

12 Jul 2006, 16:13

32. I'm a mathematician and I'm on my way to somewhere as far away as possible in the opposite direction to being an engineer.

13 Jul 2006, 01:30

33. Lol, each to their own

13 Jul 2006, 16:42

34. #### Michael Jones

Indeed. I have absolutely no objection to a mathematician (or in fact anyone else) becoming an engineer (or anything else, within reason e.g. not an axe–wielding maniac or suicide bomber) – not that I would be in any position to stop them if I had – but my experiences over the last twenty years or so have been enough to suggest that I'm probably not destined for a career involving anything remotely practical.

13 Jul 2006, 20:57

35. Well it is good to see that you are using your twenty years of experience wisely Michael =) (Interesting to see that suicide bomber is now a house hold term for an unhinged person, like axe murderer. What a wonderful world we live in)

Maths does give you great tools for a degree like Engineering, it is the best language (after Klingon) in the world after all =)

One of the things that I found frustrating whilst studying maths was my inability to relate it to the real world. Saying that, I still prefer the theoretical and more applied maths type modules in engineering. (Quick look it's a woman that doesn't know what she wants, rare)

I too have doubts about how 'practical' I really am though. No sane person should let me loose in a lab. So good luck to my group project team members next year…

14 Jul 2006, 00:27

36. #### Hero

How do you KNOW that you are bombing suicides, any more than you know you are murdering an axe? This is all getting very confusing.

14 Jul 2006, 09:18

37. #### Michael Jones

I didn't say anything about murdering an axe, just wielding it.

15 Jul 2006, 14:47