Does 0.999... = 1?
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- Опубликовано: 3 окт 2024
- Have you ever heard that 0.999... equals 1? Is that true? If so, why? In this video we prove in an extremely simple way what's going on! The steps are so simple that you could show this to an elementary school student and they would understand! So, use this to impress your friends!
Want to see some weird advanced math? If you're interested in mathematical proofs, this is the video for you! In it we explain whether or not 0.999 repeating equals 1. Check out this extremely simple video for some very cool math tricks!
Can you make a video explaining the proof of the Pythagorean theorem?
Great suggestion, will do!
Please mute neoicon mint. He’s a troll with nothing better to do
You explain very good ;)
This is not a valid proof for reasons that are quite well known. (It starts with the assumption that 0.333... = 1/3, which is almost the same as the idea that 0.999... = 1.) The key point that needs to be understood is that the recurring decimal notation must be defined as referring to the mathematical limit of the value of the decimal number with the digits continued without end. The mathematical limit is not, in principle, the same as the value it 'eventually reaches'. With that definition in place 0.999... = 1. Without it, 0.9999... < 1.
I can't agree with everything that you wrote. Without a definition 0.999 . . . cannot be assumed to represent a number at all. Its meaning does not follow from e.g. the axiomatic definition of the reals. That only deals with the sums of finitely many terms. The fact that the sum of infinitely many finite terms can be finite is paradoxical (although now known to be perfectly OK).
The sequence 0.9, 0.99, 0.999, . . . does not eventually reach anything. The terms just get bigger and bigger "forever".
For the benefit of others, the universally (!!!) accepted definition for the sum of a real infinite series (i.e. one whose terms are reals) is that it is the limit of the sequence of its partial sum (if the limit exists). That sum is the number that the series represents.
@@Chris_5318 I'll amend my final comment. Instead of 'Without it, 0.9999.... < 1' I will say 'Without it, 0.9999... < 1' no matter how many 9s you add.
You also mentioned that the definition using the mathematical limit is universally accepted. This may be true among those who are aware of the definition. But for those who are not aware of this point it is not accepted. The definition needs to be stated and used in proofs of the kind attempted in this video if all those people are to appreciate it.
@@matthewleitch1 Without a definition, 0.999 . . . is undefined (who knew?). You cannot sensibly claim that it really is a [representation of a] number, yet alone compare it to 1.
You cannot append any more 9s to 0.999 . . . on the right. There is no place to append them. You are thinking of the sequence 0.9, 0.99, 0.999, . . .. For that every term is < 1. The limit is 1 (and in fact 0.999 . . . too). The limit is not a member of that sequence. You cannot define it to be one as every term (i.e. all infinitely many of them) has a natural number of 9s.
Realistically, children (age 10 to 13 or so) will not easily grasp the actual formal definition along with all the baggage it requires. So the naive proof due to Euler) is used. The formal proof is probably not taught before age 16 or so, and only taught to children that have gone on to things like the British A-level. Even then things like the supremum of a set, the formal definition of limit and things like the Bolzano-Weierstrass theorem are not taught. That's degree territory.
The proof in the video is fine for the vast majority of people. That's why it (or variations of it) is taught in schools throughout the world.
@@Chris_5318 The 'proof' in the video is not a proof at all and presenting it as such is undermining to mathematics. It reminds me of the problem of teaching one theory at GCSE and then another at A level and then another to undergraduates, each time presenting the theory as if it is simply the truth. It undermines the credibility of the whole educational process in the eyes of students who realise they have been fibbed to.
@@matthewleitch1 You are not being realistic or reasonable. The proof is not being presented as a formal proof or a PhD thesis. Young children don't start with degree level math. Would you advocate going straight in with Einstein's field equations and quantum mechanics at kindergarten on day 1 on the grounds that anything else would be a fib?
In Euler's day, the proof would have been regarded as being rigorous.
Thanks for the clear explanation. One important but often overlooked fact about 9 in the debate is that it's the biggest valued digit in the decimal number base, which of course is the one we happen to be using. In duodecimal (base 12) for example, the biggest digit is conventionally designated as B, amounting to 11. Now B > 9, and since 0.B amounts to 11/12 and 0.9 amounts to 9/10, then 0.B > 0.9. Therefore 0.BBB... > 0.999... Yet if 0.999... = 1, then so must 0.BBB...= 1 by the same logic as given in a proof such as yours. There can't be anything special about decimal in this respect. But how can they both equal 1 if 0.BBB... > 0.999...?
I can see a couple of answers. Perhaps 0.BBB... isn't really bigger than 0.999... even if 0.B is bigger than 0.9. And/or perhaps the two equalities are each correctly derived in their respective number bases and that's all that can be demanded? I'm not trying to prove or disprove anything, I'd just like your opinion.
Chris G, for every natural B, 0.BBB . . . (base B+1) = 1
Proof (in base B+1 throughout):
(B+1) * 0.BBB . . . = B.BBB . . . = B + 0.BBB . . .
So B * 0.BBB . . . = B
So 0.BBB . . . = B/B = 1
0.999... is a never ending sequence which is NOT a number therefore it cannot be used in an equation ! a sequence can't be used with Arithmetic operation: add/sub/multipy divide
'0.999... is a never ending sequence'
0.999... is a real number. It is not a sequence.
'which is NOT a number'
Lol no. Decimals represent real numbers. '0.999...' is a decimal.
Learn some basics before you embarrass yourself next time.
amazing ,awesome
Ok but say
.999...=x
9.999...=10x
8.999...=9x
7.999...=8x
All the way to
.999...=1x
That proves nothing. And plus try saying .222...=.333... cause if .999...=1 the .999...8=.999...=1 which make every number equal
Everything was correct up until you said 0.222 . . . = 0.333 . . . and introduced 0.999 . . .8.
0.999 . . . 8 < 0.999 . . . because the first is a finite length decimal. You just didn't say how many 9s it has.
BTW it cannot have infinitely (i.e. endlessly) many 9s because it has a last digit (the 8).
What possessed you to say that 0.999 . . . 8 = 0.999 . . .? Show your steps.
it shows that 0.999...=x
anmd x=1
@@EmperorZelos You've just caused me to notice that I had omitted to say that, although his third statement is correct, he could also have subtracted the first from the second to get 9 = 9x and so x = 1. I must have been tired. Also 8.999 . . . = 9 is a bonus.
WRONG.... 3 x 1/3 = 0.99999... not 1.
You seriously believe that 3/3 != 1. Wow.
@@Chris-5318 who said that.? If you've written 1!=1 it dose.. not sure what you're getting at.
@@Longtimerolling You said it, as everyone (that's not you) can see. Perhaps you don't realise that 3 x 1/3 = 3/3 = 1.
@@Chris_5318 You're missing the point here, there are two ways to do this depending on whether you do it as a fraction, like you're more likely to in the USA, or decimal which is an option as well. If you leave it in the fraction form like you have above, then it does equals 1. But if you break it down into decimal sooner i.e 1/3 = 0.3333333333... then it would equal 0.99999999..... because decimal can only approximate "One Third".
@@Longtimerolling I have not missed the point. You said (almost directly) that 3/3 != 1. What you really meant is 3 * 0.333... = 0.999... != 1. Of course that is wrong because 0.999... = 1 exactly and 0.333... = 1/3 exactly, and no approximations are involved.
10 * 0.999... = 9.999...
expand each side
9 * 0.999... + 0.999... = 9 + 0.999...
subtract the isolated 0.999... from each side
9 * 0.999... = 9
divide through by 9
0.999... = 9/9 = 1
More formally, 0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1.
Similarly, 10 * 0.333... = 3.333...
expand each side
9 * 0.333... + 0.333... = 3 + 0.333...
subtract the isolated 0.333... from each side
9 * 0.333... = 3
divide through by 9
0.333... = 3/9 = 1/3 precisely.
More formally, 0.333... := lim n->oo 0.333...3 (n 3s) = lim n->oo (1/3) * 0.999...9 (n 9s) = 1/3 * lim n->oo 1 - 1/10^n = 1/3.
Those results are mathematical facts regardless of whether or not you agree.
You need to try PROVING your assertions. Really you should get some remedial math lessons because you haven't understood it.
I suggest that you at least glance at the "geometric series" Wiki.
sorry i disagree .333... * 3 =/= .999... its = 1
It's ok to be wrong.
Abra, you haven't proven that 0.999 . . . != 1, you have merely asserted it. Where is your proof? Be aware that the result is math fact.
Abra, what is 0.999 . . . /3 ? NB "Not 0.333 . . ." isn't an acceptable answer.
Newcastle University states : “The decimal point shows where the fractional part of a number begins. To the left of the decimal point, we have the whole number part, and to the right we have the fractional part, …”. Documented here : internal.ncl.ac.uk/ask/numeracy-maths-statistics/economics/numbers/decimals-and-rounding.html
99.9… minus 9.9… equals 90 and no fractional part.
99.9… minus 10 equals 89 and a fractional part.
Two different results because the subtracted values are different.
10 is not equal to 9.9… therefore 0.9… is not equal 1. A whole number part has greater value than a fractional part.
@@NeoiconMintNet How did your link support your assertion? Just calling the part to the right of the decimal point the fractional part in no way proves that 0.999 . . . can't be 1. Hilariously, your linked page also says *"A recurring decimal is a decimal which repeats the last digit, or last few digits, forever. For example, "one third” written as a decimal is 0.3333333…"* that being something that you totally disagree with, even though every [non-crank] mathematician, and every university agrees with it. (Spare me the agreement is not proof BS - your disagreement is not proof either - you are delusional and you are wrong).
Newcastle Uni also contradicts you here: internal.ncl.ac.uk/ask/numeracy-maths-statistics/core-mathematics/pure-maths/series-and-sequences/the-sum-of-an-infinite-series.html
They say 1/2 + 1/4 + 1/8 + 1/16 + . . . = 1. That being something that you say is wrong.
It is equivalent to 0.111 . . . (base 2) = 1
Yep, 99.999 . . . - 9.999 . . . = 90. It's a rare treat to see you get something right. You must have done it by accident.
Yep, 99.999 . . . - 10 = 89.999 . . . (why didn't you just say that!?).
10 * 9.999 . . . = 99.999 . . . = 90 + 9.999 . . .
So 9 * 9.999 . . . = 90
So 9.999 . . . = 90/9 = 10
Also 9 * 9.999 . . . = 89.999 . . . = 90 (from two lines above).
What led you to claim that 10 is not equal to 9.999 . . . ? As usual, you just assert an idi0tic opinion as if it were a fact, when it is demonstrably wrong.
No matter how many times you post your idi0tic lies, nothing is going to change the fact that 0.999 . . . = 1 and 0.333 . . . = 1/3.
0.9 repeated IS equal to 1
👍👍👍❤️❤️❤️🥰🥰🥰😊😊😊⭐️⭐️⭐️⭐️⭐️
.333333... is not equal to 1/3
.333333... is actually the closest representative to 1/3 that decimals can provide
Video, where's your proof of your (actually wrong) opinion?
By definition 0.9... is a rational number that can be expressed as an infinite series with a partial sum :
9/10, 99/100, 999/1000, ... an odd to even ratio, not equal in value to 1. The 0 before the decimal point defines the number as having no value of 1. The digits within the number are integers, that can not express a fractional value. This was explained in elementary school and repeated on this academic site :
Newcastle University state here: internal.ncl.ac.uk/ask/numeracy-maths-statistics/economics/numbers/decimals-and-rounding.html : “The decimal point shows where the fractional part of a number begins. To the left of the decimal point, we have the whole number part, and to the right we have the fractional part, …”. A whole number is for example the value 1, in this case the number 0.9... reads as 0 whole numbers. Learn to read numbers so you understand decimal values.
You two math illiterates need to go back to elementary school because you don’t qualify to discuss math.
@@login892173 , 0.3... is not equal to 1/3. Long division confirms this fact. Learn about fractions and the division process. 3 divides into 10/10 only about 3 times with a remainder each time you divide 3 into 10.. always.
@@NeoiconMintNet You are an irrational number. The fact is that 0.333 . . . = 1/3 and you know that for a fact.
@@Chris-5318 the best part is that kid claims that 0.999.. is a rational number that isn't equal to 1 but also claims that there's no rational number between the two in a different thread. Complete imbecile lmao
No. That would mean that all numbers are equal.
what?
As 0.999 . . . = 1 is a math fact, you should reconsider. Where is your proof of your absurd claim?
Prove with this method that 2=1
Yes cause then .999...8 equals .999... which equals 1 and then .999...7 and .999...6 and blah blah blah every number is equal and blah blah blah 1-90=0=1=2=3=4=5 all the way to infinity and all the way down to -infinity
@@Justme_1213 no wrong, the number 0.999..8 does not exist
What’s the difference between an integer and decimal number?
10 × 1 = 10 ; an even integer.
10 × 0.9… = 9.9… ; not an integer.
0.9… ; is not equal 1.
extranet.education.unimelb.edu.au/SME/TNMY/Decimals/Decimals/backinfo/advprops.htm#cantwo
Mathologer (Prof. Burkard Polster): ruclips.net/video/SDtFBSjNmm0/видео.html
Prof. Jordan Ellenberg: ruclips.net/video/rT1sIVqonE8/видео.html
Prof. Charles Fefferman on a similar sum: ruclips.net/video/Jwtn5_d2YCs/видео.html
Siningbanana (Dr. James Grime): ruclips.net/video/G_gUE74YVos/видео.html
Neoicon MInt (honorary 25 yard swimming certificate): ruclips.net/video/vY9jQ4p7H2o/видео.html
basic logic using the numbering system proves beyond a doubt that 1 is not equal to 0.9... (period).
1 is a whole number and 0.9... is a fraction such as 9/10, therefore since a whole number is greater than a fraction, the numbers are not equal.
Once you learn the basics of the numbering system and understand the purpose of the decimal point, then you can move onto algebra.
An actual prove is as follows: X=1 ; Y=0.9... ; X=Y when X-Y=0; the difference between X and Y is not zero therefore the numbers are not equal.
"since a whole number is greater than a fraction"
so 1>1/1 and 1>2/1?
@@jNe4l you need to go back to 5th grade math where numeric notation was taught, that you either forgot or still don't understand.
Numbers 0.9… (9s repeating) and 1
Decimal place values: Numbers contain digits representing values. Each digit within a number has values per position: value increases by 10 to the left, decreasing by 10 to the right. Each digit has a maximum value of 9.
Number 1 has a value of 1 for being in the ones place. Number 0.9… has no value of 1 because the ones place has a value of 0.
Decimal point purpose: Integers represent whole numbers, fractions to the right of the decimal point. Decimal point separates: the whole part of a number from the fractional part of the number. A whole number is greater than a part of a number.
Fractions: when the top part of a fraction (numerator) is equal to the bottom part (denominator), the value is 1, requiring both numbers to be either odd or even. Number 0.9… has an odd numerator and an even denominator because: Number 0.9… = 9/10 + 9/100 + 9/1000 + … ; Also represented by (10-1) to the exponent of n, over 10 to the exponent of n.
Adding fractions: evaluating the summation of 0.9… = 9/10 + 9/100 + 9/1000 + … results: 9/10 requires an addition of 1/10 to make 10/10 = 1, but 9/100 is not 10/100 = 1/10, and the 1/100, now required is not available because: 9/1000 is not 10/1000 = 1/100; And this pattern shall continue forever because of the rules of decimal place values. The one digit in each position can’t equal to 10 (10 times the value of the next position, because the highest value in each place is only 9.
Algebra: X = Y when X - Y = 0; X = 0.9...; Y = 1;
subtracting these two numbers doesn’t result in 0, therefore: these numbers are not equal. Subtracting 0.9… from 1.0… is not 0.
Division: dividing 3 into 1 results in the process of 0.3 plus a remainder, then the next process results in the next digit 3, but the remaining will continue to exist. Multiplying 0.3… only, still leaves the remainder, the difference in value.
Visual explanation: Take a round pizza give it a value of 1. Divide this pizza into 10 equal parts and and remove 1 of the 10 slices. The value of the pizza is 0.9. Break the removed slice into 10 parts, remove one of those 10 parts of the removed slice, return only 9 of those pieces back to where you removed the original slice. The value of the pizza is 0.99. Repeat this process for eternity and the pizza's value shall never return to a whole pizza, meaning 1.
In summary (logic): the question is: are the numbers equal in value, not if there is a number in between. The digits right of the decimal point are for counting fractions, to the left are the whole numbers, which are greater than the fractions.
If you read and understood everything stated above and you still don't understand that these two numbers can't be equal, you are a math illiterate.
@@NeoiconMintNet You haven't even got to 4th grade you trolling muppet.
6th grade math lesson: ruclips.net/video/IszQnXIbYd8/видео.html
8th grade math lesson: www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbers-operations/cc-8th-repeating-decimals/v/coverting-repeating-decimals-to-fractions-1
What a bunch of math illiterates in one discussion. 1/3 is 1 divided by 3. Divide 1 = 10/10 by 3, then multiply the result times 3 to check if the answer is correct. 3 divides into 10 approximately 3 times, with a remainder because 3 times 3/10 = 9/10; not 10/10. So every time you divide 3 into 10, the result includes a remainder, a value that needs to be added to 0.3... to equal to 1/3. You people that don’t understand why these numbers are not equal, need to go back to elementary school.
You are math illiterate, deluded (full on mentaIIy iII), thick and you only come here to lie and deceive. The world's mathematicians know for a fact that 0.999 . . . = 1 and 0.333 . . . = 1/3.
Your "argument" beggars belief. You have only, at best, considered the sequence 0.3, 0.33, 0.333, . . . and in your own muppet like way realise that it doesn't get to 1/3. You completely failed to examine 0.333 . . . at all, yet made a "conclusion" about it. The only conclusion you would have made is that the sequence doesn't get to 0.333 . . . either. That's actually because 0.333 . . . = 1/3.
1/3 = 0.333 . . . 3 (n 3s) + (1/3)/10^n
0.333 . . . = 0.333 . . . 3 (n 3s) + (0.333 . . .)/10^n
Subtracting gives
1/3 - 0.333 . . . = (1/3 - 0.333 . . .)/10^n
The LHS is constant so the RHS must also be constant. That is only possible if 1/3 = 0.333 . . .
I'd suggest that you read a texbook, but I already know that when Prof. Howard Anton says, and proves, over several pages, and many times, that 0.333 . . . = 1/3, you say that he agrees with you that 0.333 . . . isn't 1/3.
You are actually mad and you need to see a doctor.
University sources confirm I’m correct and the evaluation proves the numbers are not equal.
Series for 0.9… : 9/10 + 9/100 + 9/1000 +... = 9.../10…
Series for 1.0… : 10/10 + 0/10 + 0/100 + 0/1000 +... = 10.../10…
Sequence for 1.0… : 10/10, 100/100, 1000/1000, …
Sequence for 0.9… : 9/10, 99/100, 999/1000, …
A difference in ratio, a difference in value.
The pattern of the two sequences and ratio of the series is different, the value of the numbers are different.
Geometric series formula: a/1-r ;
a := first term of series ;
r := common ratio;
Geometric series for 0.8… ;
(8/10)/(1-1/10) ;
(8/10)/(9/10) ;
8/9 ; proof by performing long division.
9 does not accurately divide into 8, always a remainder.
8 divided by 9 = remainder + 0.8… ;
because 9 divides into 8 approximately 0.8 times.
The sum of the series can’t be larger than the sum of the sequence,
Yet the formula claims the sum is larger.
Formula is inaccurate, the formula calculates for the limit of the series.
The limit has a different value from the function.
Sequence for ⅓ : 3/9, 33/99, 333/999,...
Sequence for 0.3… : 3/10, 33/100, 333/1000,...
Definitely a different value because the ratio is difference
between 0.3… compared to 1/3
0.3... is an approximation of ⅓.
An approximation of ⅓ times 3 is an approximation of 1.
@@NeoiconMintNet LIAR. All universities say 0.333 . . . = 1/3 and 0.999 . . . = 1, and you know that for a fact. You need to see your doctor about your ridiculous obsessive lying.
@@NeoiconMintNet Your stuff with the geometric series formula was hilarious, as is all of your math. Your previous proof that 1 - 0.999 . . . = 1.999 . . . and so not 0 was pure comic genius.
@@Chris_5318 Newcastle University states : “The decimal point shows where the fractional part of a number begins. To the left of the decimal point, we have the whole number part, and to the right we have the fractional part, …”. Documented here : internal.ncl.ac.uk/ask/numeracy-maths-statistics/economics/numbers/decimals-and-rounding.html
99.9… minus 9.9… equals 90 and no fractional part.
99.9… minus 10 equals 89 and a fractional part.
Two different results because the subtracted values are different.
10 is not equal to 9.9… therefore 0.9… is not equal 1. A whole number part has greater value than a fractional part.