Just read an interesting (short) article on number theory which leads me to pose this question: Are the numbers 1 and 0.99 recurring in fact the same number? Can you explain why either way? Cheers Reachy
Here's the proof Let n=.9 repeating 10n = 9.99 repeating 10n-n = 9.99 repeating - n 10n-n = 9 (10-1)n = 9 9n = 9 n = 1
Bloody Hell! Far more elegant a solution than in the article I read!!! One of the proofs in the article was a lot more simplistic: 1 divided by 3 = 0.3333 recurring 3 x 0.3333 recurring = 0.9999 recuring Therefore 1 = 0.9999 recurring Simple! Cheers Reachy
Bad math It’s always fun to manipulate algebra/math principals to suit ones needs. Let’s take a look at the 2 proposed “answers”. VTVolfan’s answer: The fallacy here is right at the first assumption: Let n=.9 repeating If my high school algebra recollection is correct (and I can’t quote the exact words), when assigning a value to a letter, that value must be an exact number to get an exact answer. That "number" can be represented by digits, fractions, or algebraically. If that value is not exact, then the result of applying valid algebraic principals will produce results that are not exact. For example: “.3 repeating” is not exact but “1/3” is exact. They are close in value but they are not the same. If the first assumption was “Let n=.99999” then you have an exact number and the sequence would be: Let n=.99999 10n = 9.9999 10n-n = 9.9999 - n 10n-n = 8.99991 (10-1)n = 8.99991 9n = 8.99991 n = .99999 VTVolfan’s formula breaks down in line 5 when “n” is factored out to produce "(10-1)n = 9”. From this point on the equation is invalid (substitute for n and solve each side and you will find that the 2 sides are not equal) because, in effect, the first value (10n from line 4) contains one less decimal place than the second value (n from line 4). Reachy’s answer: The fallacy here is similar. The first assumption “1 divided by 3 = 0.3333 recurring” is a fallacy since “.3333 recurring” is not exactly equal to 1/3. Therefore any manipulation of that number will produce an incorrect answer. Sorry, 1 does not equal .99. But I do have a question for Reachy: Is this how you make your money, give someone 99 cents for $1.00 (both American)?
My 7th Grader Told Me... ...to convert this to a fraction. The numerator is the repeating sequence. The denominator is the number of 9's equal to the number of integers in the repeating sequence. If the repeating sequence doesn't start right after the decimal point the denominator is the number of 9's followed by a number of zeros equal to the number of decimal places to the right of the decimal point that the repeating sequences start. You have to also convert the figures to the left of the repeating sequences to a fraction and add that to the repeating sequence's fraction. For example: 0.166666.... = 6/90 + 1/10 = 1/6 0.9999.... then is 9/9 = 1.
So what about Pi? Not heard about the rule about symbols having to be numbers that are non-recurring after the decimal place and logically I can't see why either. Is it then incorrect to use the symbol for Pi in equations to represent Pi because that is an infintely recurring number? There are other numbers, universal constants if you like, that are similarly infinite and yet are represented by symbols. And what about the lemniscate? The symbol for infinity. Is that not used in algebra? I'm genuinely asking the question BTW because I don't know the answer . Another way of looking at the 0.999...... = 1 debate is to try and find a number between 0.999.... and 1. Can you? Cheers Reachy
Close but not equal OK. Maybe I didn't use the term "exact" in the correct context. And as I said, I cannot remember the exact wording if the rule, law, whatever. Or maybe it's something my math teacher said that stuck in my mind. I don't know, High School has been a long time ago. The point is that if you attempt to convert a fraction to it's decimal equivalent and that number results in an infinite number of decimal places, then the two are not exactly equal. Close, but not equal. Using a fraction instead of a decimal equivalent with an infinite number of decimal places is preferred in formulas. The question of using symbols to represent an infinite number, ie Pi (22/7), is used for convenience purposes if I'm not mistaken. When a formula containing Pi is finally calculated out, 22/7 is used rather than 3.142857142857etc. because it results in a accurate answer. Example: Pi = 22/7 = 3.142857142857etc. If we have a formula "7(pi)", using 22/7 for Pi the answer is 22, and this is the correct answer. But using 3.142857142857etc for Pi the answer is 21.9999999etc. . The 2 answers are really close but do not match exactly. So when you set up a set of algebraic formulas and at some point decide to chop off some of the infinite decimal places, inaccuracies arise. Those inaccuracies may not be relevant to what you are doing, but they are inaccuracies non the less. I'm sure a mathematician can explain this using 50 cent words and phrases but I can't.
Out of my depth again but... A further 2 toughts on this topic. Firstly, I can see why using a fraction is more accurate when you are actually using the formula to calculate a number. However when manipulating algebraic equations where the number/symbol is not actually changed in any way does it really matter whether the number that the symbol represents is an infinitely repeating decimal? Secondly, and I suppose it's just restating what I said earlier. What is the answer to the following problem? 1 minus 0.99 recurring? I would say that the answer is 0.00000..... recurring i.e. zero. If that's the case then 1=0.99.............. Cheers Reachy
??????? Don't know the answer to either question. Never got far enough in math to explore infinite numbers. Today's generation is learning math in 5th or 6th grade that my generation first learned in 2nd year of High School. I see you are on-line right now. So just in case you are wondering, it's 4am here and no this didn't keep me up all night.:laugh: I have a case of insomnia and I just woke up 20 minutes ago. Now my problem is what to do the rest of the morning. Shouldn't you be working?
Wednesdays off!!! Luckily I'm self-employed so I can decide my own hours! I own a shop and work Saturdays so Wednesdays I take off. Also I pretend to be working in the office when in fact I'm either on here, playing BJ or Poker. Easy life! Cheers Reachy
Repeating Sorry to tell you, toolman, you're wrong. .333 repeating is exactly 3/9 (or 1/3). .999 repeating is exactly 9/9. The fact that we can't write down enough 9's on paper to represent it exactly doesn't change the fact of what it is. If you plotted .333 repeating on a number line, it would be exactly (not approximately) 1/3 way between 0 and 1. I was a math major in college and I am 100% sure on this.
I see the error VTVolfan: I've been studying your response and this whole issue. I now stand corrected and see the error of my logic. I just had a hard time believing it's true. I thought I had my logic in place but I guess not. I learned from this experience and isn't that what this site is all about? On to the next problem. Take care and good luck.
let me help... Who ever has the .99, I'll give you a .01 so you will have a total of 1. and then they can be the same for sure. Now we can go on to another thread....LOL
Tell you what I'll do.... If .99 and 1. are the same, I'll give everyone that thinks so .99 cents for a $1.00 X 1,000,000 transactions. I'll be up $10,000 on the deal, then tell me if you think it is the same...LOL
Very Large Brains... I'd give some of those decimals for some Advil. Reading all of this has given me quite a headache! Damn glad I went into photography. No algebra involved. Probably have flashback nightmares of high school tonight.
I'm convinced Like toolman1, I was a skeptic, so I did some research looking to disprove the proposition and instead came to the opposite conclusion. For those who want more information (gluttons for punishment ) read on. A repeating decimal is a rational number because it can be expressed as the ratio of two integers; in other words, a fraction. The general method for finding this fraction, as described by Monkeysystem, is shown in this article from mathforum.org, Converting Repeating Decimals to Fractions UTVolfan’s method is the simple case where the repetend is just one digit and is the result of a more rigorous proof of the 0.9999… problem given in the article, 0.9999… = 1 At the bottom of that page there are links to articles with other explanations not quite so lofty. Over the years I’ve had a lot of math courses—I’m surprised I never heard of this question. As they say, live and learn. --jr
Oh, I remember the "convert repeating decimals to fractions" from high school now! The confusing thing for me now is that .3 repeating times 10 equals 3.3 repeating. Shouldn't the last digit be a 0 instead of a 3? This is how you can make .9 repeating equal exactly to 1.
