Why is 0.9999999......... not equal to 1?
In fact, [math]0.9999...[/math] is equal to [math]1[/math]. Here's the proof:[math]x = 0.9999... [/math][math]10x = 9.9999… [/math][math]10x = 9 + 0.9999... [/math][math]10x = 9 + x [/math][math]9x = 9 [/math][math]x = 1 [/math][math]0.9999... = 1[/math]Also see, Why isn't [math]0.999... = 1[/math] common knowledge?
Why is 0.999...9 equal to 1?
I see. Then the same could be applied for example to 0.8(repeating), 0.7(repeating) or 1.1(repeating) equaling to 0.9 , 0.8 and 1.2 respectively. Isn't this a false distinction between decimal numbers that are followed by an infinite number of same numbers and those that are followed by an infinite number of different numbers. The infinite number of numbers could be visualised as a numerical spectre. Like the colour spectre each number is followed with a profound normality by the next number, but it retains its unique individuality. You cannot say that two numbers are equal to each other, because they are infinitely close to each other. This is such a clumsy use of infinity.
Since 0.9, 0.99, 0.999, 0.9999 are not equal to 1, after how many 9s does 0.9999... become a 1?
I can answer this by giving an answer which is, in my view, a perfect analogy.Count all the natural numbers aloud. Say “1.” Then say “2.” Keep going indefinitely…Question: when will you be done counting this sequence aloud? The only correct answer is: Never. (That assumes, btw, that there is some minimum fixed amount of time that it takes to count a number.)And that is exactly the answer to your question. How long does it take to count an infinite number of 9s? The answer is the same as How long does it take to count to infinity?After how many 9’s will 0.9999… become 1? After an infinite number. How do you get an infinte number of anything? The general answer is that you never do.However, there is a “trick” answer to this question. In theory, you can do an infinite amount of counting if you count the first number in 1 second, the second number in 1/2 a second, the third number in 1/4 a second, and so on. This requires that you be able to count each number in an arbitrarily small amount of time. If you can, you can count an infinite number of things in a finite amount of time:1 + 1/2 + 1/4 + 1/8 + 1/16 +…. = 2 seconds.This is a trick that is not performable in this physical universe, but may be doable in some other theoretical universe. If you can somehow provide an infinite number of 9s, you get to 1.
Does 0.9999.... equal 1?
Your calculator is giving the right answer but for the wrong reason. Yes, 0.99999~ does equal 1, exactly. But no, your calculator is only rounding. You are putting in a finite string of 9's, your calculator only has so many significant digits of accuracy before it rounds. And it does round. And when it rounds it spits out the number 1 for you. What your calculator is doing is giving you the correct value for what you *intend* to enter (not what you actually do enter: a finite string of 9's), and its doing it for the wrong reason: its rounding. The number 0.9999~ does not approximately equal 1, it does exactly equal 1. Huge difference. The people that are telling you to round up are wrong. As a previous respondent noted, the figures we write, the shapes and the symbols of the numerals themselves, are only linguistic emblems if you will, that conjure up a thought, a conceptualization in the mind, of a quantity. Do not confuse the abstract notion of a quantity for the figures we use to represent them in written language. "One" does not exist. There is no tangible "one" we can pick up. That said, why is it inconceivable that we may have multiple different symbols or representations of the same quantity? Ignoring the previous paragraph, there exists literally dozens of proofs of the equality.
Does "0.9999..." equals "1"?
No.There will always be difference between 0.9999....and 1. If you do the subtraction,1 – 0.9999...will not equal zero. 1.000... – 0.999...? You'll get an infinite string of zeroes. A common objection is that, while 0.999... "gets arbitrarily close" to 1, it is never actually equal to 1.
Why doesn't 9/9 equal 0.999999...?
I've actually read about someone who uses exactly your reasoning to prove that 0.99999999... = 1. It goes like this: 0.1111... = 1/9...0.8888... = 8/90.9999... = 9/9 and 9/9 = 1, therefore 0.9999... = 1.Sorry, can't help it but here's what I hope is a more relevant answer to your question:First, think of the fraction 9/9 as 9÷9... If you know how to do division manually, you will remember that you can do the division in one single step: ____1_remainder 09| 9 | 9 |____ 0But you can also tweak the division a little to give a slightly different answer that almost means the same thing: _____0.999 ad infinitum___9 | 9 | 0 |______ |90 |81 |____ | 90 | 81 |_____ | 90 | 81 |_____ 9...This format is more consistent with the format of 1/9 = 0.11111..., 2/9 =0.2222... etc. You will realize that while 9÷9's answer of 1 can be quite wholly represented as 0.999... ad infinitum;1÷9, 2÷9... 8÷9's answers cannot be represented wholly as easily (or practically) in a different format other than the decimals version. Once you understand the above, you can relate to how 1/9 = 0.1111..., 2/9 = 0.2222... and so on, where each increase in numerator with 9 as a denominator will always have an increase by 0.11111... in the decimal format. So, we can see that 9/9's decimal format can also be 0.9999... (if we try to solve the 9÷9 statement in a forced decimal format) instead of just a simple 1... So, while it is intuitive to perceive 9/9 immediately as 1, know that you can write the number 1 in a different format that's consistent with the idea that 9/9 is equal to 9 times 1/9 = 0.1111... Some say 0.9999... is exactly equal to 1, others say it is not, but I guess that's for another question. (I may post my thoughts about this when I find the time to answer in a more appropriate post.)
Is 0.9999 and 0.99990 equal?
Yes. Absolutely… So are 0.999900 and 0.9999000.For this you should know the concept of significant figures.One of the rule of significant figures states that any number of zeros are insignificant that follow a decimal dot(.) and are not followed by any other number.Likewise any number of zeros are insignificant that occur before a decimal dot(.) and not preceded by any other number.For example 9999, 009999 are same and both are same as 0009999.000
Does 0.9 (unlimited 9s) equals to 1?
no not quite. 1/3 = 0.333....... without stopping to within an error of 1/3 /10^n 1/3 = 0.3 .....3 (n digits) so to within an error of 1/10^n 3/3 = 0.9.....9 (n digits) but however many digits you take 3 * 0.3....3 will never quite get there. You can amuse yourself by asking why, if you keep adding another place, which means adding 3*10^(-n) the number does not get infinitely large!.
If 0.999 is equal to 1, then what number approach is 0.9999?
0.999 is NOT equal to 1, it is equal to 999/1000. On the other hand, 0.999… indicates that the digit 9 repeats indefinitely. This is the result of multiplying the decimal representation of 1/9 (0.111…) by 9, which results in 0.999…The thing to realize is that 9 times 1/9 also equals 9/9 = 1 exactly.