How come when I put .3 repeated in a calculator times 3, it becomes 1, but on others it is .9?
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How come when I put .3 repeated in a calculator times 3, it becomes 1, but on others it is .9?
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That's a matter of interpretation. Some matters of math are debated. For example, if you multiply 0.3 repeated 3 times, it's 0.9 repeated, which is soooooo close to 1, but not quite, hence other calculators counting it as 0.9 (sort of an abbreviation in math terms, they wouldn't go on with the 9's forever). However, if you convert 0.3 repeated into fraction form, it is 1/3, and 3 x 1/3 is 1. It all depends on how you look at it.
Hope this helped.
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Actually, most people agree that 0.9999999999999999... is 1.
$$\begin{aligned} 1/3&=0.333333333333333... \\ \text{Multiply bo}&\text{th sides by 3}\\ 1&=0.999999999999999... \end{aligned} $$
Here's a way you can look at this. If two numbers are different, you should be able to find a number between them. Can you find a number between 0.999999999999999999... and 1? If not, then they can't be different numbers.
Another way:One final way:
$$\begin{aligned} \text{Let }x&=0.9999999999999999...\\ \text{Then }10x&=9.9999999999999999...\\ \text{Subtract the }&\text{first equation from the first.}\\ 9x&=9\\ x&=1\\ \end{aligned} $$ -
Hi @userusername,
@The-Darkin-Blade and @Potato2017 are both correct and provided great mathematical proofs of this fact. Thank you all!
Thinking about it, it is true that some calculators are "cleverer" than others, and they "know" that \(\overline{0.333...}\times 3=1.\) It could be for different reasons: some calculators have this fact stored as basic initial information for other calculations, while other calculators can transform some of the repeating digits of the fractions, so when they receive \(\overline{0.333...}\times 3\) they think of it as \(^1/_3\times 3,\) and that's why we get \(1\) as an answer. The other, less "clever", calculators just multiply these numbers and we get \(\overline{0.333...}\times 3=\overline{0.999...}\) as a result. Or, maybe, they are more sophisticated than we think, and they want to clearly show what the actual answer is.
But, in the end, for us, the answer \(\overline{0.999...}\) and \(1\) are the same. The easiest way to realize this and put it in your head as not like something that you just "know", but something that you "understand", is to realize that the number \(\overline{0.999...}\) has infinitely many \(9's,\) so it is infinitely close to \(1,\) and there is no actual difference between \(\overline{0.999...}\) and \(1.\) At least, I think of it in this way
Hope this helped you to find your own understanding of what is going on!
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so 1/3 is 0.3 repeating, right? 1/3 *3 is 1, but 0.3 repeating *3 is 0.9 repeating. So, we know that 0.9 repeating is 1.
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All a matter of interpretation
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If you put \(1/3 \cdot 3 \)into almost any calculator, 0.99999999 will come out. This is because when it calculated 1/3, it rounded 0.3333333333333333333... into 0.333333333333. When it calculated 0.3333333333333*3, it forgot that 0.33333333333333 was rounded. Good calculators will give you 1.
Edit: the asterisks made it italicized
i fixed it -
Personally I like to think of 0.99 repeating as a real small tad less than 1.
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Oh, and why when I do .3 times 2 it is 0.66666667 instead of 0.66666666?
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Hi @userusername,
It is because your calculator rounds the number \(\overline{0.66666666...}\) to the number \(0.66666667.\)
The same thing happens when you want to round the number \(\overline{0.316495...}\) to its hundredths. Instead of \(0.31\) you will get \(0.32,\) since the next digit after \(1\) is \(6\) that is \(\geq 5,\) so the rounding goes up.
