In mathematics, 0.999... (also written as 0.9, 0.. 9, or 0. (9)) is a repeating decimal that is an alternative way of writing the number 1. Following the standard rules for representing numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, ....

What is 0.999… equal to? The infinite repeating decimal 0.999… is equal to the whole number 1. This is very different from the value of, say, 0.999, which is equal to \frac {999} {1000} 1000999 (which is close to 1, but not equal to 1). The ellipsis (ell-IPP-siss) — that is, the "dot, dot, dot" — after the last 9 in 0.999… makes all the difference.

So the difference between 1 and 0.999... is zero. We can think of 0.999... as being equal to: = 0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.9×0.10 + 0.9×0.11 + 0.9×0.12 + ... This is an Infinite Geometric Series where a = 0.9 and r = 0.1 with the series being convergent because r is between −1 and +1. The formula for the sum is: So our sum is equal to:

0.999... is a repeating decimal, which means the digit "9" is repeated forever. It is different from 0.999, which only has three 9s. 0.999... can also be written as or . It is hard for many people to understand why 0.999... is the same as 1. There are many proofs that show why they are the same number, but many of these proofs are very complex. [1]

It may come as a surprise when you first learn the fact that this real number is actually EQUAL to the integer 1. A common argument that is often given to show this is as follows. If S = 0.999…, then 10*S = 9.999… so by subtracting the first equation from the second, we get. 9*S = 9.000… and therefore S=1. Here’s another argument.

0.999... = 1 is another case when two decimals that are written differently are, in fact, the same number. The fact that there are two different ways to write 1 as a decimal is a result of the role that infinite sums have in defining what non-terminating decimals mean.

2016年10月11日 · There is no error in your logic. Infinity does weird things, and one thing it does is that the infinitely long decimal expression 0.999... 0.999... turns out to be equal to 1. The implication is that there are actually many real numbers with two equivalent decimal expressions for them (so 0.5829999... = 0.583 0.5829999... = 0.583, for example).

There are many proofs that the repeating decimal 0.999... is equivalent to the number one, some more rigorous than others. Also with counter-proofs. The notion that 0.999... represents an infinitely repeating decimal—that is, the digit 9 repeats itself without end to the right of the decimal place. [1] Therefore: QED. Likewise:

2023年12月29日 · It includes select topics in real analysis, calculus, and the arithmetic of infinite decimals, culminating in several proofs of the identity itself. The material cuts across the study of pure mathematics, mathematics for scientists and …

2017年7月20日 · I’m going to show you how with basic math you can show your friends that the repeating decimal 0.999… is in fact equal to 1. Yep, you heard me right. We can show this is …