How is 0.999 repeating the same as 1.00?

Answer 1

Answers: (1) It is not, 0.9 repeating is an infinitesimal, it will get infinitely close to 1 but never reach it.

(2) You sir, are incorrect. What is 1 divided by 3? 0.3 repeating. What is 0.3 repeating times 3? 0.9 repeating. So how can you say they are any different?

(3) The first 2 answerers have illustrated a classic philosophy puzzle. According to Zeno's dichotomy paradox (see link below), when moving to Point A to Point B, before you get there you must move halfway to Point B, then half of the remaining distance, and half of the now remaining distanceand so on. The paradox is that by this logic you will never get to point B, but we see in the real world that you do in fact reach point B.

Thus, answer (1) above is the literal logical interpretation of strict truth of computer programming, in which such exactitude is both possible and necessary. Answer (2) is the practical interpretation as is applicable in our daily lives, in which humans cannot visualize infinite decimal places(or infinitely smaller distances) and so we round numbers up or down to a more graspable number of decimal places.

Both are correct. It is also possible that both are merely being purposely obtuse for the sake of argument. (4) If two numbers are not equal, there exists another number between them. (e.g. 1 and 2 have 1.5 between them, 1.0001 and 1.0002 have 1.00015 between them) What number is between 0.999... and 1? There isn't one, so 0.999... = 1. (5) Subtract 1 - 0.999 = 0.000... = 0. No, the answer isn't 0.000...1 because that number doesn't exist. You can't have an infinite series of 0s that ends with a 1. The infinite series of 0s doesn't end, so there is no last digit to put the 1. (6) If you don't like the subtraction in (5), try it this way:

0.999... * 10 = 9.999...

Subtract 0.999... from both sides:

0.999... * 10 - 0.999... = 9.999... - 0.999...

Simplify the left side: 0.999... * 9 = 9.999... - 0.999...

Simplify the right side: 0.999... * 9 = 9

Divide both sides by 9: 0.999... = 1

(6) Why shouldn't they be equal? There are many ways to write the same number.

1 = 1/1 = 5/5 = 1.0 = 1.000... = 0.999...

(7) "0.9 repeating is an infinitesimal", erm, how can 0.999... be infinitely small? There are lots of number that is smaller than that, such as 0.5. Also, "it will get infinitely close to 1 but never reach it." is wrong. 0.999... is a number, it has an exact value and doesn't approach another number.