does 0.999999 (repeated) = 1??

Question

0.9999999999 (repeated) = 1



1/3 = 0.333333333 (repeated)

0.3333333333 (repeated) * 3 = 0.999999999

you agree?

Answer 1

Yes: 0.999... = 1.

The way you have shown is one good method.

Here's another way to show it:

Let x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1


Another way would be to calculate the infinite sum (which is essentially the same thing as above):
0.999... = 9(1/10 + 1/100 + 1/1000 + 1/10000 + ...)

Let x = 1/10 + 1/100 + 1/1000 + ...
10x = 1 + 1/10 + 1/100 + 1/1000 + ...
9x = 1
x = 1/9

0.999... = 9(1/9) = 1

In fact, we can do this in any base for the above reasons. That is:

0.1111111... base 2 (binary) = 1
0.2222222... base 3 = 1
0.3333333... base 4 = 1
etc.

And because of this, we see that any terminating decimal can be rewritten as a repeating decimal:

0.25 = 0.249999999...
0.021 = 0.02099999...
0.9 = 0.8999999999...
etc.

Answer 2

It equals exactly one. Anyone that believes otherwise will learn why it is true after learning about infinite geometric series as early as high school. The source link demonstrates several ways to look at it.

One simple way to look at it:

Every repeating decimal can be written as a fraction. This is a fact (.333...=1/3; .818181...=54/66). How do you write .999... as a fraction? Well, let's take a look at how one can convert repeating decimals to a fraction. The following is an example from my textbook.

Convert 4.372372372... to a fraction.
Let x = 4.372372. Call this equation #1.
Count how many numbers there are in the repeating part. In this example, the repeating part is 372. So there are 3 numbers in the repeating part.
Multiply both sides by 1000, because 1000 has 3 zeroes. We get 1000x = 4372.372372... Call this equation #2.
Subtract equation #1 from equation #2.
999x=4360.000... The zeroes are negligable.
Solving for x, we get x = 4368/999.
Reduce the fraction, and we get x = 1456/333.

Applying the same concept to .999... we get the equation in lordofdrgn's answer.

So the answer is 9/9, which equals one. If .999... did not equal one, there would be no fractional form of .999... but any number that can be written as a fraction is a rational number, and repeating decimals are always rational. See the problem here? .999... = 1 exactly.

Answer 3

Nope. Your only proof is wrong. People who think 0.99999...=1 might want to think: It's obvious 1/3 is NOT, I repeat- NEVER 0.33333.....! In fact, it's 0.33333333333...... and a 4 at the end! 10(0.999999.....)-0.99999.. IS LARGER THAN 9!!!!! Why do all websites use this as their proof? Because they neglected to use notations correctly!
The Rule of Infinity states that any digit put after a mathematical operation done to a repeating number (ex. 0.99999... times 10) will have ANY DIGIT STAY THERE, even if it's a zero! In that case, for all other infinite numbers, there will be TWO OF THE SAME, NOT one!
10(0.99999..)=9.99999....0
0.99999999...=0.999999..99 which leaves 9.00000...11, NOT 9! Is that your only proof? This paradox is fake! Clearly 0<1!
Also: 1/3 is not 0.333333..., it's 0.33333....4, due to the Rule of Infinity.

Answer 4

No. In pure mathematics, a rational number is one that can be written as a/b, where a and b are integers. If b is not divisible by two or five, the division will yield a decimal number where digits repeat themselves. You put an example of 1/3, where the digit 3 is repeated eternally. For proper multiplication, you would have to put your number in the form a/b. Your 0.333333333 is not 1/3, it is 333333333/1000000000

In practical mathematics, ideal numbers need to be converted to something we can work with. 1/3 can be equivalent to 0.3333333...How many digits would this number have, depends on the application. In the real life, there is always a way to write the decimal that is so close to the ideal that the difference is not significant. It is not obvious, but there are rules pertaining operations of this type, but we apply them so naturally we are not aware of them, in the same way we use grammar in normal conversation.

Answer 5

It does equivalent a million. And by way of the various responses which say they do no longer, in easy terms proves the wikipedia article splendid: *** The equality has long been huge-unfold by way of professional mathematicians and taught in textbooks. in the previous couple of a protracted time, researchers of arithmetic training have studied the reception of this equation between pupils. a great many question or reject the equality, a minimum of initially. Many are swayed by way of textbooks, instructors and arithmetic reasoning as under to settle for that the two are equivalent. despite if, they are in lots of circumstances uneasy adequate that they grant added justification. the scholars' reasoning for denying or declaring the equality is usually in accordance with considered one of a few easy erroneous intuitions with regard to the authentic numbers; for example, a concept that each and each unique decimal growth ought to correspond to a different selection, an expectation that infinitesimal parts ought to exist, that arithmetic could be broken, an lack of ability to comprehend limits or basically the thought that 0.999… ought to have a final 9. those techniques are fake with appreciate to the authentic numbers, which would be shown by way of explicitly development the reals from the rational numbers, and such structures could additionally practice that 0.999… = a million rapidly. on the comparable time, a number of those pupils' intuitive expectancies can happen in different selection structures. *** There are various counsel on a thank you to symbolize a million: e^0, 50/50, sin(pi/2) .... Why is it so no longer ordinary to settle for 0.9999(repeating) equals a million as nicely?

Answer 6

yes because it can be expressed as the sum of an infinate geometric series

sum[ (1 to infinity) 9*10^-n ]

S = n1/ (1-a)

where a is the common difference 1/10 and n1 is the first term.
therefore

S =.9/(1-.1) = .9/.9 = 1

Answer 7

Yes
Two real numbers are seperate if we can find another real number between the two numbers .

there is no real number between 0.999 repeated and 1, since for any given real d the difference of 1 and 0.999.... .is less than d.

So essentially there is no difference between 1 and 0.999...

Answer 8

0.999999999999999999999... does equal one, as long as it repeats over and over infinitely. It can be proven using the sum of geometric series and by other methods (shown by others).

Answer 9

Only if infinitely repeated.

Answer 10

I guess it would because 1/3=0.3333333(repeated)
2/3=0.6666666(repeated)
so i guess ........3/3=1=0.9999999(repeated)