1.Introduction:
I read an interesting post in a forum few days ago. The topic is “Is 0.999… (a recurring decimal) equal to 1?” It was quite surprised for me that, less than 30% people agree that 0.999… equal to 1. Although Algebraic proof and Fraction proof were given, most of the people ignored the proofs or rejected to accept the proofs without any reason. Some people tried to use logical proof to show that 0.999… is not equal to 1, but all of them misunderstood the definition of recurring decimal or the meaning of infinity. They just showed a proof with a wrong assumption.
In this post, I shall discuss the characteristic of recurring decimal and fraction. After then, I shall explain why there are so many people believe that 0.999… is not equal to 1. Then it is more likely for me to give example for persuasion.
2. Recurring decimal and fraction
A recurring or repeating decimal is a number which when expressed as a decimal has a set of “final” digits which repeat an infinite number of times. For example:
0.123123123…… a set of number (123) is repeated infinitely
0.999456456456…… a set of number (456) is repeated infinitely
0.333333333…… a set of number (3) is repeated infinitely
A fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. For example
a quarter can be represented numerically as 1⁄4.
two quarters is one half can be represented numerically as 2⁄4 or 1/2.
3. Is 0.999… (a recurring decimal) equal to 1?
For the supporter, two simple methods are used commonly. One is algebraic proof and the other one is fraction proof.
Algebraic Proof:
Let x = 0.999…
10x = 9.99…
10x – x = 9.999… – 0.999…
9x = 9
x = 1
Fraction proof:
0.333… = 1/3
0.666… = 2/3
0.999… = 3/3 = 1
For the algebraic proof, there are lots of opposite view. Nearly all of the people pointed out that, x and 10x have different decimal. For instance, assume y = 0.25, then 10y = 2.50. All of the number should have different decimal after multiplication. As a result, 9x should be equal to 8.999……9991, not 9.
For the fraction proof, most of the people said that 0.333… is not equal to 1/3. There is no suitable decimal figure to represent the fraction 1/3. 0.333… is just an approximate number to represent 1/3 in decimal figure.
Finally, people tried to show that, if 1 = 0.999…999, then 0.999…999 = 0.999…998, …… ,0.000…001 = 0.000…000 = 0. That means 1 = 0. As a result, 0.999… is not equal to 1.
Obviously, we can see that many people do not fully understand the meaning of recurring decimal. It seems that most of the people do not have any conception of infinite and uncountable.
I shall say that 0.999… is a tricky number. We may be misled by this number easily, and started to think that there should be a final decimal 9. This illusion can simply be eliminated by given another recurring number.
Lets see 25/99, 0.252525……, can anyone tell me that the final number is 2?? or 5??
Again, 1357/9999, 0.135713571357……
Apparently, there is no final decimal in recurring decimal. As I have said before, a recurring decimal is a number which when expressed as a decimal has a set of “final” digits which repeat an infinite number of times. Before making any assumption, keep your mind that the digit of a recurring decimal is uncountable.
For the same reason, it is clear that 0.333… is not an approximate number of 1/3. 0.3 is exactly equal to 1/3. They are two different symbols to represent a same value. Similar to 0.125 and 1/8, it is undoubtedly two different symbols can represent a same value. Again, you should always remember the definition of recurring decimal.
4. Addition between special form of recurring decimal and fraction
In the following, I am going to do addition with two numbers. Go throught the mathematical formula step by step slowly. You shall then understand the basic meaning of recurring decimal and fraction more.
Let we see 25/99 and 74/99
It is no doubltly that 25/99 is equal to 0.252525…. and 74/99 is equal to 0.747474….
Add them together, What would happened??
25/99 + 74/99 = 99/99
0.252525…. + 0.747474…. = 0.999999….
Now, We can have a similar addition with another two numbers, 33/99 and 66/99
33/99 + 66/99 = 99/99
0.333333…. + 0.666666…. = 0.999999….
As you can see , the characteristic of 0.3333…. is changed a little bit.
0.333333…. become a set of number (33) repeated infinitely now, not with (3) any more.
I do not make any change to the value, just simply represent 1/3 in a different form. However, you can see that everything is changed. Thing is changed when you see a question in different view.
5. Miracle proof with wrong assumption
Lets see two miracle numbers, 09/99 and 90/99
Obviously, 09/99 + 90/99 = 99/99 = 1
09/99 = 0.090909……
90/99 = 0.909090……
This time, I am not going to add them together. instead, I attempt to simplify 09/99 and 90/99, then I shall use some simple mathematical formula and operation to show that, even if recurring decimal is finite, we can still show that 0.999… is equal to 1.
09/99 = 1/11 and 90/99 = 10/11
let z = 1/11 , then 10z = 10/11
case1
if z = 0.090909……090909 <- 9 is final digit
then 10z = 0.909090……90909 <- 9 is final digit
z + 10z = 0.999999……999999 <- 9 is final digit
case2
if z = 0.090909……0909(0) <- 0 is final digit, so 0 can be omitted
then 10z = 0.909090……909(0) <- 0 is final digit, so 0 can be omitted
z + 10z = 0.999999……9999(0) <- 0 is final digit, so 0 can be omitted
A little bit tricky. Although the proof is wrong, it really amazing!!
6. Conclusion
In this post, I do not attempt to proof that 0.999…( a recurring decimal) equal to 1. Actually, much more conscientious proofs or analyses have been done by many mathematicians.
Here, I attempt to restate the definition of recurring decimal (repeating decimal). Trying to estimate that why many people consider that 0.999… is not equal to 1. Attempt to state that, some people may misunderstand the meaning of recurring decimal and infinite.
If you still do not accept 0.999… is equal to 1 after you have read this post. Maybe, you will accept it one day, or maybe you will find out another miracle proof to demolish my belief in the future. Maybe……
Finally, I should say, post something about natural number “1″ is really like an idiot do something without any reason……