Why is 0.9 recurring 1

If , then. Since , then so does , thus :. This is saying that if two numbers are not equal, there is a third number that is also unequal and that can fit in between them on the number line.

Regardless of the type of real number or the difficulty in computing its value or representing its value, from a purely abstract perspective, there does exist a number that is larger than one but smaller than the other.

It is impossible to find a "next higher" number that is both larger to a given value but couldnt have been smaller see argument from averages. If , then what number could exist in between them such that? Since there is no conceivable number that can exist in between the two, they must be equal according to the definition of the real numbers as a continuum.

There could also be numbers between 0. One might argue that after the infinitely many zeros, there is going to be a 1. Negatives gave us the conception that every number can have an opposite. And you know what? Expanding our perspective with strange new ideas helps disparate subjects click. When writing, I like to envision a super-pedant, concerned more with satisfying and demonstrating his rigor than educating the reader.

This mythical? I really should give Mr. Rigor a name. My goal is to educate, entertain, and spread interest in math. Can you think of a more salient way to get non-math majors interested in the ideas behind analysis?

Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter. A famous joke illustrates my point: A man is lost at sea in a hot air balloon. With these rules, 0. Do Infinitely Small Numbers Exist? So, Do Infinitesimals Exist? The Traditional Approach: 0. Most mathematicians see the problem like this: 0.

The first number in the sequence is 0. For each number in the sequence, we attach another 9 at the end of the previous number's expansion.

Here we are no longer dealing with a weird infinitely long decimal expansion, but instead a collection of nice, basic, easy to understand terminating decimals. As we go through our sequence, sticking more 9's at the end of our terminating decimals, we are getting closer and closer to 1. The distance between two real numbers is just the difference of the larger number minus the smaller number.

So, the distance between the first number in our sequence and 1 is 1 - 0. The distance between 1 and our second number is 1 - 0. The distance between 1 and our third number is 1 - 0.

Each term of our sequence gets closer and closer to 1, and for any tiny little distance we want, we can find a number in our sequence — some finite pile of 9's after a decimal point — that is closer to 1 than that tiny little distance.

Since we saw above that the numbers in our sequence are always getting closer to 1, this further means that all the subsequent numbers in the sequence will also be closer to 1 than our tiny distance. So the "there's always a difference" argument betrays a lack of understanding of the infinite. That's not a "criticism", per se; infinity is a messy topic. Proof by geometric series. The number " 0. In other words, each term in this endless summation will have a " 9 " preceded by some number of zeroes.

This may also be written as:. Since the size of the common ratio r is less than 1 , we can use the infinite-sum formula to find the value:. So the formula proves that 0. Note: Technically, the above proof requires that some fairly advanced concepts be taken on faith. If you study "foundations" or mathematical philosophy way after calculus , you may encounter the requisite theoretical constructs.

Other pre-calculus arguments. Reasonably then, 0. But 3 0. Then 0.