Why does 9999 equal 1

Then 0. Argument from arithmetic: When you subtract a number from itself, the result is zero. So what is the result when you subtract 0. For shorter subtractions, you get:. Then what about 1. You'll get an infinite string of zeroes. The zeroes go on forever.

And 0. Argument from philosophy: If two numbers are different, then you can fit another number between them, such as their average. But what number could you possibly fit between 0. Argument from algebra: The expression 0. Call this numerical value " x ", so 0.

Multiply this equation by ten:. There is no "end" after which to put that zero. It seems to me like assuming the something which is already known. There are genuine conceptual difficulties implicit in this question. The transition from the rational numbers to the real numbers is a difficult one, and it took a long time and a lot of thought to make it truly rigorous. There are questions to be addressed about what we mean when we write down or pretend to an infinite decimal, or an infinite series.

Either of those devices is just a shorthand notation which mathematicians agree will represent some numbers, given a set of ground rules. Okay I burned a lot of reputation points at least for me on MathOverflow to gain clarity on how to give some intuition into this problem, so hopefully this answer will be at least be somewhat illuminating.

To gain a deeper understanding of what is going on, first we need to answer the question, "What is a number? There are a lot of ways to define numbers, but in general numbers are thought of as symbols that represent sets. This is easy for things like the natural numbers. So 10 would correspond to the set with ten things -- like a bag of ten stones. Pretty straight forward. The tricky part is that when we consider ten a subset of the real numbers, we actually redefine it.

This is not emphasized even in higher mathematics classes, like real analysis; it just happens when we define the real numbers. So what is 10 when constructed in the real numbers? Well, at least with the Dedekind cut version of the real numbers, all real numbers correspond to a set with an infinite amount of elements.

This makes 10 under the hood look drastically different, although in practice it operates exactly the same. So let's return to the question: Why is 10 the same as 9. Because the real numbers have this completely surprising quality, where there is no next real number. So when you have two real numbers that are as close together as possible, they are the same. I can't think of any physical object that has this quality, but it's how the real numbers work makes "real" seem ironic.

With integers bag of stones version this is not the same. When you have two integers as close to each other as possible they are still different, and they are distance one apart. Put another way, 10 bag of stones are not the same as 9. The bottom line is that the real numbers have these tricky edge cases that are hard to understand intuitively.

Don't worry, your intuition is not really failing you. Here is one coming from social choice theory. The first two functions are plotted here:. What happens for different sets of weights?

Where does this stem from? For more, take a look at this question and this manuscript. The problem is being convinced that every argument you are making actually is valid and makes sense, and not having a sinking feeling you aren't just falling for some parlor trick.

The obvious objection is: does it even make sense to talk about adding an infinite number of terms? How can we talk about taking and adding an infinite number of terms? So when we learn math in elementary school we are told: Every real number can be written as a decimal expansion maybe infinite and every possible decimal expansion is a real number.

And this is true. But we are not told why and we are expected to take it on faith, and we usually do. We've got the Integers. We use them to count discrete measurements.

We hope. But the Rationals still have infinite precision. At this point, we hope we can say "we can't measure it with any finite power of 10 but we can always go one more significant measure, so if we go through infinite powers of 10 we will measure it to precision" and we hope that explanation will be convincing.

But it isn't really. We have these "missing numbers" and we can get infinitely close the them, but what are they really? Well, we decide to become math majors and in our senior year of college we take a Real Analysis course and we find out.

We can view numbers as sets of rational numbers. We can split the rational numbers at any point into two sets. It often indicates a user profile. Log out. US Markets Loading H M S In the news. Andy Kiersz. Sign up for notifications from Insider!

Stay up to date with what you want to know. Loading Something is loading. Email address. Deal icon An icon in the shape of a lightning bolt. There are many proofs that the repeating decimal 0. Also with counter-proofs.

The only refutation this example proof has is that which questions whether or not. Both one-third as an infinitely repeating set of three's and one as an infinitely repeating set of nine's are equally exact, and both must be taken on a little bit of faith. If one-third and one-ninth in decimal form are taken without question to be equal to their fractional counterparts, then why cant one as a decimal of nines?

The only refutation this example has is as follows. One might argue that when multiplying by ten, the right hand side shifts to the left a decimal place, leaving a terminating zero at the end of the infinite string of 9's. And, therefore, when subtracting with a one after the last of infinite nines. But it is important to realize the meaning of infinite.