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. But it is important to grasp what "infinite" means. The 9's are infinite, there is no terminating number at the end. The zeros are also infinite, there is no 1 at the end.
There is no "end", there will also be another 9 or another 0. Since any number subtracted from an equal value is zero:. Math Wiki Explore. Browse content. Register Don't have an account? Proof:The Decimal 0. View source. Infinitely long decimals are a shorthand way to write out a sequence of finitely long terminating decimals. 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. To see this in action, here are the first five numbers in our sequence, all getting really close to When an infinite sequence of numbers has this property of getting arbitrarily close to some number — when, for any tiny little distance we choose, we can find some point in the sequence so that every number in the sequence after that point is closer to the target number than that tiny little chosen distance — we say that the sequence converges to that number, called the limit of the sequence.
In the case of an infinite decimal, again standing in for the kind of infinite sequence of terminating decimals we saw above, we identify the sequence with its limit.
This is what we mean when we say that 0. The same idea works for any rational number with a repeating infinite decimal expansion.
Something similar happens with irrational numbers that have non-repeating decimal expansions. In general, any infinitely long decimal expansion is thought of as the limit of the sequence of terminating decimals that make up the infinite expansion. So, the reason why 0. For you. World globe An icon of the world globe, indicating different international options.
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