binary sequences which are eventually zero

In other words, the cardinality of the continuum is two to the power of aleph-zero.

Using this method, every real number can be generated from a subset (finite or infinite) of the natural numbers, and the real set is the power set of the natural numbers. One can also choose the first element of the subset to represent the whole number part, which is always a finite natural number for all positive reals. This is not a precise mathematical way of doing this, but it serves as an intuitive glimpse into the cardinality of the continuum. These numbers can be collected into a set of their own, called A, the first few members of which are, which will provide an infinite decimal expansion. Only numbers with terminating decimal expansions can be expressed in the two ways shown above, and in binary, the only such numbers are those whose denominators involve only a power of 2. Therefore, each real number does not quite have a unique decimal expansion. with infinite 1's is actually equal to another, namely. When extending this back to our real number problem, there are a few slight glitches in this system, one of which being that an infinite binary decimal expansion such as. Sets with cardinalities greater than aleph-zero are also known as uncountable. Therefore, the set S has a cardinality greater then aleph-zero. Therefore, assigning a natural number to each infinite binary sequence does not cover all such sequences, and the function from the natural numbers to S is not a bijection. However, by definition, it is different from any sequence in the set (S1,S2,S3.) because for any sequence S n, with n a natural number, the nth element of the sequence is different from that of S x. The resulting sequence is clearly an infinite binary sequence, and therefore is a member of the set S. The nth element of every nth sequence is bolded, and for each bolded element, the opposite one is placed in the sequence S x. For the next part of the proof, consider a sequence S x, which is constructed by taking the nth element of each S n and reversing it, i.e. The actual ordering of this set is arbitrary, since if the cardinality of S is aleph-zero, all sequences with be covered eventually. The first few elements of the set are shown below: If the cardinality of the real numbers is aleph-zero, then each element in the set can be numbered S n, with n being a natural number. Each element S n is then an infinite binary sequence. To understand this argument, consider all possible infinite binary sequences as making up a set, called S. Georg Cantor was the first to devise this method, and through infinite binary sequences found a very elegant way to find the cardinality of the real numbers, through proof by contradiction. whether it has a cardinality of aleph-zero. We have the just simplified the problem to determining whether the set of all infinite sequences consisting of 1's and 0's is countable, i.e. However, no clarity is lost if these real numbers are converted to binary, and each number still as a unique infinite decimal expansion, this time only incorporating 1's and 0's. Since all values in this sequence are place values, each must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. Therefore, each of these numbers is defined by an ordered n-tuplet, with n being infinite, and of the form All of these are defined uniquely (almost, as we will see below) by their infinite decimal expansion. It is simpler to just ignore the whole number part, and focus on the real numbers on the interval (0,1). For example, the real number π has a whole number part of 3, and a decimal expansion of. The general real number has a finite whole number part, followed by an infinite decimal expansion. This is done by considering the construction of an arbitrary real number. To determine the cardinality of the real numbers, this problem can be again simplified to a problem involving ordered n-tuplets. Numbers such as e and π are transcendental.

All of the numbers that are real but non-algebraic are irrational, and are specifically known as transcendental. However, not all real numbers fall under the umbrella of algebraic numbers. In the previous posts of this series, it was established that the sets of natural numbers, integers, rational numbers, and even algebraic numbers have an equivalent cardinality: aleph-zero. Before reading this post, make sure you have read Infinity: The First Transfinite Cardinal, and Infinity: Countable Sets.

Binary sequences which are eventually zero