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Not all infinite numbers are same

Many ancient cultures have ideas about infinite, but most of them defined the infinite as a philosophical concept instead of mathematics concepts. The symbol did not exist until 17th century. The symbol was firstly used by John Wallis in 1655 when he wanted to divide a region into infinitesimal strips, and each strip is . That is why mathematicians need the infinite numbers. They need an arbitrary large number for some certain situation. Given a large number , there must be a larger number L+1 exists. To move forward for a single step, if mathematicians take L+1 as the new large number, there also be a large number L+2 exist. In this way, no one could express the largest number. To express ‘large number’, the notation and definition of infinite were invented.

However, a new question will be elicited: whether all infinite numbers are same. It is hard to image and convince other people without using mathematical proof.

Many infinite numbers are related to limit. Expression (1) and expression (2) are two limits and their values are all infinite, If we just compare their result, the two positive infinites, seem to be same. However, expression (2) could be rewritten as

Obviously, when we know . Thus, it is enough to say although .

So, infinite numbers may not have to be same to each other. Some are relatively larger, and some are relatively smaller, even though all of them are denoted by the same notation . In my understanding, infinite numbers,, represent a tendency of increasing and never decrease. Just like , as n increase toward positive direction, the value of the expression will increase. Similarly, in , the will also increase as n increase toward being large. However, the difference from is the value of increase faster than the value of . Although mathematicians cannot express their value by an exact number, they are not same number. The infinite numbers are numbers that is larger than a certain boundary (likes a supremum). If a number inside the boundary, we could express it in a normal way, such as for r and q are natural numbers.

In addition, we could consider the infinite in another way. is the set of natural numbers and is the set of real numbers. is a proper sub-set of , so the cardinal number of is larger than cardinal number of . However, if we consider and respectively. Both and have infinite elements. If we just compare the result, the cardinal number of should be equal to cardinal number of . But it does not. Thus, infinite,, does not means a certain number, it represents some very large number. And not all infinite numbers are same.

In a real-world analogy, if we ask an average person to describe the wealth of Bill Gates and Warren Buffett, he or she may just say billions of dollars instead of $86B and $75.6B. Why? Because whatever $86B or $75.6B, both numbers are too far from an average person, he or she just thinks there is no difference for her or him. Thus, he or she just called $86B and $75.6B as billions of dollars, because $86B and $75.6B are higher than his or her boundary. However, $86B and $75.6B are not same.

Similarly, in mathematical number system, some numbers are too large for us to describe and we denote them as infinite, but they are not same.

Above all, infinite numbers are same as normal numbers. The small difference is that we just denote their value by the symbol . In my understanding, the symbol just means their value are above our expression ability and does not mean their value are same.

Reference

1. Cajori, Florian . A History of Mathematical Notations. 1993.