January 9, 2017
Hiroshi Michiwaki
Infinity, the Infinity Point, and Division by ZeroGiven a real number 𝑥,

based on the fundamental theorem of division by zero, 𝑧/0=0, upon reaching the absolute point where𝑥 = 0, rather than getting arbitrarily close to 0 as in 𝑥 → ±0,This is equivalent to the infinity point 1/0 corresponding to 0. This also shows thatand upon reaching the absolute point where 𝑥 = [±∞], rather than getting arbitrarily close to ±∞ as in 𝑥= ±∞,It should be noted that here, [±∞] denotes the infinity point; that is, the absolute value increases
without limit, with brackets to distinguish it from ±∞, which represents infinity.
Considering the above,
Theorem 1 For 𝑎>1, given

the following is true:

Theorem 2 For 𝑎>1, given

the following is true:where 𝑛 = [–∞] is equivalent to Theorem 1. QEDThis immediately implies the following.
Corollary 1 In the limit as 𝑛 → +∞, the sum contains a term without a limit but does not reach theinfinity point. As such,

However, if the sum is up to the infinity point 𝑛 = [+∞], then by Theorem 2,

which shows that it equals exactly 1.
Therefore, the following is true:

Corollary 2 Given an infinitesimally small positive number ε greater than 0 and smaller than any
positive constant δ (0 < ε < δ) no matter how small,

However, if the sum is up to the infinity point 𝑛 = [+∞], then the following is true:Pot of Infinitely Many Numbers
How can one express a number 𝑠 that is smaller than 1 yet greater than any constant smaller than 1?
Expressing this with a single number requires an infinite decimal because the following would be true
if 𝑠 were a finite decimal:

which would clearly imply that𝑠 is an infinite decimal. (This can also be abbreviated as 𝑠 = 0.9̇.)However, according to modern mathematics,

Is this actually correct? The representative example of the basis for this derivation is to multiply bothsides of (2) by 10 and then subtract both sides of (2) from the resulting equation, namely

Dividing both sides by 9 would then lead to the result 𝑠=1.
However, is this operation strictly mathematically correct? 𝑠 is supposed to be smaller than 1 yetgreater than any constant smaller than 1, which is a contradiction. What is the cause of such acontradiction? It may lie in the fact that this operation treats the infinite series of a limit as if it were afinite series. Can it be stated that the infinitely small terms that continue indefinitely were treated insome clever manner so that the difference between them was 0 after the subtraction?
Strictly speaking, 𝑠 is an infinite series, expressed thus:

As this is a limit, the partial sum becomes arbitrarily close to 1 but never reaches 1. Generally, ininfinite series changing the order of terms or performing operations term by term is not permitted.
That is, is it true that multiplications performed term by term, as in (4), or decimal place by decimal
place yield the expected result in the limit?
To answer this question, the problem should be considered from a different perspective. Here is athought experiment involving a mysterious pot that represents a particular number. If you reach into
the pot, you can pull out the digit at each decimal place of this number indefinitely. Even if you donot, the pot still contains the source of each digit. The digit at the first decimal place is 9, and so is thesecond, the third, and so on, so that you can keep on taking out 9s at all successive decimal places.
The following is a visualization of this pot:

Then, would it be possible to perform an operation such as that in (4), that is, multiplying both sides
by 10? Without going through every number in the pot, this may be carried out as follows:.

Dividing both sides by 9 naturally yields

Here, the subscript number 5 on the pot represents the fact that the digits corresponding to fivedecimal places were taken out of the pot. Below, a subscript 𝑛 on a pot will denote that there are digits
corresponding to 𝑛 places in the pot.
The result of this calculation suggests that extending the multiplication of each decimal place toinfinity is probably not valid.
This implies that based on this thought experiment with the mysterious pot, multiplying both sides by
10 simply yields 10S on the left-hand side, and a 5-decimal-place number and ten mysterious pots on
the right-hand side. Furthermore, the difference between a pot with 𝑛+1 digits and 𝑛 digits isexpressed asThat is, although letting 𝑛 tend to the limit will cause the difference to get arbitrarily close to 0, thisdoes not imply that the pot will cease to exist at that point; rather, for all intents and purposes, thisdifference will never be 0.