『無限関係』(30歳の時に書いたもの)

The Logic of the “Remainder”

In 1999, when I was 30 years old, a college student posted on my website’s message board. He was deeply puzzled by what he had learned in math class: the notion that 1 = 0.999… This paradox stems from the fact that while multiplying (1 ÷ 9) by 9 should logically result in 1, a calculator gives 0.999…

Being “occasionally a genius,” I pondered why this fallacy occurs. To put it simply: mathematics tends to confuse “Count” (the number of steps/items) with “Magnitude” (the actual quantity).

The Conventional Explanation:
“If you calculate 1 ÷ 9 × 9 on a calculator, you get 0.999… (limited digits). If a calculator with infinite digits existed, you would get 0.999… (infinity). Since the left side is clearly 1, then 1 = 0.999…”

My Rebuttal:
Consider the standard algebraic proof:
Let a = 0.999…
Then 10a = 9.999…
10a – a = 9a, which means 9a = 9, so a = 1.
However, this contradicts the starting premise that a = 0.999…

Why does this contradiction occur? It is because the setting of “infinity” allows for a certain kind of “cheating.”
In a finite set, 0.999 × 10 = 9.99. Every time you multiply by 10, the number of decimal places decreases by one. But with the “Infinity” setting, no matter how many times you multiply by 10, the number of digits after the decimal point never changes.

It’s arbitrary, isn’t it?
If we calculate 10a – a as 9.99 – 0.999, we get 9a = 8.991. Thus, 9 × 0.999 = 8.991, and there is no contradiction at all.
In the “Infinity” setting, the number of decimal places is constantly increasing (changing). Therefore, multiplying 0.999… (infinity) by 10 does not shift the digits in the traditional sense; it relies on the “cheat” that the supply of 9s is inexhaustible, masking the structural change of the number.

II. The Logic of the “Remainder”
I also considered why 10 ÷ 3 = 3.333…, yet 3.333… × 3 = 9.999…

When you divide 10 by 3, the answer is 3 with a remainder of 1.
3 × 3 = 9; adding the remainder 1 gives 10.
If you divide that remainder 1 by 3, you get 0.3 with a remainder of 0.1.
3.3 × 3 = 9.9; adding the remainder 0.1 gives 10.
Continuing this, 3.33 × 3 = 9.99; adding the remainder 0.01 gives 10.

In other words, saying “10 ÷ 3 = 3.333… (infinity)” is merely another way of saying “3.333… plus a remainder of 0.000…1”.
If you multiply this by 3, you get 9.999… + 0.000…1, which equals 10. There is no contradiction.
The “paradox” is nothing more than an illusion caused by forgetting that the “…” represents the ongoing process of dividing the remainder.

The Paradox and the Scale

The paradox of “Achilles and the Tortoise,” proposed by the ancient Greek philosopher Zeno of Elea, is as follows: A fast Achilles and a slow tortoise race. The tortoise starts slightly ahead. However, by the time Achilles reaches the point where the tortoise started, the tortoise has moved a little further. When Achilles reaches that next point, the tortoise has again moved forward. Mathematically, it seems Achilles can never overtake the tortoise.

In my view, Zeno failed because he forgot that while one can subdivide space to increase the “Count” (the number of segments) infinitely, the actual “Magnitude” (the total distance) remains unchanged.

Consider this: If the tortoise starts 10 units ahead, and Achilles moves at 1 unit per second while the tortoise moves at 0.1, Achilles will reach the tortoise’s starting point in 10 seconds. At that moment, the tortoise is 1 unit ahead. Normally, when Achilles has moved 20 units from the start, the tortoise would be 8 units behind him. The reason this was considered a paradox is that Zeno assumed we must redefine the distance into smaller and smaller segments every time Achilles reaches the tortoise’s previous position.

Segmentation can be infinite, but Zeno’s failure was confusing the “Count of segments” with the “Actual Magnitude of distance.” This paradox is an illusion stemming from this confusion.

IV. The Analogy of the Apple and the Watermelon
To illustrate the difference between “Count” and “Magnitude”: “Saying that an apple divided into a hundred pieces has more quantity than a watermelon divided into ten pieces is fundamentally flawed.” In terms of “Count,” the apple has 100 pieces and the watermelon only 10, but in “Magnitude,” the watermelon is always larger.

The core of Zeno’s paradox is the illusion that if you make objects infinitely small, their “Infinitesimal Magnitudes” become equal. However, if an apple’s magnitude is 1/10th that of a watermelon, any segment of the apple must always be compared to a segment of the watermelon that is ten times larger. This ratio of magnitude never changes, even at infinity.

V. Defining the “Infinitesimal” as a “State”
The infinitesimal is not a fixed final number; it is merely a “state” in which no matter how much you divide, further division is always possible.

Apple: 1st: 1/2 → 2nd: 1/4 → 3rd: 1/8 … → ∞
Watermelon: 1st: 1/2 → 2nd: 1/4 → 3rd: 1/8 … → ∞

The illusion that the infinitesimal magnitudes of an apple and a watermelon are the same is like comparing 1/4th of an apple to 1/16th of a watermelon at arbitrary points in the process. But since the infinitesimal is a process of eternal division, we must always compare their volumes at the same number of divisions.

The moment you assume the infinitesimal is a fixed “destination,” it is no longer infinitesimal—it becomes finite. Consider an apple and the Earth: both can create two distinct infinities through division. If you divide an apple infinitely, the “Count” is infinite, but the “Total Magnitude” never exceeds the size of the apple. If you divide the Earth infinitely, the “Count” is also infinite, but the “Total Magnitude” never exceeds the size of the Earth. To claim that the size of an infinitesimal piece of an apple and an infinitesimal piece of the Earth become “equal” is, once again, a mathematical “cheat.”