- Visualization of infinitesimal(ε)
In order to understand the image of infinitesimal, can use the area graph, the ε, ∞, ε² and ∞² and other clear display (one-dimensional length ε-∞ can be mapped to the two-dimensional area through the unit width of the area).
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Fig. 1. The area corresponding to ε, ∞, ε² and ∞²
The unit area of a small square 1:
1 ×1, the same area deformation, the height of 1 compressed to ε, the width will be increased to infinity, the area of ε×∞，is also 1;
high 1, wide ∞ strip, 1×∞=∞;
high ∞, wide ∞ square, ∞×∞=∞², actually can not see the infinite boundary, with a dotted line marked “theoretical boundary”;
small strip with high ε and width 1, ε×1=ε;
small square of high ε and wide ε, ε×ε=ε².
After defining infinity and infinitesimal,
take the product of two dimensions,
the product of three dimensions，
the width and density of the number are increased continuously. It’s also an endless cycle.
∞² has to be outside of the real numbers, ε2 has to be between the real numbers and 0. This contradicts the statement that the width of a real number is infinite and corresponds to a point on the number line.
The “Theoretical Gap” of “Initial Numbers”
A ray, extending infinitely forward, starting from point 0, defines a unit length 1 on the ray. By repeated superposition of 1 (continuous addition movement), infinite values can be marked on the ray and the length of the corresponding line segment.
There is no end beyond the ray. But for convenience, people give the “theoretical range”, infinity, for decimal numbers, there are 10 sign variations on each bit, and an infinite number of bits contains 10－∞ of integers.
Infinite is the furthest boundary of the theoretical integer width, it is in the width range, contains all the integers, it is the cage of addition.
If you invert all integers, the corresponding 1/∞ (infinitesimal ε) is the “space” between the “theoretical” “Initial Numbers” and 0. In other words, there is “theoretical” “Initial Numbers” between ε and 0.
However, infinitesimals preserve the property of distance and have two endpoints, which makes it impossible for a ““Initial Numbers” to “correspond” to a point on the poin.
Dedekin segmentation method, which proves that 1 and 0.999…… There is no “real number” between them, asserting that the two numbers correspond to the same point on the point axis. The theoretical flaw lies in this: they are not a point, and there is no “real number” between them because there is a “theoretical gap” between the “Initial Numbers”. In theory, an infinite number of decimal digits is used to express numerical values, but the results tend to slip through the gaps between these decimals.
Decimals and fractions
On the number line, the length of the unit 1 is defined, and it is bisected. Each fraction has a unique point on the point line.
It is known from the division with remainder that if the division is not completed, there must be a remainder (or induction). It is deduced that:
1/1 = 1.0/1 = 0.9…… 0.1 (remainder 0.1),
1 = 1/1 = 0.999…… + 10(-∞), ,that is, 1 = 0.999…… +ε(tentatively agreed infinitesimal ε=10(-∞)).
Thus, the decimal 0.999… And the fraction 1, they correspond to a different point on the point axis, they’re two different numbers.
In other words, is it impossible to express fractions accurately as decimals?
Now, if we look at the decimal number within 1,
1 decimal place, there’s 10 decimal places,
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0
Two decimal places, 102 decimal places,
0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00
It follows that,
So infinity of the decimal, 10∞ of the decimal.
If all of these 10∞ decimal places correspond to a point on the point axis, it’s clear that they’re equidistant from each other, with a theoretical gap, space is 10－∞ between eachother. In other words, between adjacent decimals (e.g., 1 and 0.999… There is always a “theoretical distance”, that is, a point on the point axis is not fully represented by a decimal.
The infinite repeating decimal is only an approximate point that is close to the fraction (e.g., 0.333… The difference from 1/3 is 10－∞/3, which is less than infinitesimal. “Irrational decimals” can be thought of as special repeating decimals that increase endlessly until there are no repeating decimals.
The proof of irrational numbers and fractions, which does not involve decimals, explains that decimals, not only cannot accurately express fractions, but also cannot accurately express irrational numbers.