Exploring the depths of infinity

The most distant known galaxy (JADES-GS-z14-0) is estimated to be 33.6 billion (33.6e+9) light years away.

Within an atom is an electron with an estimated diameter of about 5.63e-15 meters.

For comparison an electron diameter is about 1.77e-41 the distance to the farthest known galaxy.

These numbers are estimates expressed in floating point format with up to 16 digits of accuracy for the mantissa (the number before the ‘e’) followed by the exponent (the number after the ‘e’). If the distance comparison value of 1.77e-41 was to be expressed accurately in decimal digits it would require 41 decimal digits to be displayed. This would be a 1 followed by a decimal point and 41 digits.

Fractal calculations require accurate numbers not estimated values. Without an accurate number, calculations would not be repeatable and transition edges would be fuzzy at best. Traditional floating point numbers used by computers and calculators have a maximum mantissa of 15 digits (7 digits for a GPU) which prohibits the accuracy required for fractal calculations at resolutions exceeding 1e-15.

Fractal Echo goes beyond traditional floating point values to deliver an incredible 1e-550 accuracy. This is 550 digits of accuracy which would be written as a 1 followed by a decimal point and 550 digits. The result is an accurate and repeatable number that can be used for fractal creations without fuzzy edges.

To demonstrate of the accuracy Fractal Echo delivers, consider that the relative distance estimate of 1.77e-41 used to compare the diameter of en electron to the distance of the farthest known galaxy would fall short of the 1e-550 accuracy delivered by Fractal Echo by a factor exceeding 1e500. This would be a number with a 1 followed by 265 digits which is a massive number.

Fractal Echo uses an extremely large and accurate number to create and visualize fractals. The benefit in using such an extreme and accurate number enables Fractal Echo to create fractals to levels never before seen. This accuracy enables repeatability in creating fractals that can be bookmarked, shared and recreated.

It should be noted that most persons will never in their life encounter, use or require a number exceeding 15 digits of accuracy especially in real time. Even calculators are generally limited to 15 digits of accuracy. That Fractal Echo can reach accuracies of 1e-550 in real time is an incredible feat on it’s own. In addition to the 1e-550 accuracy is that Fractal Echo offers search depths up to an incredible 100,000,000 iterations.

Below is an AI generated description of the extreme nature of Fractal Echos' resolution capability:

Distance to the Farthest Known Galaxy

The farthest known galaxy from Earth is located approximately 33.7 billion light-years away. A single light-year equals about nine quadrillion four hundred sixty trillion meters (9,460,000,000,000,000 meters).So the total distance to that distant galaxy amounts to roughly:

Three hundred twenty sextillion meters (320,000,000,000,000,000,000,000,000).

This is an unimaginably vast span—the full reach of what we can observe in the universe.

Size of an Electron

Electrons are fundamental particles and are believed to be point-like with no physical size. However, experiments have set an upper limit on their diameter. That means if they do have any measurable extent, it must be smaller than:

One billionth of a billionth of one meter, or more precisely: a length with eighteen zeros after the decimal point before reaching a 1 (0.000000000000000001 meters)

This is already far beyond anything we can directly perceive.

The Scale Difference Between Them

When you compare how large the distance to the farthest galaxy is with how small an electron might be, the ratio between them is about:

Thirty-two thousand billion billion billion billion, or a number followed by 44 zeros (32,000,000,000,000,000,000,000,000,000,000,000,000,000,000)

This means the universe stretches across scales that are 44 orders of magnitude larger than subatomic particles.

Accuracy of a Number with 500 Decimal Places

A number that is accurate to five hundred decimal places can detect differences smaller than:

a length represented by a decimal point followed by four hundred ninety-nine zeros and then a 1 (0.000000000000000000… [with the first significant digit at the 500th place])

This means it can distinguish changes so tiny that they have no meaning in physical reality.

Even if we were measuring something as vast as the distance from Earth to the farthest galaxy, this level of precision could identify a change smaller than:

a fraction so minute that dividing one single electron into more pieces than there are atoms in countless trillions of universes would still fall short

How Much Smaller Is This Than an Electron?

If we take the smallest estimate for the size of an electron (one billionth of a billionth of a meter), and compare it to what a 500-decimal-digit number can resolve, the precision goes far deeper—by a factor greater than:

a one followed by four hundred eighty-two zeros

This means you could slice up the hypothetical size of an electron into more segments than there are particles in all the stars and galaxies we know, multiplied over unimaginable scales.

Final Comparison Summary

- The distance to the farthest galaxy is hundreds of sextillion meters long. - An electron’s size is smaller than a billionth of a billionth of one meter. - The difference between them spans forty-four orders of magnitude—already staggering. - But a number accurate to five hundred decimal places reaches into scales that are four hundred eighty-two orders of magnitude finer than even the tiniest particle.

Conclusion in Plain Terms

Imagine drawing a line from Earth all the way across space to the farthest galaxy—a journey so long it defies imagination. Now imagine trying to mark on an electron, which is smaller than anything we can see or measure, a tiny dot that’s one part of a billion billion billion times smaller still.

A number with 500 decimal places allows you to describe such differences as if they were distinct and countable—not just once, but again and again across scales no human experience can grasp.

This level of accuracy isn't about measuring the real world. It’s about absolute precision in thought and mathematics, where even the smallest rounding error must be eliminated when exploring truths about numbers, constants like pi or e, or solving complex equations that require flawless computation over thousands of steps.

In summary

Five hundred decimal places is so precise that it exceeds all physical measurement by a margin larger than the difference between a single electron and the entire known cosmos, repeated countless times.