4
Jan 16

A Radical '123' proposal for Real Numbers

Mathematics has been defined as an understanding of patterns and relationships involving Real Numbers.

So how accurate do we need Real Numbers to be?

In Universal Arithmetick, Isaac Newton said: "By Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity."

So how does the longest measure in the universe understood compare to the shortest thing? That could be acceptable as a practical range, from which a real world 'universal' set of numbers can be 'cut out' of the set of reals numbers.

Based on the speed of light since the big bang, and the subsequent expansion of the universe, the upper distance bound is 8.8×10²⁶ meters. 880,000,000,000,000,000,000,000,000 m

The lower bound may reasonably be 'Planck length' (theoretically) calculated as 1.616×10⁻³⁵ meters. 0.000000000000000000000000000000000001616 m

So the largest abstracted ratio with a common unit of meter is 8.8×10²⁶ / 1.616×10⁻³⁵ which is 5.4455445544554455445544554455445544554455445544554455445544554... × 10⁶¹

The smallest abstracted ratio is 1.616×10⁻³⁵ / 8.8×10²⁶ which is
1.8363636363636363636363636363636363636363636363636363636363636... × 10⁻⁶²

To me, these biggest (non infinite) and smallest (non-zero) numbers are enough, given 'negative length' in only a vector pointing in the opposite direction.

To put this range into perspective, we only need know pi to 32 places to calculate the size of the known universe within the accuracy of one proton.

So why not 'cut' numbers after these 123 orders of base 10 magnitude? It seems ironically simple to say '123' in the context of how big or small we need real numbers to be.

Changing the unit from meters to millimeters, or even kilometers, does not of course, change the meaning behind such numbers. Any real number outside the above proposed range, could reasonably be written about in a purely theoretical or religious context, as was popular in Indian culture 3000 years ago.

In the meantime, the best supercomputers in the world, do our real-world calculations with rational numbers, as a result of having a finite number of decimal places for calculation.

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