How can we use a base e system to count and represent numbers?

[jamie.j1989]

I once heard a mathematician (Adrian Banner of Princeton I believe), say the most natural way to count would be in units of $e^1$, I've been thinking about this recently and can't think of how this would work, and how it would be more natural. Does anyone have any ideas on where to start?

 

[jbriggs444]

The group of positive integer powers of e under multiplication is clearly isomorphic to the group of positive integers under addition. As bonus, the natural logarithms of all of these "natural numbers" are themselves natural numbers. What could be more natural?

 

[Ssnow]

In biology or chemistry there are a lot of laws that are described by $\ln$-functions, so in this sense counting in unit of $e$ is more "natural" ...

 

[mathman]

What would 2+2=4 look like in such a system?

 

[FactChecker]

I once heard a mathematician (Adrian Banner of Princeton I believe), say the most natural way to count would be in units of $e^1$, I've been thinking about this recently and can't think of how this would work, and how it would be more natural. Does anyone have any ideas on where to start?

Suppose that the number system we already use is really in units of e in some other number system. What property or calculation would change? I don't think anything would change.

 

[DrClaude]

What would 2+2=4 look like in such a system?

2 + 2 = 11.020011200001...

 

[jamie.j1989]

What would 2+2=4 look like in such a system?

This is the type of calculation I don't understand, it seems a bit messy?

I aslo don't know how to write the number 2 in units of e without using the number 2?

 

[jbriggs444]

This is the type of calculation I don't understand, it seems a bit messy?

Dr Claude is writing down the place value expansion of the number 4 using a radix of e. The system is described here: https://en.wikipedia.org/wiki/Non-integer_representation

The algorithm to convert a number to a base e representation is tedious but straightforward.

Edit: This will yield a base e expansion. Some tweaks would be required to produce the canonical base e expansion.

Divide your number by e. If the result is greater than e, keep dividing by e until you have a number between 1 and e. Keep track of how many times you divided. This is the number of places you will need to move the radix point to the right. For small numbers you will multiply by e instead and keep track of the number of places to move the radix point to the left.

In this case, 4 divided by e is 1.41930... That result is between 1 and e. So there will eventually be a one place shift of the radix point.
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Write down the integer part of this number (a digit which will be either 1 or 2).

In this case, we write down "1".
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Subtract the integer part and multiply the remainder by e.

In this case, we multiply .41930... by e giving 1.13978...
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Repeat, writing down the integer part of the number (a digit which will be 0, 1 or 2), subtracting and multiplying the remainder by e.

As per Dr. Claude, the resulting digit string is 1102001...​

When you have as many digits as you please, insert a radix point and shift it the appropriate number of digits. The default position is to the right of the first digit.

In this case, one division by e to start means a one place right shift. 11.02001...
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