## Tuesday, March 5, 2024

### No, e is not a number

Euler himself used the word "quantitas" and not "numerus" ...

e (Euler's Number) - Numberphile
Numberphile | 19 Dec. 2016

May I protest against the title?

Euler never called "e" a number. He only knew arithmetic numbers, as far as I can tell, but he knew both arithmetic and geometric quantities.

So, you mean, I presume, Euler's quantity, it's a geometric one, and so not a number.

Before 0:24.

Now, "e" is not a number. It is irrational, which implies it is not a number.

dhkatz
@dhkatz_
Irrational numbers are numbers. Irrational means a number than cannot be represented in the form a / b where a and b are integers.

I can't think of a name
@water_is_wet
even a 6 grader knows irrational numbers are NUMBERS

so share your wisdom and knowledge somewhere else

Hans Georg Lundahl
@hglundahl
@water_is_wet "even a 6 grader knows irrational numbers are NUMBERS"

6 graders have probably heard it — along with lots of other BS.

You can have one apple or two apples or three or four. You cannot have π or e apples.

ISaveNewspapers
@isavenewspapers8890
@hglundahl I imagine that you are not much for having your mind changed, considering what little that half a decade has done in that regard. But regardless, your view of mathematics has inspired much curiosity in me. I would like to learn more.

It seems we can agree that the natural numbers (1, 2, 3, and so on) are numbers—possibly to the exclusion of zero, depending on your acceptance of the concept of having "zero apples".

You also seem to have implied an acceptance of numbers such as 1/2 and 3/5. These are not natural numbers, of course; however, by definition, they are rational numbers.

If I say, "I have 1 and 3/5 apples," does that mean anything to you? I think the logical interpretation here is that I had two apples, and I sliced off two-fifths of one of my apples with a knife. (In the real world, I could not actually slice off exactly that amount, but we are assuming a mathematically ideal situation.)

What about distances? If I say, "I have traveled 1 and 3/5 kilometers away from home," does that mean anything to you? I imagine it should, since by definition, that means I have traveled 1600 meters. Do you accept the usage of numbers to measure distances?

Now here's a situation: let's say I'm standing on one vertex of a square with side lengths of 1 kilometer. I walk straight to the opposite vertex of the square. How far have I traveled? (Again, we are assuming a mathematically ideal situation.) If you accept that I really have traveled some distance in this situation, then how do you measure that distance? Does that measurement involve a number?

Hans Georg Lundahl
@isavenewspapers8890 "You also seem to have implied an acceptance of numbers such as 1/2 and 3/5."

Not as numbers. As proportions.

e is unlike them irrational, i e cannot be expressed like a number to number proportion, but like them a proportion.

@isavenewspapers8890 "Do you accept the usage of numbers to measure distances?"

In distances they don't function as numbers.

3 km doesn't mean "three distance items of a km each" but "one distance item 3:1 of a km".

3 is a number, but 3:1 is a proportion.

@isavenewspapers8890 "I think the logical interpretation here is that I had two apples, and I sliced off two-fifths of one of my apples with a knife."

There are more interpretations than that one, but OK. Thing is, once you cut, you are no longer dealing with "apples" but apple parts (and the apple part has a certain, wait for it, proportion to the size of the previous apple).

ISaveNewspapers
@hglundahl This has been very informative. I see now that you simply use the word "number" in a different way than everyone else. But that's only a matter of terminology, after all; a rose by any other name would smell as sweet.

I'm still curious, though: why do you want children to be taught about the way that you specifically use the word "number", instead of how everyone else does? It seems that the latter is much more important.

Hans Georg Lundahl
@isavenewspapers8890 "in a different way than everyone else."

Than every Mathematician else. AND since about the time of Euler, or perhaps rather a bit later.

Mathematicians before that distinguished "quantitas discreta" or "numerus" from "quantitas continua" or "magnitudo" (which while it has absolute values is only measured by proportion to standard values).

The payoff is in thinking about infinity, or especially rejecting the number line as a parallel to time changes of infinite duration.

Discrete quantity begins with "one" and adds up "in infinitum" meaning "towards infinity" and continuous quantity begins with "whole" and divides down "in infinitum" ... meaning both infinities are unrealised, they are only ever finitely realised.

Pays off in the proofs for God, and some other scholastic contexts, like in proving God's infinity is not a material one.

0:38 I think e actually is defined by geometry.

Take a logarithmic graph, and somewhere in it, you will find e as a proportion.

dhkatz
That isn't geometry.

Hans Georg Lundahl
@dhkatz_ It is.

6:20 "if the fraction goes on forever it's an irrational" quantity, noted.

How do calculate values of fractions the go on forever?

7:15 Here you have the graph!

I e, "e" is related to geometry!