# The random function's universality of multiplicative identity

#### Loren Booda

The multiplicative identity of the random function r(x) is any other function f(x) except where f(x)=0.

#### Loren Booda

In other words, a field in x of random values r(x) each multiplied by a corresponding value for a (non-)random mapping f(x), always retains a random mapping r(x), except where f(x)=0.

Can you imagine this situation?

Also, may this be a great way (in theory) to obtain 0's of f(x)?

#### bogdan

What if f(x)=k/r(x) ? Huh ?
Maybe you wanted to say f(r(x))...or I just don't understand...

#### plus

Originally posted by Loren Booda
In other words, a field in x of random values r(x) each multiplied by a corresponding value for a (non-)random mapping f(x), always retains a random mapping r(x), except where f(x)=0.

Can you imagine this situation?

Also, may this be a great way (in theory) to obtain 0's of f(x)?

Does it always maintain the properties of the random function r(x). For example you might have specified that if the domain is the real numbers, the function r(x) returns values in the interval [0, 1], multiplying this by 2 does not preserve this property.

Do you need to make your definition of randomness more tight, or am I missing something.

As to Bogdan's point, f(x)=k/r(x) will be a random function.......and thus not allowed by the Loren Booda's initial statement.

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#### bogdan

In a set X equipped with a binary operation called a product, the multiplicative identity is an element e such that e * x = x * e = x, for all x in X...got it...

So e = f(x)...hmmm... r(x)*f(x)=r(x)...hmmm...still don't get it...

#### plus

Originally posted by bogdan
What if f(x)=k/r(x) ? Huh ?
Maybe you wanted to say f(r(x))...or I just don't understand...
Certainly he does not mean f(r(x)) as you could just set f(x) = 1 for all x.

What he appears to be saying that there if r contains no information, i.e. it returns a value on the range with equal probability, then multiplying it by any conventional (not including random) and not equal to zero at x ,function returns a result which contains no information as defined above.

Perhaps his statement is more general?

#### bogdan

I don't know for sure...maybe he'll explain better...

#### Loren Booda

plus
Does it always maintain the properties of the random function r(x). For example you
might have specified that if the domain is the real numbers, the function r(x)
returns values in the interval [0, 1], multiplying this by 2 does not preserve this
property.

Do you need to make your definition of randomness more tight, or am I missing
something.
What is returned is the property of randomness, apparently not the function itself as first defined. Indeed I need to "loosen" my definition of random r(x). Perhaps I should require its returned interval to be [-[oo],[oo]]. Thanks for your feedback.

I hope this helps your understanding of the problem also, bogdan?

#### bogdan

Yeap...if (non-)random means not random... ...those brackets...

#### Loren Booda

bogdan-

A random number times a random number, or times a non-random number, is a random number.

#### bogdan

But why ?
if f(x)=1/r(x), then both are random...
Or am I just stupid ? Or worse ?

#### plus

What is the probability distribution of your random function on the real line going to look like?

#### Loren Booda

Actually, plus, I don't know if r(x) can be "seen," as its average magnitude may be infinite. Can you think of a finite random function that returns the property of randomness to similar bounds (or better yet, as I initiated, the random function itself?) Perhaps I should require normalization of r(x) and r(x)f(x)?

bogdan, I appreciate your interest too. Think of a random function with magnitude r(x) returning values from 0 to [oo], multiply each by a nonzero arbitrary number f(x), and one returns the random function along x. The cardinality of "randomness" apparently is greater than that of the real number line.

#### plus

The only way I could think of to get this in a finite range would be to have modular multiplication. so if it was mod 10,

8.18* 10 = 0.9 (mod 10)

#### Loren Booda

Beautiful idea. The credit is yours. Do your see any application for the infinite range r(x) (like finding zeroes of a function)?

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