An alternative mathematical system

Where integers are positive or negative by factor

Classical math

In classical math, we group and name numbers according to their characteristics as illustrated in the diagram below.

  • ℕ={a: (a>0 ⋀ ∄b/c: [a=b/c ⋀ b≠c ⋀ c≠1 ⋀ c≠0])}
  • ℤ={a: (∄b/c: [a=b/c ⋀ b≠c ⋀ c≠1 ⋀ c≠0])}
  • ℚ={a: (∃b/c: [a=b/c ⋀ b∈ℤ ⋀ c∈ℤ ⋀ c≠o])}
  • ℝ={a:a²≥0} 
  • ⅈ={a:a²<0}
  • ℂ={a: (∃b⋀c: [ b∈ℝ ⋀ c∈ⅈ ⋀ a=b+c])}
Or in English:
  • Natural numbers: ℕ, are all whole (not fractioned) positive numbers excluding zero.
  • Integer numbers: ℤ, are all whole (not fractioned) numbers including positive, negative and zero.
  • Rational numbers: ℚ, are all numbers that can be represented by a fraction
  • Real numbers: ℝ, are all numbers which multiplied by themselves give a positive product.
  • Imaginary numbers: ⅈ, are all numbers which multiplied by themselves, give a negative product.
  • Complex numbers: ℂ, are all numbers consisting out of the sum of both a Real as well as an Imaginary term.

A different axiom

Mathematics are often seen as a universal truth. I disagree, math isn't something intrinsic to nature. Mathematics is a language. The language of science. And just like Rene Magritte pointed out with "Ceci n'est pas une pipe."  I could argue that the symbol; 1 isn't really a number. Instead It's a symbol representing the concept. And just like how in a language, we have agreed rules for conjugation and syntax and grammar, so too in math do we find agreements on how to deal with these numbers. We call these axioms. Axioms are presumptions on which a mathematical concept is build. But who is to say which axiom is right or wrong? Well looking at math as a language, there is no such thing as right or wrong. Just as in language there no such thing as a wrong grammar, just as long as everybody agrees on the same set of rules for practical reasons, what matters most is how pragmatic and useful the theory is. 

The Image below is a numerical scale of integers under classical math. The scale is linear. The second image gives a slightly different representation. In classical math the absolute value of a number is considered equal to the positive value: |1| = (+1)
This characteristic of Integers is axiomatic. It is as such only because we defined it to be so. The alternative that I would like to present to you today, is a theory where Natural numbers are not a section of Integer numbers. In such a theory, the absolute value of a number would not be equal to the positive value of said number, nor to the negative value for that matter. |1| ≠ (+1); |1| ≠ (-1)
The logic behind it being that the absolute value is raw, deficient of either positive or negative values. 

However, if negativity or positivity is a value by factor, then we should expect exponential values to. For if two numbers, each with their value-factor are multiplied with each other, it would logically follow that their respective factors are multiplied as well, thus resulting in exponential value-factors.

These representations aren't very good though, for reasons I will discuss later on. You can find a more proper representation here

Alternative notation

I think I don't need to tell you that this whole method of notation where you always place a plus or minus sign to express the value of the number, and even put exponents on them can be very confusing. So I suggest an alternative notation like this:

1 = |1|
(+1) = p1
(-1) = m1
(+n1) = pn1
(-n1) = mn1

Diagram

Our diagram of types of numbers would then be different as well.

 

 

Defenitions

Classical math

Definitions of sets in symbols:

  • ℕ={a: (a>0 ⋀ ∄b/c: [a=b/c ⋀ b≠c ⋀ c≠1 ⋀ c≠0])}
  • ℤ={a: (∄b/c: [a=b/c ⋀ b≠c ⋀ c≠1 ⋀ c≠0])}
  • ℚ={a: (∃b/c: [a=b/c ⋀ b∈ℤ ⋀ c∈ℤ ⋀ c≠o])}
  • ℝ={a:(a²≥0 ⋀ ∄b,c:[b²<0  ⋀ a=b+c)]}
  • ⅈ={a:a²<0}
  • ℂ={a: (∃b⋀c: [ b∈ℝ ⋀ c∈ⅈ ⋀ a=b+c])}

Definitions of sets in language:

  • Natural numbers: ℕ, are all whole (not fractioned) positive numbers excluding zero.
  • Integer numbers: ℤ, are all whole (not fractioned) numbers including positive, negative and zero.
  • Rational numbers: ℚ, are all numbers that can be represented by a fraction
  • Real numbers: ℝ, are all numbers which multiplied by themselves give a positive product.
  • Imaginary numbers: ⅈ, are all numbers which multiplied by themselves, give a negative product.
  • Complex numbers: ℂ, are all numbers consisting out of the sum of both a Real as well as an Imaginary term.

 

Alternative math

Definitions of sets in symbols:

  • ℕ={a: (a≥0 ⋀ a=|a| ⋀ ∄b/c: [a=b/c ⋀ b≠c ⋀ c≠1] ⋀ c≠0 )}
  • V=(a: (a=pxmy ⋀ x∈ℕ ⋀ y∈ℕ)
  • ℤ={o ⋀ a: (∄b/c: [a=b/c ⋀ b≠c ⋀ c≠1 ⋀ c≠0])}
  • ℚ={a: (∃b/c: [a=b/c ⋀ b∈ℤ ⋀ c∈ℤ ⋀ c≠o])}
  • ℝ={a:∃a}

Definitions of sets in language:

  • Natural numbers: ℕ, are all whole (not fractioned) numbers including zero. These numbers are neither positive nor negative, they are neutral in value.
  • Values: V, are all factors of either p or m.
  • Integer numbers: ℤ, are all whole (not fractioned) numbers including positive, negative, neutral numbers, values and zero.
  • Rational numbers: ℚ, are all numbers that can be represented by a fraction
  • Real numbers: ℝ, are all existing numbers.

Note that we no longer have or need imaginary nor complex numbers.