# Question #afd88

##### 1 Answer

First, note that if we assume negative numbers do not exist, as is supposed in the question, then

Negative numbers are an abstraction, and their existence stems from the *axioms* (properties accepted without proof) which dictate the rules elementary arithmetic.

Some of the axioms are called *order axioms*, and dictate how the *less than* and *greater than* relations operate. Once those are in place, we say that *positive* if *negative* if

The full set of axioms used in elementary arithmetic can be found here.

Using only axioms, we can prove some elementary results, including

#-1 xx x = -x# for any real#x# #-(-x) = x# for any real#x#

Using those, together with commutativity and associativity of multiplication and

#=4 xx (-1 xx 4) xx -1#

#=4 xx (4 xx -1) xx -1#

#= (4xx4)xx(-1xx-1)#

#=(4xx4)xx(-(-1))#

#=(4xx4)xx1#

#=4xx4#

That

The incongruity of having two empty spaces combine to create a visible space comes from using negative numbers to model empty space, and using multiplication to model combining them. Depending on what is meant by "empty space," it may be more appropriate to model empty space using

This question highlights an important difference between math and physics.

In physics, we change our models to match reality. If our models say that a ball should float up into the air, but it falls down, then we must change the model.

In math, we start from a system, and work within it. The physical world has no bearing on whether a mathematical statement is true or not. It is no coincidence that the real numbers are useful for modeling reality, but a physical observation or hypothesis cannot change the properties of an abstract mathematical system.