# Question #afd88

Dec 9, 2016

First, note that if we assume negative numbers do not exist, as is supposed in the question, then $- 4 \times - 4$ is not considered to be $16$, as $- 4$ does not exist within our system.

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 $x$ is positive if $x > 0$ and that $x$ is negative if $x < 0$. The number $0$ is neither positive nor negative.

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

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

• $- 1 \times x = - x$ for any real $x$
• $- \left(- x\right) = x$ for any real $x$

Using those, together with commutativity and associativity of multiplication and $1$ as the multiplicative identity, we get

$- 4 \times - 4 = \left(4 \times - 1\right) \times \left(4 \times - 1\right)$

$= 4 \times \left(- 1 \times 4\right) \times - 1$

$= 4 \times \left(4 \times - 1\right) \times - 1$

$= \left(4 \times 4\right) \times \left(- 1 \times - 1\right)$

$= \left(4 \times 4\right) \times \left(- \left(- 1\right)\right)$

$= \left(4 \times 4\right) \times 1$

$= 4 \times 4$

That $4 \times 4 = 16$ results from how we construct the integers and define addition and multiplication.

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 $0$. Depending on how they combine, it may be more appropriate to use addition or a different operation altogether. The real numbers may not be useful for modeling the given space at all, depending on the circumstances.

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.