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I know this seems like a dumb question, but hear me out. Whenever you multiply a number by itself, you get a positive number, and whenever you divide a number by itself, you also get a positive number.

For some reason, the operations of multiplication and division seem to have some sort of connection to the positive numbers. It seems like there should be some reason for this. I know how multiplication and division work, but I still feel dissatisfied. It seems like there is something special about positive numbers in relation to these operations. Maybe this is more of a philosophical question than anything.

If this still sounds dumb I completely understand. Thanks for any thoughts.

Tdonut
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  • Positive numbers are closed under multiplication, while negative numbers are not. – Crostul Mar 21 '19 at 16:18
  • Building up the number system from scratch starts with $1$ and building up positve integers. Negatives come later. – herb steinberg Mar 21 '19 at 16:20
  • It is a worthwhile observation, even if it has a philosophical nature. Leopold Kronecker famously said, "God made the integers, all else is the work of man." One can go beyond this to say that the counting numbers (positive integers) are even more fundamental (since one needs to count in order to verify arithmetic facts and check that formulas are in agreement).. – hardmath Mar 21 '19 at 16:23
  • The fact that $(-1,1)$ forms a group under multiplication called $\mathbb{Z}_2$ is probably what you are noticing. That may not mean anything to you if you are not familiar with group theory but the basic idea is that “all the positive numbers” have this “positiveness” propert that makes them act as if they are the “original identity” element of the set of reals, whereas the negative numbers, carry more unusual behavior. – Sidharth Ghoshal Mar 21 '19 at 16:27
  • @frogeyedpeas But why do positive numbers have this "original identity" property? Is it just because we defined them first? – Tdonut Mar 21 '19 at 16:44
  • No it’s deeper than that, it basically has to do with the fact that $a(b+c) = ab + ac$, $(-1)(0) = (-1)(1-1) = (-1)(1) + (-1)(-1) = 0$ If you insist that multiplication by positive 1 act the normal way: “1 times anything stays the same” then you are forced to accept that $-1\times -1 = 1$. We could define our negative numbers first, but if we define multiplication by positive 1 as $1\times m = m$ then this will happen. In fact whatever symbol $S$ has the property that $S \times m = m$ for all $m$, inherits the “fundamental” property. – Sidharth Ghoshal Mar 21 '19 at 16:49
  • Whichever half of the number line centered at 0, that $S$ resides on then will become the “fundamental” side. – Sidharth Ghoshal Mar 21 '19 at 16:50

2 Answers2

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There is no strictly mathematical answer to this question. It has to do with the long history of mathematics. For thousands of years, negative numbers were not used in mathematics and hence did not exist. All numbers were positive and the positive versus negative number concept did not arise until a few centuries ago. For example, from Wiktionary article positive

Of number, greater than zero. [from the 18th c.]

More information from Latin Stack Exchange question 1438 "Negativus and Positivus".

The use of “positive” and “negative” as opposites is surprisingly modern. It seems to have originated (or at least been popularised) in the context of modern (i.e., 17th-century) mathematics with the distinction of positive and negative numbers. “Positive” numbers are numbers that you can posit, put on the table, observe as objective reality. “Negative” numbers negate the corresponding positive numbers (positive 1 plus negative 1 makes zero). A century later we get also "positive" and "negative" electricity.

So, it seems safe to state that "positive" numbers are more fundamental than "negative" numbers. If the word "real" had not already been used in the context of numbers, we might have called numbers greater than zero "real" numbers or "actual" numbers, and to borrow a phrase from John Napier, we might have called numbers less than zero "artificial" numbers. Of course, all this is just giving ideas conventional names. The "imaginary" numbers are slightly more imaginary than negative numbers. The "complex" numbers are slightly more complicated than real numbers, but all this is just a naming game and changes over time.

To state the obvious, when we refer to or write the number seven we write just $7$. To refer to the additive inverse we need to use "negative" seven and we need to write $-7$ which is strong evidence that the positive numbers are more fundamental than the so called "negative" numbers since we can only refer to them in terms of positive numbers.

Somos
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The appealing properties of the positive numbers (e.g., being closed under multiplication) are likely an artifact of our cognitive structure. We are good at perceiving and processing what is present (we model this presence with positive quantities), because such presence can be tested by experiment directly; i.e., without altering the state of the system.

In that same context, what do negative numbers model? Sometimes, they model "potential disappearance" of a substance or reduction of a quantity--such as resistance to a force that is doing work. But, direct experimental tests no longer suffice: to find out the stiffness (mechanical resistance) of a spring, one has to try compressing the spring--i.e., to alter the state of the system and observe the response. (Classical mechanics has the concept of a virtual displacement, a worthwhile one.)

For that same reason, negative numbers are irreplaceable: without them, we would not have such complete physical theories as we do (though none is absolutely complete). https://www.quora.com/Why-does-a-negative-number-multiplied-with-another-negative-number-give-a-positive-number-as-a-product/answers/128775469

avs
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