Negative numbers were popularized in Europe by Michael Stifel's 1544 book Arithmetica Integra, where he called them "numeri absurdi." The concept emerged gradually, as mathematicians found they were useful for solving equations as a kind of "notional convenience," even though they did not think they were real in a Platonic sense.

The same book contains an extraordinary number of nascent mathematical ideas. For example, he talks about "circular numbers," which today we would call modulo arithmetic. He gives a method of multiplication involving a cross that gives rise to our modern "X" symbol for multiplication, but was the first to use algebraic juxtaposition (simply putting two letters next to each other to denote multiplication) and the concept of an "exponent:" `E = mc^2` would look a lot different without Stifel's work!

The most amazing thing in the book, in my opinion, is the extraordinary connection between arithmetic progression and geometric progression he mentions in an almost offhand way[1] (link goes to the Internet Archive version of the book.)

    | -3  | -2  | -1  | 0 | 1 | 2 | 3 | 4  | 5  | 6  |
    |-----|-----|-----|---|---|---|---|----|----|----|
    | 1/8 | 1/4 | 1/2 | 1 | 2 | 4 | 8 | 16 | 32 | 64 |

Here, he is using his new negative number notation and exponent concepts together to illustrate that there is some deep connection between addition and multiplication. As far as we know, this was the first mention of the concept that led Napier to invent the logarithm.

[1]: https://archive.org/details/bub_gb_ywkW9hDd7IIC/page/n539/mo...

I have five friends in debt five dollars each. That’s 5-5-25.

Now I have five people that hate me in debt five dollars each.

The weird thing is since they are negative friends, that is enemies, their loss is kind of like a gain for my team. So -5-5=25