Finally, negative numbers and imaginary numbers

http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

Quote from the  above link:Negatives were considered absurd, something that “darkened the very whole doctrines of the equations” (Francis Maseres, 1759). Yet today, it’d be absurd to think negatives aren’t logical or useful. Try asking your teacher whether negatives corrupt the very foundations of math.

What happened? We invented a theoretical number that had useful properties. Negatives aren’t something we can touch or hold, but they describe certain relationships well (like debt). It was a useful fiction.

Rather than saying “I owe you 30” and reading words to see if I’m up or down, I can write “-30” and know it means I’m in the hole. If I earn money and pay my debts (-30 + 100 = 70), I can record the transaction easily. I have +70 afterwards, which means I’m in the clear.

The positive and negative signs automatically keep track of the direction — you don’t need a sentence to describe the impact of each transaction. Math became easier, more elegant. It didn’t matter if negatives were “tangible” — they had useful properties, and we used them until they became everyday items. Today you’d call someone obscene names if they didn’t “get” negatives.