If you, or someone you know, is having difficulty “getting” negative numbers, be happy. They’re a very strange concept and an understanding, as distinct from a rote application of rules, is going to be hard to come by because they’re one of the earliest mathematical leaps from the concrete to the abstract.
Negative numbers are a great concept, but that’s what they are, a concept. Negative quantities do not exist in our physical experience. I can point to two apples or two dollars, but I can’t point to minus two apples or minus two dollars.
Math can be a source of wonder, but too often we’re in too much of a rush and it’s like that with negative numbers. We present them to students as if they’re obvious and difficulty with them in the sign of a shallow mind, when actually the opposite is true. If they don’t seem weird, you’re probably not really thinking about them.
Here’s what some great mathematicians thought about them:
“Pascal regarded the subtraction of 4 from 0 as utter nonsense.” (Kline, Morris, Mathematical Thought form Ancient to Modern Times, Oxford University Press, New York, 1972, p 252)
Antoine Arnauld, a theologian and mathematician, invoked ratios and “questioned that -1:1 = 1:-1 because, he said, -1 is less than +1; hence, How could a smaller be to a greater as a greater is to a smaller?” (Kline, Morris, Mathematical Thought form Ancient to Modern Times, Oxford University Press, New York, 1972, p 252)
The person generally credited with the idea of the number line, John Wallis, “rejected as absurd the … idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity — a viewpoint shared by Swiss mathematician Leonhard Euler…” (http://en.wikipedia.org/wiki/John_Wallis)
“As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions…” (http://en.wikipedia.org/wiki/Number)
“In 1759, Francis Maseres, an English mathematician, wrote that negative numbers ‘darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple’. Because of their dark and mysterious nature, Maseres concluded that negative numbers did not exist…” (http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml)
“In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers as exponents and referred to them as ‘absurd numbers’” http://en.wikipedia.org/wiki/Negative_and_non-negative_numbers
So, if there’s something in you that thinks negative numbers are strange, you’re in great company!