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My question is bit naive, Related to the above question is: Add, Subtract, Multiply and Divide are fundamental "operations", then why we try to provide a sign to numbers as +ve or -ve, to say why are these two (+,-)symbols overloaded (they represent the type of number and operation also)?

Going further the concept of negative number itself gives rise to absurdities like the square root of a negative number and then we try to invent one more type of a number i.e. Complex Numbers.

Fundamental question is will the world be any different if there is no negative number concept. Or Where do we encounter complex numbers in real world?

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  • $\begingroup$ Can you explain why you think negative numbers are not needed? I can understand your question for complex numbers, but negative numbers seem to have applications wherever you look. $\endgroup$ – Naman Kumar Feb 8 at 11:52
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    $\begingroup$ For complex numbers in "real world", see quora - related questions $\endgroup$ – Mauro ALLEGRANZA Feb 8 at 12:01
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    $\begingroup$ And see Negative numbers : bank account with negative quantities is a real real example. $\endgroup$ – Mauro ALLEGRANZA Feb 8 at 12:02
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    $\begingroup$ See dam.brown.edu/people/mumford/beyond/papers/… $\endgroup$ – Barry Cipra Feb 8 at 12:11
  • $\begingroup$ Complex numbers are often a great tool by simplification of (complex) problems. $\endgroup$ – drhab Feb 8 at 12:22
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Like any mathematical tool, it's a tool. It's there, in your belt, to help you solve problems. Nobody "needs" a hammer (not when you could swing a solid rock), but nothing is going to get that nail into wood any faster, safer, or more efficiently.

If you are, say, modelling the acceleration of a train along a rail, you would reach for negative (and positive) numbers. When the train goes faster (or at least, faster in a particular direction), we assign the acceleration to be positive number, but when it slows down, we assign it a negative number.

Could we get around using negative numbers here? Absolutely! We can make acceleration and deceleration fundamentally different processes. Either acceleration or deceleration is $0$, and when we wish to "add" $r$ acceleration we must first check if this added acceleration is less than the deceleration or not. If it is, then subtract it from the deceleration. If it's not then we subtract the deceleration from $r$, make this the acceleration, and make the deceleration $0$.

For example, our train might have deceleration $2ms^{-2}$ due to a hill, but we need it to have $3ms^{-2}$ acceleration instead. Our front car has the ability to add up to $10$ acceleration and our rear car can add up to $15$ deceleration. How much acceleration and deceleration should we add? Bear in mind, there's no negative numbers to help you here; you'd have to work it on a case by case basis. If you add less than $2$ acceleration, this will be one operation, but adding more than $2$ acceleration is a different operation.

(Imagine teaching school kids this! They would hate it even more than they currently do.)

So, we can get around using negative numbers here, but why would we? Clearly having a consistent number system in which negative numbers is permitted gives us elegant, easy to remember tools, such as $-(a - b) = b - a$. It helps us realise that acceleration and deceleration are different sides of the same coin: the derivative of the velocity function (be it positive or negative). It neatly models the observable fact that deceleration, when left constant long enough, eventually turns movement in one direction into movement into the opposite direction.

We have nothing but gain when we allow ourselves more elegant tools to help us model and understand the world.

I'll leave you with a thought: al-Khwarizmi (from whom we get the word "algorithm") was the first to start solving quadratics, and did so without negative numbers. He had to deal with several cases, such as $x^2 = bx + c$, and $x^2 + c = bx$, where the numbers are assumed to be positive (including $x$!). Imagine trying to work out the quadratic formula without negative numbers! It's irritating enough without complex numbers. :-)

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