Fri September 21, 2012 By:
Dear Sir,

Can zero has a negative?

Expert Reply
Sat September 22, 2012
Negative zero does exist, and is equal to zero.

Let's think in terms of a number line. We start off with a 0. To the right of 0 are all the positive numbers in order. 0, 1, 2, 3, 4, etc.

Now let's go ahead and add in the negative numbers to the left of 0.

..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

Now think about what the relationship is between any number and its negative. We see that given a number, n, and its negative, -n, we see that they are both n spaces away from 0 on the number line. It's just the positive one is to the right and the negative one is to the left.

Given this information, we need to look at negative zero. This number will be the same number of spaces away from zero as positive zero is. Confused? Let's simplify it...

How far away from zero is zero? It's 0 numbers away.

How far away from zero is negative zero? It's 0 numbers away.

From this it should be more clear that zero and negative zero are the same number.

How Negative Zero Is Used

So, we know that negative zero and positive zero are equal to each other. Why, then, do we have both and when is each one used?

We'll look at two common uses (the only ones I've come across, personally).

The first is rounding. Let's say we want to round to the nearest whole number.

0.5 rounds to 1
10 rounds to 10
0.1 rounds to 0
-0.1 rounds to -0 <- Ah ha!

This first use is to show from which direction we have reached zero. If we're rounding down, it's a positive zero; if we're rounding up, it's a negative zero.

The second use is very similar, in that it shows the direction from which we approach zero. (Those of you who have not yet learned any calculus may want to skip this part.)

Let's say we have a limit that we want to evaluate.
How about the limit as n approaches infinity of 1/n? This should be one of the earliest limits you learn, and so you should know this approaches zero. For our purposes let's clarify and say this approaches positive zero.

The other example is, of course, the limit as n approaches infinity of -1/n. This value would approach negative zero.

In Computing

Most modern computer languages use the IEEE 754 floating point standard for storing decimal numbers. You don't need to know what that means, you just need to know that due to the way numbers are stored in computer languages, there can exist both a positive zero and a negative zero. Furthermore, these two values may not be equal to each other.

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