# How should one rank a combination of positive and negative numbers from smallest to largest?

I thought size was the distance from zero, in which case ordering {-1, 2, -3} would be {-1, 2, -3}, but I saw it ordered as {-3, -1, 2}. Which method is correct?

• Rank the temperatures $-1, 2,-3$ from coldest to warmest. – André Nicolas Jan 11 '15 at 2:00
• Perhaps the terminology 'least to greatest' is preferable (clearer). – paw88789 Jan 11 '15 at 2:07

## 2 Answers

While "size" is not a formally defined property for real numbers, we do have the notions of order (whether $x<y$) and magnitude (whether $|x|<|y|$). Whether size should refer to order or magnitude is somewhat open to preference, but in my experience everyone considers it to mean magnitude.

I think the most convincing argument for this interpretation is that we often talk about large or small complex numbers, or vectors in $\mathbb{R}^n$. In these settings, we're clearly talking about magnitude--no order exists. Since the real numbers are embedded in these spaces in a very natural way, it makes sense to define "size" on the reals to be consistent with its meaning in the other spaces.

The second. -3 is less than -1 which is less than +2. (At the risk of bad analogies, think money. If you earned two pounds, you have more money than if you lost one pound, which is more money than losing three pounds.) The first method is not ordering them by magnitude but by their absolute values or, as you said, "distance from zero", which is different from their "size".

• How is {-1, 2, -3} not ranked by magnitude? Isn't the magnitude equivalent to the distance from zero? That's what it seems to imply on Wikipedia. – Kelmikra Jan 11 '15 at 13:36