Interpretation of how to define “bigger” and “smaller” real numbers

Question

This is a variant on the question small real numbers.

I have a disagreement with someone about the meaning of "bigger" real numbers.

Say we have the real number $-1.$ Is $0$ "bigger" or "smaller" than $-1$? In other words, should we interpret "bigger" to be a synonym of "greater than"?

This may be considered an "opinion-based" question, but is there a consensus about how to assign meaning to "bigger" or "smaller" (real) numbers?

My interpretation is that "bigger" real numbers have larger magnitude, irrespective of sign. For example, I consider that $-2$ is "bigger" than $-1.$ To avoid ambiguity (we are mathematicians!), I think it better to say/write "$-2$ has a larger magnitude/size than $1$".

Further, I try to avoid using "bigger" or "smaller" with respect to comparing numbers unless it is clear we are comparing their magnitude (e.g. vectors or complex numbers), where we are only comparing non-negative values. I would stick to "greater / less than" or "closer to / further from zero".

Regards,

Answer 1

Since "bigger" and "smaller" are ambiguous, it is best to avoid them, as you mention. The methods you mention seem reasonable, though I am not native English speaker.

Someone may be able to refer to a credible source or official standard, but regardless, you can only know how someone else understands the terms by asking them. So you might as well do that and dispel the confusion.

A mathematician should know that definitions are matters of convenience and communication. As such, a mathematician can easily adapt to the terminology the other person prefers, for the purposes of a given discussion. Most people have less practice with working with explicit and sometimes arbitrary definitions than mathematicians (and philosophers) do.

Answer 2

The words "big" and "small" are relative. Asking a student: "Is $1$ small?" is confusing, and meaningless. Likewise, "bigger" and "smaller" are relative. If you consider six numbers, $1, 2, 4, 5, 7, 8$, then each of $2, 4, 5, 7, 8$ is bigger than $1,$ and each of them is "smaller" than $8$.

What you seem to want to convey, when comparing two real numbers and calling one "bigger than" the other, is that one is "bigger in magnitude" or one is "smaller in magnitude", where magnitude can be thought of as the distance of a number from zero. So $-2$ is bigger (greater) in magnitude than is $1$, where the magnitude of a real number $x$ is given by $|x|$. But without adding the qualifier in magnitude, I'm afraid you will not be well understood.

So, indeed, $|-2| > |1|$.

The more common and well understood terms used to compare two real numbers, 'greater than' and 'less than', (or, 'equal to'), are used to refer to the stricter interpretation in which the one further to the left on the real number line is less than the other. (And the number further to the right on the real number is greater than the other.)

Hence we write, e.g., $-83\lt -3 \lt 0 \lt 1$.

I'd suggest that the use of "greater than" or "less than," when comparing magnitudes of two numbers, that, e.g., "greater in magnitude" ("lesser in magnitude") is more widely used than is "bigger than, smaller than."