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I would appreciate any help with the tags. I'm afraid I don't know enough about what I'm asking to know what tags to apply!

As I've understood things, smaller describes a value as being closer to 0, so:

5 is smaller than 6
5 is smaller than -6
-5 is smaller than 6
-5 is smaller than -6

I'm slightly confused though about 'less than'. The word in natural use appears to just be a synonym of 'smaller than'.

5 is less than 6

This statement is fine, but the following statement:

-5 is less than -6

In this statement, the magnitude of -6 is more than the magnitude of -5. Yet, I can't find any program that states (-5 < -6) as true. Is this because the symbol < is actually 'lower than' rather than 'less than'?

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    $\begingroup$ In mathematics, "further to the left on the number line" is so much more useful and so much easier to work with that "closer to $0$" doesn't even have its own established name. Both "smaller than" and "less than", along with any other such, means "closer to $-\infty$". $\endgroup$
    – Arthur
    Oct 8, 2016 at 17:53

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"smaller" does not mean "closer to zero" it simply means less positive, which is why 5 is NOT less than -6

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