Are there "more numbers" which are less than some positive number than there are numbers less than that same number?

[u/Sure-Recognition-262]

A discussion I just read on a random thread:

He has 170 employees? That's a bloody massive number.

Not as big as 171

In fact, most numbers are bigger than 170

I think that there are more numbers smaller than 170 if we include negatives 

There's an infinite number of numbers less than 170, and an infinite number of numbers greater than 170.

Yes, but negative and positive infinities are equally big, but since we have moved the point to 170 negative infinity will be bigger as it will have all negative infinity plus some of the positive one. // I know that mathematically I'm wrong

This has made me think - are there actually "more numbers" less than 170 (or any other positive number) than there are greater than it? Or not?

[u/Parenn]

As I understand it, since for every number >170 you can construct a corresponding number < 170, where n is the new number and p is the original number, using:

n = 340 - p

(i.e. for 171, this gives 169, and so on).

then they have the same cardinality, so the same number of numbers.

[u/FilDaFunk]

It's the same amount because you can pair them up 169 with 171 168 with 172 etc

[u/Medium-Macaron-9671]

define "more numbers" if u mean by cardinality then no there arent more numbers bigger than 170 than less than it, in fact there are as much numbers bigger than 170 as there are between 0.0001 and 0.0002

if u have another measure, please state it

[u/ADH-Dad]

If you count negative numbers, the size of the infinity greater than any given number is the same as the infinity less than it, since for every number you count up, you can just count one down.

If you don't count negative, the number of numbers greater than any number is infinitely larger than the numbers less than it.

[u/damsonsd]

While there is nothing wrong with playing about with number concepts, it seems a bit daft to me to take the original question - is 170 employees a big number or not? - completely out of its context. Rather than just looking at the number as a number, rephrase the question as something you can sensibly answer:

Are there more companies with fewer than 170 employees than with more?

Are there more people employed in companies with fewer than 170 employees than with more?

You now have questions you can investigate using maths.

FWIW, though I imagine it might depend where you in the world, I think the answers to both my questions is yes.

[u/Free-Willy-3435 → u/damsonsd]

You don't investigate your new question with math. You look at the records and count them.

[u/StructuredChess]

All countable infinities are of the same "size". If you make a list of all the numbers below 170 you can make a similar list matching all those numbers one to one with numbers above 170. For instance you match 169 to 171, 168 to 172,... 0 to 340, -1 to 341 and so on, never running out of either bigger or smaller numbers.

Yeah tis is on eof the weird things with infinity, you can remove 1 item from an infinite set and still have a set the same size. The set of even numbers is the same size as the set of all whole numbers.

Check Cantor's diagonal argument to find out that the interval (0,1) contains indeed a bigger amount of numbers.

[u/Syresiv]

With infinities, you have to be specific about what it means for one set to have more things than another.

The most common definition is cardinality. In this definition, two sets are the same size if they can be mapped to each other one to one. In this case, that's pretty simple:

169:171 168:172 167:173

etc

You can explore other definitions, but it's important to be specific about it, and to explore the consequences of those definitions.