Why do we need zeroes when writing some of those kinds of numbers?

Question

I've looked up that zero is a placeholder in numbers that have this digit. I also figured out what would happen if there's no zero in 5,074 in a math journal. The number would be 5,74, which would look silly if it's written with a comma. Maybe that's why it's a placeholder, which makes it a helper. Is this why we need zeroes when writing these numbers like 4,307,222, 308, and 60,275? I think I'm on the right track. Tell me what you think about why we need zeroes in those numbers I just mentioned and those other numbers that have zero as a placeholder and if I'm on the right track!

Answer 1

When we talk about the number (for instance) $5,267$, what we really mean is

$$5\cdot 1000 + 2\cdot 100 + 6\cdot 10 + 7\cdot 1$$ Why is this the case? It's just our definition of base 10 numbers.

If we work now with the number $506$, our expansion will be

$$5\cdot 100 + 0\cdot 10 + 6\cdot 1$$

If we didn't have the zero there (call the number $5\, 6$?), then we have

$$5\cdot 100 + ()\cdot 10 + 6\cdot 1$$

and who knows what that means?

Answer 2

The reason that you need 0 is the issue of ambiguity, in the example which you provided with the number 4,307,222,308 we have the issue that if it were written like 4,37,222,38 that this could also represent 4,037,222,038 as well (along with several other numbers). The other reason is that each digit represents a power of 10, i.e. 5,074 = $4*10^0 + 7*10^1 + 0*10^2 + 5*10^3$ so, as you can see the 0 indicates the coefficient on its respective power of 10. They used to do math without 0's with Roman Numerals, however those were unwieldy and difficult to work with when doing large calculations.