# Rounding issue

## Main Question or Discussion Point

When we are rounding to the nearest order of magnitude and faced with a number like 3.2*10^2 should we round it to 10^2 or 10^3?

At first thought, I would round it to 10^2 for obvious reasons like 320 is closer to 100 than 1000.

But 320=10^2.505 and if we round 2.505 it becomes 3 so following this reasoning 320 would be rounded to 10^3.

$$3.2 \cdot 10^2$$

That number has two significant figures. The answer depends on how many significant figures you are suppose to round to. If it is one, then you would obviously round it to $3 \cdot 10^2$. If it is the nearest order of magnitude, I think it is safe to say that 320 is rounded to 100.

HallsofIvy
Homework Helper
That apparent "discrepancy" is due to the fact that 10x is not linear. There is no reason to bring 10x into the picture for something as basic as rounding.

This issue occured in my physics textbook and it said that 320 should be rounded to 10^3 for the nearest order of magnitude by the argument in my OP. Are people suggesting it is incorrect?

cristo
Staff Emeritus
If you wanted to round 320 to the nearest order of magnitude, it would be rounded to 100.

If you wanted to round 320 to the nearest order of magnitude, it would be rounded to 100.
Well, according to my physics textbook, they teach us to round 320 to 1000 for the reason given in my OP. You think it is not correct?

uart
Well, according to my physics textbook, they teach us to round 320 to 1000 for the reason given in my OP. You think it is not correct?
Geometrically 320 is closer to 1000 than 100.

Arithmetically 320 is closer to 100 than 1000.

So in some sense they are both correct. It's been a while since I've dealt with "order of magnitudes" but I think you are correct to round geometrically (that is to 1000 in this case). An order of magnetude simply means a multiple of 10. If two quanities are "of the same order of magnetude" then it means that they will both round geometrically to the same power of ten. So I agree with the textbook on this one.

What do you mean by geometrically?

If someone said 2*10^2 and 3.2*10^2 both had the same order of magnitude, I would have believed them and thought it was 10^2. But if going with the way in the textbook, they wouldn't as the former would be 10^2 and latter 10^3 to the nearest order of magnitude.

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What do you mean by geometrically?
Geometric and arithmetic in roughly the same sense of geometric progression and arithmetic progression. So geometric means compairing the power.

uart
Good point, I didn't think that one through properly. 