# Rounding issue

1. Jan 20, 2007

### pivoxa15

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.

2. Jan 20, 2007

### Moridin

$$3.2 \cdot 10^2$$

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.

3. Jan 20, 2007

### HallsofIvy

Staff Emeritus
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.

4. Jan 22, 2007

### pivoxa15

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?

5. Jan 22, 2007

### cristo

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

6. Jan 23, 2007

### pivoxa15

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?

7. Jan 23, 2007

### uart

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.

8. Jan 24, 2007

### pivoxa15

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.

Last edited: Jan 24, 2007
9. Jan 24, 2007

### yenchin

Geometric and arithmetic in roughly the same sense of geometric progression and arithmetic progression. So geometric means compairing the power.

10. Jan 24, 2007

### uart

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

Anyway I've mostly seen "order of magnetudes" used only in a comparative sense. The basic definition is that two qualities are "the same order of magnetude" if the larger is less than ten times the value of the smaller.

Anyway the whole concept of order of magnetudes is one of comparing ratios of quanties (less then ten times) rather than comparing their difference, so rounding geometrically makes sense.