How many orders of magnitude does $345,632$ differ from $567,123,423$?

Question

If you just divide you get: $$\frac{567123423}{345632}=1641(nwn)$$

This number is closer to $10^3$ than $10^4$ so I would have answered $3$ orders of magnitude.

But the book says to first express the numbers in scientific notation and then round the decimal number like so:

$$\frac{567123423}{345632}=\frac{5.7*10^8}{3.5*10^5}=\frac{10*10^8}{1.0*10^5}=\frac{10^9}{10^5}=10^4$$

The book gives an answer of 4 orders of magnitude.

What is the correct answer here?

Answer 1

Both, I'd say. Orders of magnitude is a fuzzy subject. Although your approach is more correct, as in closer to the unrounded answer, the book's approach is a lot easier to do (I can't divide by $345\,632$ in my head), and in the end isn't that one of the reasons to use a concept like "orders of magnitude" in the first place?

That being said, $\frac{5.7}{3.5}$ is clearly seen to be between $1.5$ and $2$, so in the end I would prefer to call it three orders of magnitute, personally.

Answer 2

The correct answer depends on definition. The reason why the book gets $10^4$ is because they round to the closest power of ten. If it is the right choice to round $5.7$ to ten and $3.5$ to one is debatable, but this is how the book does it.

So there is not "the" correct answer, there are different ones, depending on how you handle it. Both your answer and the one in the book can be considered "right" if they are properly justified. If this is part of a lecture, you might want to ask your professor about it.

Answer 3

I would like to offer a different perspective to the question: Wikipedia gives the definition of an order of magnitude as the base $10$ logarithm of a number, or the exponent when that number is represented in scientific notation.

I think the book has a mistake, as @Arthur said, the number is roughly $$\frac{5.7*10^8}{3.5*10^5},$$ which in scientific notation it is $a * 10^3$, where $a$ obviously is between $1$ and $10$. Since this form is valid, I think that there are $3$ orders of magnitude.

I think that rounding $a$ up or down can create an unnecessary order of magnitude, since in a case such as $\frac{5.0001 * 10^8}{4.9999 * 10^5}$, the book's method still gives the answer as $4$ orders of magnitude.

That being said, order of magnitude does not directly mean the number of powers of $10$ in a given number, so I think that the vagueness of this term is causing the confusion.

Answer 4

$$567123423 = 5,67123423 \cdot 10^8$$ $$345632 = 3,45632 \cdot 10^5$$

Taking the ratio between the two numbers written like this: $$\frac{567123423}{345632}=\frac{5,67123423}{3,45632}\cdot 10^3 \approx 1,6408\cdot 10^3$$ IMO, the correct answer in this case should be $3$. However, the ambiguity of the term may cause some degree of opinion. It all mainly depends on the approximations made, and, in the end, both should be considered correct answers. Still, if you multiply $345632$ by $10^3$, it is still smaller than $567123423$, so I think that's why the book suggests that the solution is four orders of magnitude.