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I'm reading a book in which the author compares two pairs of numbers $(0.31, 0.39)$ and $(6.10,0.39)$ and multiplies the second member of each pair by a factor $R_1 = 2$ and $R_2 = 20$ so that both members of the pair "have the same order of magnitude". Consequently the pairs of numbers become $(0.31, 2 \times 0.39)$ and $(6.10 , 20 \times 0.39)$. How do these factors ensure that both numbers in the pair have the same order of magnitude?

Detailed answers appreciated please. Thanks.

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(1) It would help to know the context in which this change in the order of magnitude occurs. – amWhy Dec 1 '12 at 14:22
If it helps, the question arises in the context of constrained multiobjective optimization (with evolutionary algorithms). The first number in the pair is an objective function value and the second is a constraint violation. – Olumide Dec 1 '12 at 14:32
@amWhy I'm not struggling with the optimization bit. Besides that part is done. Its just the part where the results are factored that I'm having difficulty with. Going into details about optimization would complicate the question. – Olumide Dec 1 '12 at 14:48

"Order of magnitude" can be just a synonym for "approximately equal" or it can be "the same number of digits in front of the decimal point". I don't know why the author multiplied $0.39$ by $2$-it was already close to $0.31$ and this takes it farther away. Certainly $20 \times 0.39 = 7.8$ is much closer to $6.1$ than without multiplying by $20$.

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