# Order of magnitude

#### beauty-eyes

What is the order of magnitude?

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#### billiards

It is a measure of size, in a numerical sense. To increase a number by an order of magnitude is to add a zero to the end (thereby multiplying by 10), likewise to decrease by an order of magnitude is to move the decimal point to the left (divide by 10). At least that is how I have come to understand it.

#### beauty-eyes

Thank you billiards for helping me.

what is the benifit of the order of magnitude?

#### cristo

Staff Emeritus
what is the benifit of the order of magnitude?
What do you mean by "benefit?" The question, as it stands, doesn't make any sense to me.

#### beauty-eyes

I mean why we use it?

#### billiards

Thank you billiards for helping me.

what is the benifit of the order of magnitude?
It is often the case in some branches of physics, that in attempting to solve an equation numerically it is necessary to guess the value of some of the variables. A common approach is to make an "order of magnitude estimation". Although this approach won't return a precise number, it can put a constraint on the numerical size of the quantity of interest, which may provide some further qualitative understanding.

#### jtbell

Mentor
More generally, it is often useful to estimate the order of magnitude (the power of ten) of an answer before diving into a detailed calculation. This gives you one way to check the validity of your final, more precise, result.

Enrico Fermi was very good at this, and liked to pose offbeat order-of-magnitude estimation problems as practice for his students. Physicists call these "Fermi problems." The most famous one is probably: How many piano tuners are there in Chicago?

#### rewebster

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(50 weeks/year)×(5 days/week)×(8 hours/day)×(1 piano tuning per 2 hours per piano tuner) = 1000 piano tunings per year per piano tuner

(http://en.wikipedia.org/wiki/Fermi_problem)

That was before the unions came along, I'd guess.

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Order of magnitude:

http://en.wikipedia.org/wiki/Orders_of_magnitude

(they didn't include the Richter scale on that page as a 'type')

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#### rbj

in my opinion, the reason we speak of "order of magnitude" is to quickly classify the size of a problem, phenomenon, or thing. it doesn't have to be physics or even science.

for instance, in gauging the degree of a terrorist attack, we might ask: how many people were killed in it? if i nor any of my loved ones are not directly involved, will i gauge the attack any differently if 50 people were killed vs. 53? but what if it were 5 people? or 500 people, or, to make it closer to how we were perceiving the Sept. 11 attacks, 5000 people? or 50,000 as in the terrorist attack on Hiroshima on August 6, 1945? or 5 million people in Cambodia during Pol Pot. even though the 3 extra people are important, i do not gauge the magnitude of the attack killing 53 people any differently than if it had killed 50. but i do gauge it as seriously worse than the one that killed "only" 5 and much "better" than the one that killed 500.

that is what order of magnitude is about. it is magnitude on a log10 scale.

one question i would ask about it is: why a base-10 logarithm (other than the anthropometric fact that most of us have 8 fingers and 2 thumbs on our hands)? why not base-2 or base-e? those seem pretty natural bases to define this concept of order of magitude. it would mean that if something was an order of magnitude worse, it was twice as bad in quantity.

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#### robphy

Homework Helper
Gold Member
http://www.inference.phy.cam.ac.uk/sanjoy/oom/
http://www.stanford.edu/class/ee353/SanjoyMahajanIntro-01-1.pdf
http://www.its.caltech.edu/~oom/homework.html [Broken]

http://astro.berkeley.edu/~echiang/oom/oom.html

related material:
http://www.vendian.org/envelope/
http://physics.guc.edu.eg/activities/numbers.asp [Broken]

So, with "order of magnitudes", should we "round up" at 4?

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#### Integral

Staff Emeritus
Gold Member
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