Is an order of magnitude simply 10x bigger or smaller and 2 orders of magnitude simply 100x bigger or smaller. If this is correct why not simply say 10x or 100x bigger/smaller?
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Yeah that’s literally what it is.
If someone says something is orders of magnitude bigger, if whatever they’re starting with is, I don’t know, 1000-and-something of whatever it is, then the bigger thing will be 10,000 or 100,000.
It’s just a short way of saying very imprecisely that something is a lot bigger, but giving more information than just ‘a lot bigger’
If something is 100 feet tall, something 150 feet tall **is** a lot bigger, depending on your perspective. Hell, 200 feet. 500 feet.
Someone says it’s orders of magnitude bigger, then you know instantly that it’s way more than that.
Our number system is based on powers of 10. 100 = 10^2, 1000 = 10^3 and so on. Orders of magnitude are just the difference between the powers of 10 involved.
As for why they are used, they are rather useful when comparing really large numbers and/or very rough estimates, since saying “these two things differ by 13 orders of magnitude” is less clunky than saying “thing A is ten trillion times larger than thing B2”.
Some examples:
* An order of magnitude larger than 7, would mean it could be any value between 10-99
* From 2,500, will range between 10,000 and 99,000
* Technically, if you reduce something from 1100 down to 900 you’ve decreased by an order of magnitude
It’s most helpful in situations with geometric growth (exponential growth, logarithmic growth). [Moore’s law](https://en.wikipedia.org/wiki/Moore%27s_law) could be very easily restated that over X period of time it will increase by an order of magnitude
> If this is correct why not simply say 10x or 100x bigger/smaller?
It is a matter of precision and accuracy.
If I say something is 10x bigger, but it is really 11x bigger, then 10x for many uses is wrong.
If instead I say something is an order of magnitude bigger and it works out to be 8x or 10x or 15x or even 35x, it’s still in range.
Similarly, saying something is 200x bigger but it works out to be 100x or 300x those answers are inaccurate, imprecise, or both, and in many uses wrong. But if I say something is a few orders of magnitude bigger and it works out to be 100x or 200x or 300x, they’re still in range.
Precision and accuracy matter to some people, especially those in technical fields. The more precise you make your answer, the more precise it is assumed to be. A quick way to rile up a bunch of techies is to throw around made-up numbers and make up random statistics. If you tell me something is 73% bigger, I might be fine with it being 72% or 74%, but if I end up checking and it works out to be 47% bigger then your answer is significantly wrong and I won’t find you reliable in the future.
In many situations the range you need to consider is enormous.
Let’s look at the sound that we measure in decibels. The scale is in principle the magnitude*10. Deci mean 1/10 so a decibel is 1/10 of a bel. Soe the bell is a order of magnitude but it is multiplied with 10 to reduce the need for decimal number. So a increase of 10 dB is 10x the power and 20dB is 100x the power
Lets look at the sound examples at [https://www.britannica.com/science/sound-physics/The-decibel-scale](https://www.britannica.com/science/sound-physics/The-decibel-scale)
It is in intensity in decibles, intensity in Watts/m^2m and wa example of what it is.
* 130 10 artillery fire at close proximity (threshold of pain)
* 120 1 amplified rock music; near jet engine
* 110 10^−1 loud orchestral music, in audience
* 100 10^−2 electric saw
* 90 10^−3 bus or truck interior
* 80 10^−4 automobile interior
* 70 10^−5 average street noise; loud telephone bell
* 60 10^−6 normal conversation; business office
* 50 10^−7 restaurant; private office
* 40 10^−8 quiet room in home
* 30 10^−9 quiet lecture hall; bedroom
* 20 10^−10 radio, television, or recording studio
* 10 10^−11 soundproof room
* 0 10^−12 absolute silence (threshold of hearing)
When you listen to sound how loud is sound to you is like in the dB scale but sound energy scale a lot faster.
There is may situation where you have differences like that. For 1 or 2 magnitudes the large number is quite easy to say but when you start to the to larger number it gets more and more complicated so taking of the magnitude is simpler.
The large number is in science if often written in what is called scientific notation, an exponential format so 12,000,000 is 1.2*10^7 because it is simpler to read. You only have a single-digit before the decimal point,
Because number form the real world is often limited in precision you only have a few significant digits. you will not have a number like 12,003,324 that often.
So talking the difference of magnitude is the same as the difference in the exponent you use for the 10
Technically this is not exactly how the order of magnitude works,
You should write number N as a* 10^b where a >1/sqrt(10)=0.31 and a<sqrt(10)= 3.1 and the magnitude of the number is b
3.5= 0.35*10^1 instead of 3.5*10^0
30= 3*10^1
300=3*10^2
The result is that 3.5 and 30 have the same magnitude and 300 is only 1 magnitude larger the both are not 2 larger than 3.5 and 1 larger than 30 as one might expect.
In practice and more informal usage will just be the difference in the exponent in scientific notation even if that is not technically what order of magnitude is.
https://en.wikipedia.org/wiki/Order_of_magnitude
Describing salaries by number of zeros is a good example. Six figure salary v five figures would be an order of magnitude difference.
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