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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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
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