What exactly is order of magnitude? I am little bit confused with what order of magnitude is. In my book it says, when we write approximate values of quantities in powers of ten i.e $10^b$, then $10^b$ is the order of magnitude. But later it says that the exponent of the 10 i.e b, is the order of magnitude. I searched many websites some of them say $10^b$ is the order of magnitude and some of them say $b$ is the order of magnitude. Please clear my doubt.
 A: It's both. If you asked me to give the order of magnitude of some quantity that we're talking about, perhaps to get an idea of the physical scales involved, I would use $10^b$. So if I was talking about, say, galaxies, and you asked me about the typical order of magnitude of the masses, I would say "of the order of $10^{12} M_{\odot}$".  But if one galaxy is 100 times more massive than the other, I would say that the masses are two orders of magnitude apart.
A: Order of magnitude is used for describing roughly how big something is. Sort of like distinguishing between really big and ginormous. For example, the galaxy is much bigger than the solar system, but it isn't obvious how much bigger.
The radius of the solar system is $4.5 \cdot 10^9$ km. The radius of the galaxy is $4.7 \cdot 10^{17}$ km.
Which means the galaxy is about $10^8$ times bigger, or 8 orders of magnitude bigger. The difference between $4.5$ and $4.7$ doesn't really matter when talking about approximate numbers like this.
It is also useful when talking about tiny and microscopic numbers. A typical diameter of an atom is 1 Angstrom or $10^{-10}$ m. The diameter of a nucleus is $4$ to $15$ femtometers, or $4$ to $15 \cdot 10^{-15}$ m. The nucleus is about $10^4$ times smaller, or $4$ orders of magnitude smaller.
A: If we write $N=a×10^b$, $b$ is the order of magnitude. For example, the number 1 has 0 orders of magnitude. However, this concept of orders of magnitude is more useful while comparing numbers. For example, to compare $3$ and $1000$ we could say that $1000$ is $997$ more than $3$, but sometimes, it’s more convenient to say that $1000$ is $3$ orders of magnitude greater than $3$.
