# 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.

• Exactly? The entire point of the concept is that it's inexact ツ May 10 at 22:19

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.

• When you're asked about the typical order of magnitude of galaxies and you say "of the order of $10^{12} M_{\odot}$", you are saying the desired order of magnitude is the same as the order of magnitude of the quantity $10^{12}$, which is $12$. The order of magnitude is the exponent. May 11 at 2:15
• -Javier, Now I got it, Thanks for the help. May 11 at 9:29
• @electronpusher I disagree. If someone asks me for the order of magnitude and I'm being really brief, I'm answering $10^{12}$, but never 12. May 11 at 16:01
• Well, that doesn't demonstrate what it is, just what your impression of it is. If they asked me, I'd say 12 (and that doesn't make it right either). There's several answers here and a curious lack of citations of authoritative sources on terminology. May 11 at 22:16
• @Javier If you have a $100 investment that increases in value to$1000, how many orders of magnitude has it increased? May 11 at 22:24

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.

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 $$3$$.

• I would also mention that $1\leq a < 10$ in your notation.
– noah
May 10 at 12:51
• Oh yes, I forgot to mention that. Thanks for pointing that out. May 10 at 12:52
• Indeed, a factor of ~1000 is 3 orders of magnitude. May 10 at 13:10
• As far as I know, 10^b is not the “order of magnitude”. It’s b. When they say 10^b is the order of magnitude, they’re wrong (?). Where exactly did you come across 10^b being the order of magnitude? May 10 at 13:41
• @Heisenberg, my two cents: I've seen people refer to $10^b$ as "the magnitude" of a number in standard form. So maybe the nuance here is "order of magnitude" vs simply "magnitude". May 10 at 16:30