Editing Order of magnitude (section)

==Calculating the order of magnitude==
Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number.<ref>{{cite web|title=Order of Magnitude|url=http://mathworld.wolfram.com/OrderofMagnitude.html|website=Wolfram MathWorld|access-date=3 January 2017|quote=Physicists and engineers use the phrase "order of magnitude" to refer to the smallest power of ten needed to represent a quantity.}}</ref> To work out the order of magnitude of a number <math>n</math>, the number is first expressed in the following form:
:<math>n =a\times10^b</math>
where <math>\frac{1}{\sqrt{10}}\leq a<\sqrt{10}</math>, or approximately <math>0.316\lesssim a \lesssim 3.16</math>. Then, <math>b</math> represents the order of magnitude of the number. The order of magnitude can be any [[integer]]. The table below enumerates the order of magnitude of some numbers using this definition:
{| class="wikitable"
|-
! Number <math>N</math>!! Expression in <math>N =a\times10^b</math>!! Order of magnitude <math>b</math>
|-
| 0.2 || 2 × 10<sup>−1</sup> ||−1 
|-
| 1 || 1 × 10<sup>0</sup> ||0 
|-
| 5 || 0.5 × 10<sup>1</sup> ||1
|-
| 6 || 0.6 × 10<sup>1</sup> ||1
|-
| 31 || 3.1 × 10<sup>1</sup> || 1
|-
| 32 || 0.32 × 10<sup>2</sup> || 2
|-
| 999 || 0.999 × 10<sup>3</sup>|| 3
|-
| 1000 || 1 × 10<sup>3</sup>|| 3
|}
The [[geometric mean]] of <math>10^{b-1/2}</math> and <math>10^{b+1/2}</math> is <math>10^b</math>, meaning that a value of exactly <math>10^b</math> (i.e., <math>a=1</math>) represents a geometric ''halfway point'' within the range of possible values of <math>a</math>.

Some use a simpler definition where <math>0.5\leq a< 5</math>.<ref>{{Cite web |last=Shaalaa.com |title=Answer the following question. Describe what is meant by order of magnitude. - Physics {{!}} Shaalaa.com |url=https://www.shaalaa.com/question-bank-solutions/answer-the-following-question-describe-what-is-meant-by-order-of-magnitude-significant-figures_171864 |access-date=2023-06-04 |website=www.shaalaa.com |language=en-IN}}</ref> This definition has the effect of lowering the values of <math>b</math> slightly:
{| class="wikitable"
|-
! Number <math>N</math>!! Expression in <math>N =a\times10^b</math>!! Order of magnitude <math>b</math>
|-
| 0.2 || 2 × 10<sup>−1</sup> ||−1 
|-
| 1 || 1 × 10<sup>0</sup> ||0 
|-
| 5 || 0.5 × 10<sup>1</sup>||1
|-
| 6 || 0.6 × 10<sup>1</sup> ||1
|-
| 31 || 3.1 × 10<sup>1</sup> || 1
|-
| 32 || 3.2 × 10<sup>1</sup> || 1
|-
| 999 || 0.999 × 10<sup>3</sup>|| 3
|-
| 1000 || 1 × 10<sup>3</sup>|| 3
|}