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Two numbers are within an order of magnitude of each other if their ratio is between 1/10 and 10. In other words, the two numbers are within a factor of 10 of each other ^[1]. This concept helps articulate that 1000 and 1010 are close to each other in the same way that 1.00 and 1.01 are, even though the absolute differences are 10 and 0.01, respectively.

For example, 1 and 1.02 are within an order of magnitude. So are 1 and 2, 1 and 5, or 1 and 9. However, 1 and 15 are not within an order of magnitude, since their ratio is 15/1 = 15 > 10. The reciprocal ratio, 1/15, is less than 0.1, so the same result is obtained.

Differences in order of magnitude can be measured on a base-10 logarithmic scale in "decades" (i.e., factors of ten).^[2] For example, there is one order of magnitude between 2 and 20, and two orders of magnitude between 2 and 200. Each division or multiplication by 10 is called an order of magnitude.^[3]

Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers).

Below are examples of different methods of partitioning the real numbers into specific "orders of magnitude" for various purposes. There is not one single accepted way of doing this, and different partitions may be easier to compute but less useful for approximation, or better for approximation but more difficult to compute.

An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is 10, the order of magnitude can be understood as the number of digits minus one in the base-10 representation of the value. Similarly, if the reference value is one of some powers of 2 since computers store data in a binary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value.

Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number.^[4] To work out the order of magnitude of a number ${\displaystyle N}$, the number is first expressed in the following form:

${\displaystyle N=a\times 10^{b}}$

where ${\displaystyle {\frac {1}{\sqrt {10}}}\leq a<{\sqrt {10}}}$, or approximately ${\displaystyle 0.316\lesssim a\lesssim 3.16}$. Then, ${\displaystyle b}$ 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 in light of this definition:

Number ${\displaystyle N}$	Expression in ${\displaystyle N=a\times 10^{b}}$	Order of magnitude ${\displaystyle b}$
0.2	2 × 10−1	−1
1	1 × 100	0
5	0.5 × 101	1
6	0.6 × 101	1
31	3.1 × 101	1
32	0.32 × 102	2
999	0.999 × 103	3
1000	1 × 103	3

The geometric mean of ${\displaystyle 10^{b-1/2}}$ and ${\displaystyle 10^{b+1/2}}$ is ${\displaystyle 10^{b}}$, meaning that a value of exactly ${\displaystyle 10^{b}}$ (i.e., ${\displaystyle a=1}$) represents a geometric halfway point within the range of possible values of ${\displaystyle a}$.

Some use a simpler definition where ${\displaystyle 0.5\leq a<5}$,^[5] perhaps because the arithmetic mean of ${\displaystyle 10^{b}}$ and ${\displaystyle 10^{b+c}}$ approaches ${\displaystyle 5\times 10^{b+c-1}}$ for increasing ${\displaystyle c}$.^{[citation needed]} This definition has the effect of lowering the values of ${\displaystyle b}$ slightly:

Number ${\displaystyle N}$	Expression in ${\displaystyle N=a\times 10^{b}}$	Order of magnitude ${\displaystyle b}$
0.2	2 × 10−1	−1
1	1 × 100	0
5	0.5 × 101	1
6	0.6 × 101	1
31	3.1 × 101	1
32	3.2 × 101	1
999	0.999 × 103	3
1000	1 × 103	3

Orders of magnitude are used to make approximate comparisons. If numbers differ by one order of magnitude, x is about ten times different in quantity than y. If values differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value. The growing amounts of Internet data have led to addition of new SI prefixes over time, most recently in 2022.^[6]

In words	Prefix (Symbol)	Decimal	Power of ten	Order of magnitude
nonillionth	quecto- (q)	0.000000000000000000000000000001	10−30	−30
octillionth	ronto- (r)	0.000000000000000000000000001	10−27	−27
septillionth	yocto- (y)	0.000000000000000000000001	10−24	−24
sextillionth	zepto- (z)	0.000000000000000000001	10−21	−21
quintillionth	atto- (a)	0.000000000000000001	10−18	−18
quadrillionth	femto- (f)	0.000000000000001	10−15	−15
trillionth	pico- (p)	0.000000000001	10−12	−12
billionth	nano- (n)	0.000000001	10−9	−9
millionth	micro- (μ)	0.000001	10−6	−6
thousandth	milli- (m)	0.001	10−3	−3
hundredth	centi- (c)	0.01	10−2	−2
tenth	deci- (d)	0.1	10−1	−1
one		1	100	0
ten	deca- (da)	10	101	1
hundred	hecto- (h)	100	102	2
thousand	kilo- (k)	1000	103	3
million	mega- (M)	1000000	106	6
billion	giga- (G)	1000000000	109	9
trillion	tera- (T)	1000000000000	1012	12
quadrillion	peta- (P)	1000000000000000	1015	15
quintillion	exa- (E)	1000000000000000000	1018	18
sextillion	zetta- (Z)	1000000000000000000000	1021	21
septillion	yotta- (Y)	1000000000000000000000000	1024	24
octillion	ronna- (R)	1000000000000000000000000000	1027 Zdroj:https://en.wikipedia.org?pojem=Order_of_magnitude >Text je dostupný pod licencí Creative Commons Uveďte autora – Zachovejte licenci, případně za dalších podmínek. Podrobnosti naleznete na stránce Podmínky užití.  Navigace: Módní doplňky > Šperkařství Šperky Chytré hodinky Galanterie Hodinářské firmy Hodinky Hodiny Módní doplňky Minerály Pokrývky hlavy Sklářství Zlatníci Zlatnictví Příbuzné výrazy : Řetízek (ozdoba) Šátek Beadweaving Bižuterie Bolo Dřevěný motýlek Deštník Digitálky Faléra Fiží Galanterie Hodinky Kšandy Kapesník Korálek Kravata Lanyard Modista Motýlek (oděvní doplněk) Odznak Pléd Placka (odznak) Rukávník Spona na kravatu Sponka do vlasů Tabulky (kolárek) Vějíř Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia. Copyright.sk © 2024