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| An '''order of magnitude''' is a scale of [[numbers]] with a fixed [[Geometric progression|ratio]], often [[rounded]] to the nearest ten.
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| The orders of magnitude are written in [[powers of ten]]. For example, the order of magnitude of 1500 is 3, since 1500 may be written as 1.5 × 10<sup>3</sup>.
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| [[Differences]] in order of magnitude can be [[measured]] on the [[logarithmic scale]] in "[[Decade (log scale)|decades]]" (i.e., factors of ten).<ref>{{cite web
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| |url = http://public.wsu.edu/~brians/errors/orders.html
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| |title = Orders of Magnitude
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| |last = Brians
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| |first = Paus
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| |accessdate = 5/9/2013
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| }}</ref> Examples of numbers of different magnitudes can be found at [[Orders of magnitude (numbers)]].
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| {{quotation|We say two numbers have the same order of magnitude of a number if the big one divided by the little one is less than 10. For example, 23 and 82 have the same order of magnitude, but 23 and 820 do not."<ref>[[John_Baez|John C. Baez]], 11/28/2012</ref> — [[John Baez]]}}
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| ==Uses==
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| Orders of magnitude are used to make approximate comparisons. If numbers differ by 1 order of magnitude, ''x'' is ''about'' ten times different in quantity than ''y''. If values differ by 2 orders of magnitude, they differ by a factor of about [[100 (number)|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.
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| {| class="wikitable"
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| ! In words<br>([[Long and short scales|long scale]])
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| ! In words<br>([[Long and short scales|short scale]])
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| ! Prefix
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| ! Symbol
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| ! Decimal
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| ! [[Exponent#Powers of ten|Power]]<br />of ten
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| ! Order of<br />magnitude
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| |-
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| | quadrillionth
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| | septillionth
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| | yocto-
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| | y
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| | 0.000,000,000,000,000,000,000,001
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| | 10<sup>−24</sup>
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| | −24
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| |-
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| | trilliardth
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| | sextillionth
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| | zepto-
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| | z
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| | 0.000,000,000,000,000,000,001
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| | 10<sup>−21</sup>
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| | −21
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| |-
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| | trillionth
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| | quintillionth
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| | atto-
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| | a
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| | 0.000,000,000,000,000,001
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| | 10<sup>−18</sup>
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| | −18
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| |-
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| | billiardth
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| | quadrillionth
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| | femto-
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| | f
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| | 0.000,000,000,000,001
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| | 10<sup>−15</sup>
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| | −15
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| |-
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| | billionth
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| | trillionth
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| | pico-
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| | p
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| | 0.000,000,000,001
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| | 10<sup>−12</sup>
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| | −12
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| |-
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| | milliardth
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| | billionth
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| | nano-
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| | n
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| | 0.000,000,001
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| | 10<sup>−9</sup>
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| | −9
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| |-
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| | millionth
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| | millionth
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| | micro-
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| | <nowiki>µ</nowiki>
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| | 0.000,001
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| | 10<sup>−6</sup>
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| | −6
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| |-
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| | thousandth
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| | thousandth
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| | milli-
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| | m
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| | 0.001
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| | 10<sup>−3</sup>
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| | −3
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| |-
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| | hundredth
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| | hundredth
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| | centi-
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| | c
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| | 0.01
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| | 10<sup>−2</sup>
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| | −2
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| |-
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| | tenth
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| | tenth
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| | deci-
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| | d
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| | 0.1
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| | 10<sup>−1</sup>
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| | −1
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| |-
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| | one
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| | one
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| | –
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| | –
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| | 1
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| | 10<sup>0</sup>
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| | 0
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| |-
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| | ten
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| | ten
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| | deca-
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| | da
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| | 10
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| | 10<sup>1</sup>
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| | 1
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| |-
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| | hundred
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| | hundred
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| | hecto-
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| | h
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| | 100
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| | 10<sup>2</sup>
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| | 2
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| |-
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| | thousand
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| | thousand
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| | kilo-
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| | k
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| | 1,000
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| | 10<sup>3</sup>
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| | 3
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| |-
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| | million
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| | million
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| | mega-
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| | M
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| | 1,000,000
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| | 10<sup>6</sup>
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| | 6
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| |-
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| | milliard
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| | billion
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| | giga-
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| | G
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| | 1,000,000,000
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| | 10<sup>9</sup>
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| | 9
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| |-
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| | billion
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| | trillion
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| | tera-
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| | T
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| | 1,000,000,000,000
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| | 10<sup>12</sup>
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| | 12
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| |-
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| | billiard
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| | quadrillion
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| | peta-
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| | P
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| | 1,000,000,000,000,000
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| | 10<sup>15</sup>
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| | 15
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| |-
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| | trillion
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| | quintillion
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| | exa-
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| | E
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| | 1,000,000,000,000,000,000
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| | 10<sup>18</sup>
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| | 18
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| |-
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| | trilliard
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| | sextillion
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| | zetta-
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| | Z
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| | 1,000,000,000,000,000,000,000
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| | 10<sup>21</sup>
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| | 21
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| |-
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| | quadrillion
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| | septillion
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| | yotta-
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| | Y
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| | 1,000,000,000,000,000,000,000,000
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| | 10<sup>24</sup>
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| | 24
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| |}
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| The '''order of magnitude of a number''' is, intuitively speaking, the number of powers of 10 contained in the number. More precisely, the order of magnitude of a number can be defined in terms of the [[common logarithm]], usually as the [[integer]] part of the logarithm, obtained by [[truncation]]. For example, the number 4,000,000 has a logarithm (in base 10) of 6.602; its order of magnitude is 6. When truncating, a number of this order of magnitude is between 10<sup>6</sup> and 10<sup>7</sup>. In a similar example, with the phrase "He had a seven-figure income", the order of magnitude is the number of figures minus one, so it is very easily determined without a calculator to be 6. An order of magnitude is an approximate position on a [[logarithmic scale]].
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| An '''order-of-magnitude estimate''' of a variable whose precise value is unknown is an estimate [[Rounding|rounded]] to the nearest power of ten. For example, an order-of-magnitude estimate for a variable between about 3 billion and 30 billion (such as the [[human]] [[population]] of the [[Earth]]) is 10 [[1000000000 (number)|billion]]. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus 4,000,000, which has a logarithm (in base 10) of 6.602, has 7 as its nearest order of magnitude, because "nearest" implies rounding rather than truncation. For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten (about 3.162). For example, the nearest order of magnitude for 1.7 × 10<sup>8</sup> is 8, whereas the nearest order of magnitude for 3.7 × 10<sup>8</sup> is 9. An order-of-magnitude estimate is sometimes also called a [[zeroth order approximation]].
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| An '''order-of-magnitude difference''' between two values is a factor of 10. For example, the mass of the planet [[Saturn]] is 95 times that of [[Earth]], so Saturn is ''two orders of magnitude'' more massive than Earth. Order-of-magnitude differences are called '''[[decade (log scale)|decades]]''' when measured on a [[logarithmic scale]].
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| ==Non-decimal orders of magnitude==
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| {{see also|Logarithmic scale}}
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| Other orders of magnitude may be calculated using [[Radix|bases]] other than 10. The ancient Greeks ranked the nighttime brightness of celestial bodies by 6 levels in which each level was the fifth root of one hundred (about 2.512) as bright as the nearest weaker level of brightness{{citation needed|date=July 2013}}, so that the brightest level is 5 '''orders of magnitude''' brighter than the weakest, which can also be stated as a '''factor''' of 100 times brighter.
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| The different [[decimal]] [[numeral systems]] of the world use a larger base to better envision the size of the number, and have created names for the powers of this larger base. The table shows what number the order of magnitude aim at for base 10 and for base 1,000,000. It can be seen that the order of magnitude is included in the number name in this example, because bi- means 2 and tri- means 3 (these make sense in the long scale only), and the suffix -illion tells that the base is 1,000,000. But the number names billion, trillion themselves (here with [[Long and short scales|other meaning]] than in the first chapter) are not names of the ''orders of'' magnitudes, they are names of "magnitudes", that is the ''numbers'' 1,000,000,000,000 etc.
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| {| class="wikitable"
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| ! order of magnitude !! is [[Common logarithm|log<sub>10</sub>]] of !! is log<sub>1000000</sub> of !! short scale !! long scale
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| |-
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| | 1 || 10 || 1,000,000 || million || million
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| |-
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| | 2 || 100 || 1,000,000,000,000 || trillion || billion
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| |-
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| | 3 || 1000 || 1,000,000,000,000,000,000 || quintillion || trillion
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| |}
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| [[SI]] units in the table at right are used together with [[SI prefix]]es, which were devised with mainly base 1000 magnitudes in mind. [[Binary prefix#IEC standard prefixes|The IEC standard prefixes]] with base 1024 were invented for use in electronic technology.
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| The ancient [[apparent magnitude]]s for the brightness of stars uses the base <math>\sqrt[5]{100} \approx 2.512</math> and is reversed. The modernized version has however turned into a logarithmic scale with non-integer values.
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| ===Extremely large numbers===
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| For extremely [[large numbers]], a generalized order of magnitude can be based on their [[Logarithm#Inverses of other exponential functions|double logarithm]] or [[super-logarithm]]. Rounding these downward to an integer gives categories between very "round numbers", rounding them to the nearest integer and applying the inverse function gives the "nearest" round number.
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| The double logarithm yields the categories:
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| : ..., 1.0023–1.023, 1.023–1.26, 1.26–10, 10–10<sup>10</sup>, 10<sup>10</sup>–10<sup>100</sup>, 10<sup>100</sup>–10<sup>1000</sup>, ...
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| (the first two mentioned, and the extension to the left, may not be very useful, they merely demonstrate how the sequence mathematically continues to the left).
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| The super-logarithm yields the categories:
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| : <math>0-1, 1-10, 10-10^{10}, 10^{10}-10^{10^{10}}, 10^{10^{10}}-10^{10^{10^{10}}}, \dots</math>, or
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| :negative numbers, 0–1, 1–10, 10–1e10, 1e10–10<sup>1e10</sup>, 10<sup>1e10</sup>–<sup>4</sup>10, <sup>4</sup>10–<sup>5</sup>10, etc. (see [[tetration]])
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| The "midpoints" which determine which round number is nearer are in the first case:
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| :1.076, 2.071, 1453, 4.20e31, 1.69e316,...
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| and, depending on the interpolation method, in the second case | |
| :−.301, .5, 3.162, 1453, 1e1453, <math>(10 \uparrow)^1 10^{1453}</math>, <math>(10 \uparrow)^2 10^{1453}</math>,... (see [[Large numbers#Standardized system of writing very large numbers|notation of extremely large numbers]])
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| For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but of course the generalized order of magnitude of the [[Reciprocal (mathematics)|reciprocal]] can be considered.
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| Similar to the [[logarithmic scale#Graphic representation|logarithmic scale]] one can have a double logarithmic scale (example provided [[Graphical timeline from Big Bang to Heat Death|here]]) and super-logarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the [[generalised f-mean]] with ''f''(''x'') the corresponding function log log ''x'' or slog ''x''. In the case of log log ''x'', this mean of two numbers (e.g. 2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log ''x'' ([[geometric mean]], 2 and 8 giving 4), but unlike in the case of log log log ''x'' (4 and 65536 giving 16 if the base is 2, but, otherwise).
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| ==See also==
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| * [[Big O notation]]
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| * [[Decibel]]
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| * [[Names of large numbers]]
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| * [[Names of small numbers]]
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| * [[Number sense]]
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| * [[Orders of approximation]]
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| * [[Orders of magnitude (numbers)]]
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| ==References==
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| <references />
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| ==Further reading==
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| * [[Isaac Asimov|Asimov, Isaac]] ''The Measure of the Universe'' (1988)
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| ==External links==
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| *[http://htwins.net/scale2/ The Scale of the Universe 2 ] Interactive tool from [[Planck length]] 10<sup>−35</sup> meters to universe size 10<sup>27</sup>
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| *[http://www.shekpvar.net/~dna/Publications/Cosmos/cosmos.html Cosmos – an Illustrated Dimensional Journey from microcosmos to macrocosmos] – from Digital Nature Agency
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| *[http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html Powers of 10], a graphic animated illustration that starts with a view of the [[Milky Way]] at 10<sup>23</sup> meters and ends with [[subatomic particle]]s at 10<sup>−16</sup> meters.
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| *[http://www.vendian.org/envelope/TemporaryURL/what_is_oom.html What is Order of Magnitude?]
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| {{orders of magnitude wide}}
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| {{DEFAULTSORT:Order Of Magnitude}}
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| [[Category:Orders of magnitude| ]]
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| [[Category:Elementary mathematics]]
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