In aratio scale based onpowers of ten, theorder of magnitude is a measure of the nearness of two figures. 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 about a factor of 10 of each other.[1]
For example, 1 and 1.02 are within an order of magnitude. So are 1 and 2, 1 and 9, or 1 and 0.2. 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-10logarithmic 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] This phrasing helps quickly express the difference in scale between 2 and 2,000,000: they differ by 6 orders of magnitude.
Examples of numbers of different magnitudes can be found atOrders 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.
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, the number is first expressed in the following form:
where, or approximately.[5] Then, represents the order of magnitude of the number. The order of magnitude can be anyinteger. The table below enumerates the order of magnitude of some numbers using this definition:
| Number | Expression in | Order of magnitude |
|---|---|---|
| 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 |
Thegeometric mean of and is, meaning that a value of exactly (i.e.,) represents a geometrichalfway point within the range of possible values of.
Some use a simpler definition where.[6] This definition has the effect of lowering the values of slightly:
| Number | Expression in | Order of magnitude |
|---|---|---|
| 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, one number is about 10 times larger than the other. 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 newSI prefixes over time, most recently in 2022.[7]
| 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 | 27 |
| nonillion | quetta- (Q) | 1000000000000000000000000000000 | 1030 | 30 |
| In words | Prefix (Symbol) | Decimal | Power of ten | Order of magnitude |
The order of magnitude of a number is, intuitively speaking, the number of digits in it which are above the ones place. More precisely, the order of magnitude of a number can be defined in terms of thecommon logarithm, usually as theinteger part of the logarithm, obtained bytruncation. For example, the number4000000 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 106 and 107. In a similar example, with the phrase "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 alogarithmic scale.
An order-of-magnitude estimate of a variable, whose precise value is unknown, is an estimaterounded 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 humanpopulation of theEarth) is 10billion. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus4000000, 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 thesquare root of ten (about 3.162).[8] For example, the nearest order of magnitude for1.7×108 is 8, whereas the nearest order of magnitude for3.7×108 is 9. An order-of-magnitude estimate is sometimes also called azeroth order approximation.
An order of magnitude is an approximation of thelogarithm of a value relative to some contextually understood reference value, usually 10, interpreted as thebase 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 abinary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value.
Other orders of magnitude may be calculated using bases other than integers. In the field ofastronomy, the nighttime brightnesses of celestial bodies are ranked by"magnitudes" in which each increasing level is brighter by afactor of greater than the previous level. Thus, a level being 5 magnitudes brighter than another indicates that it is a factor of times brighter: that is, two base 10 orders of magnitude.
This series of magnitudes forms a logarithmic scale with a base of.
The differentdecimalnumeral 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 base1000000. It can be seen that the order of magnitude is included in the number name in this example, because bi- means 2, tri- means 3, etc. (these make sense in the long scale only), and the suffix -illion tells that the base is1000000. But the number names billion, trillion themselves (here withother meaning than in the first chapter) are not names of theorders of magnitudes, they are names of "magnitudes", that is thenumbers1000000000000 etc.
| Order of magnitude | Islog10 of | Is log1000000 of | Short scale | Long scale |
|---|---|---|---|---|
| 1 | 10 | 1000000 | million | million |
| 2 | 100 | 1000000000000 | trillion | billion |
| 3 | 1000 | 1000000000000000000 | quintillion | trillion |
| 4 | 10000 | (1 000 000)4 | septillion | quadrillion |
| 5 | 100000 | (1 000 000)5 | nonillion | quintillion |
SI units in the table at right are used together withSI prefixes, which were devised with mainly base 1000 magnitudes in mind.The IEC standard prefixes with base 1024 were invented for use in electronic technology.
Two quantities A and B which are within about a factor of 10 of each other are then said to be "of the same order of magnitude," written A∼B.
Physicists and engineers use the phrase "order of magnitude" to refer to the smallest power of ten needed to represent a quantity.
The magnitude of a physical quantity, whose order of magnitude is to be determined, is written in the formN x 10, whereN is a number between 1 and 10 andx is a positive or negative integer. If N is equal to or smaller than√1x10 = 3.16, then the order of magnitude of the quantity is 10x. But, ifN is greater than 3.16, then the order of magnitude of the quantity is 10x+1.
The magnitude of a physical quantity, whose order of magnitude is to be determined, is written in the formN x 10, whereN is a number between 1 and 10 andx is a positive or negative integer. If N is equal to or smaller than√1x10 = 3.16, then the order of magnitude of the quantity is 10x. But, ifN is greater than 3.16, then the order of magnitude of the quantity is 10x+1.
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