Use
Orders of magnitude are generally used to make very approximate comparisons, and reflect very large differences. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they 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 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 106 and 107. 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.
An order-of-magnitude estimate of a variable whose precise value is unknown is an estimate 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 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 × 108 is 8, whereas the nearest order of magnitude for 3.7 × 108 is 9. An order-of-magnitude estimate is sometimes also called a zeroth order approximation.
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 decades when measured on a logarithmic scale.
| In words (long scale) |
In words (short scale) |
Prefix | Symbol | Decimal | Power of ten |
Order of magnitude |
|---|---|---|---|---|---|---|
| quadrillionth | septillionth | yocto- | y | 0.000,000,000,000,000,000,000,001 | 10−24 | −24 |
| trilliardth | sextillionth | zepto- | z | 0.000,000,000,000,000,000,001 | 10−21 | −21 |
| trillionth | quintillionth | atto- | a | 0.000,000,000,000,000,001 | 10−18 | −18 |
| billiardth | quadrillionth | femto- | f | 0.000,000,000,000,001 | 10−15 | −15 |
| billionth | trillionth | pico- | p | 0.000,000,000,001 | 10−12 | −12 |
| milliardth | billionth | nano- | n | 0.000,000,001 | 10−9 | −9 |
| millionth | millionth | micro- | µ | 0.000,001 | 10−6 | −6 |
| thousandth | thousandth | milli- | m | 0.001 | 10−3 | −3 |
| hundredth | hundredth | centi- | c | 0.01 | 10−2 | −2 |
| tenth | tenth | deci- | d | 0.1 | 10−1 | −1 |
| one | one | – | – | 1 | 100 | 0 |
| ten | ten | deca- | da | 10 | 101 | 1 |
| hundred | hundred | hecto- | h | 100 | 102 | 2 |
| thousand | thousand | kilo- | k | 1,000 | 103 | 3 |
| million | million | mega- | M | 1,000,000 | 106 | 6 |
| milliard | billion | giga- | G | 1,000,000,000 | 109 | 9 |
| billion | trillion | tera- | T | 1,000,000,000,000 | 1012 | 12 |
| billiard | quadrillion | peta- | P | 1,000,000,000,000,000 | 1015 | 15 |
| trillion | quintillion | exa- | E | 1,000,000,000,000,000,000 | 1018 | 18 |
| trilliard | sextillion | zetta- | Z | 1,000,000,000,000,000,000,000 | 1021 | 21 |
| quadrillion | septillion | yotta- | Y | 1,000,000,000,000,000,000,000,000 | 1024 | 24 |
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