Order of magnitude

An order of magnitude is an approximate measure of the number of digits that a number has in the commonly-used base-ten number system. It is equal to the whole number floor of logarithm (base 10). For example, the order of magnitude of 1500 is 3, because 1500 = 1.5 × 103. An order of magnitude is an approximate measure of the number of digits that a number has in the commonly-used base-ten number system. It is equal to the whole number floor of logarithm (base 10). For example, the order of magnitude of 1500 is 3, because 1500 = 1.5 × 103. Differences in order of magnitude can be measured on a base-10 logarithmic scale in “decades” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers). Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number. To work out the order of magnitude of a number N {displaystyle N} , the number is first expressed in the following form: where 10 10 ≤ a < 10 {displaystyle {frac {sqrt {10}}{10}}leq a<{sqrt {10}}} . Then, b {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: The geometric mean of 10 b {displaystyle 10^{b}} and 10 b + 1 {displaystyle 10^{b+1}} is 10 × 10 b {displaystyle {sqrt {10}} imes 10^{b}} , meaning that a value of exactly 10 b {displaystyle 10^{b}} (i.e., a = 1 {displaystyle a=1} ) represents a geometric 'halfway point' within the range of possible values of a {displaystyle a} . Some use a simpler definition where 0.5 < a ≤ 5 {displaystyle 0.5<aleq 5} , perhaps because the arithmetic mean of 10 b {displaystyle 10^{b}} and 10 b + c {displaystyle 10^{b+c}} approaches 5 × 10 b + c − 1 {displaystyle 5 imes 10^{b+c-1}} for increasing c {displaystyle c} . This definition has the effect of lowering the values of b {displaystyle b} slightly: Yet others restrict a {displaystyle a} to values where 1 ≤ a < 10 {displaystyle 1leq a<10} , making the order of magnitude of a number exactly equal to its exponent part in scientific notation. 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 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 4000000 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 6. An order of magnitude is an approximate position on a logarithmic scale.