Order of magnitude

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 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. 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 , the number is first expressed in the following form: where , or approximately . Then, 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 and is , meaning that a value of exactly (i.e., ) represents a geometric halfway point within the range of possible values of . Some use a simpler definition where , perhaps because the arithmetic mean of and approaches for increasing . This definition has the effect of lowering the values of slightly: 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.