Order of magnitude

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Order of magnitude is a concept used to discuss the scale of numbers in relation to one another.

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-10 logarithmic 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 at Orders 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.