# Order of Magnitude Calculations

In order to express quantitative magnitudes only approximately, the concept of order-of-magnitude is often employed. For example, the height of a small insect, say an ant, might be 8 x 10¯⁴ m ≈ 10¯³. We would say that the order of magnitude of the height of an ant is 10¯³m. Similarly, though the height of most people is about 2 m, we might round that off and say that the order of magnitude of the height of a person is 10⁰ m. By this we do not mean to imply that a typical height is really 1 m but that it is closer to 1 m than to 10 m or to 10¯¹m = 0.1 m. We might say that a typical human being is 3 orders of magnitude taller than a typical ant, meaning that the ratio of their heights is about 10³.

The order-of-magnitude of a given number is the nearest power of ten to which it is approximated.

The operational definition for the order of magnitude (x) of a number (n) is 0.5 < n/10ͯˣ ≤ 5

The order-or-magnitude of some numbers is given below.

 Number (n) Order-of-magnitude (x) 8 1 as 8/10 = 0.8 < 0.5, i.e., 8 ≈ 10¹ 147 2 as 147/10² = 1.47 < 5, i.e., 147 ≈ 10² 499 2 as 499/10² = 4.955 < 5, i.e.,  499 ≈ 10² 501 3 as 501/10³ = 0.501 > 0.5 i.e., 501 ≈ 10³ 7853 4 as 7853/10⁴ = 0.7853 > 0.5 i.e., 7583 ≈ 10⁴ 0.187 -1 as 0.187/10¯¹ < 5 i.e., 0.187 ≈ 10¯¹ 0.05 -2 as 0.05/10¯² = 5 i.e., 0.05 ≈ 10¯²