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Perform the divisions.

5. 4.19 ÷ 0.5

6. 4.09 ÷ 0.21

7. 353 ÷ 2.5

8. 0.004 ÷ 0.002

9. 29.997 ÷ 0.01

4.10 Order of magnitude, Scientific notation

In the sciences, we often see very large and very small numbers. Forexample, the distance that light
travels in one year (known as a “light-year”) is approximately

5, 870, 000, 000, 000 miles.

At the other extreme, the mass of an electron is approximately

0.000 000 000 000 000 000 000 000 000 91 grams.

The order of magnitude of a number N is defined to be the unique integer m such that the absolute
value of
N × 10 −m
is a number between 1 and 10 (including 1 but excluding 10.)

Example 172. Find the orders of magnitude of (a) the distance light travels in a year; (b) the mass of
an electron.

Solution. (a) The decimal point in 5, 870, 000, 000, 000 must be moved left 12 places (count them!) to
obtain a number between 1 and 10 (namely, 5.87). This is equivalent to multiplying by 10 −12 .Thus,
one light year has order of magnitude 12. (b) The decimal point in

0.000 000 000 000 000 000 000 000 000 91

must be moved right 28 (!) decimal places to obtain a number between 1 and 10 (namely, 9.1). This
is equivalent to multiplying by 10 28 =10 −(−28) . So the tiny mass of an electron has negative order of
magnitude −28.

‘Small’ positive numbers (less than 1) have negative order of magnitude; ‘large’
numbers (greater than or equal to 10) have positive order of magnitude. Numbers
between 1 and 10 but not equal to 10 have order of magnitude 0. (10 itself has
order of magnitude 1.)

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