Power-of-10 Notation for Physics Help

Orders Of Magnitude

As you can see, power-of-10 notation makes it possible to easily write down numbers that denote unimaginably gigantic or tiny quantities. Consider the following:

2.55 × 10 ^45,589

−9.8988 × 10 ^−7,654,321

Imagine the task of writing either of these numbers out in ordinary decimal form! In the first case, you’d have to write the numerals 255 and then follow them with a string of 45,587 zeros. In the second case, you’d have to write a minus sign, then a numeral zero, then a radix point, then a string of 7,654,320 zeros, and then the numerals 9, 8, 9, 8, and 8.

Now consider these two numbers:

2.55 × 10 ^45,592

−9.8988 × 10 ^−7,654,318

These look a lot like the first two, don’t they? However, both these new numbers are 1,000 times larger than the original two. You can tell by looking at the exponents. Both exponents are larger by 3. The number 45,592 is 3 more than 45,589, and the number −7,754,318 is 3 larger than −7,754,321. (Numbers grow larger in the mathematical sense as they become more positive or less negative.) The second pair of numbers is three orders of magnitude larger than the first pair of numbers. They look almost the same here, and they would look essentially identical if they were written out in full decimal form. However, they are as different as a meter is from a kilometer.

The order-of-magnitude concept makes it possible to construct number lines, charts, and graphs with scales that cover huge spans of values. Three examples are shown in Fig. 2-1. Part a shows a number line spanning three orders of magnitude, from 1 to 1,000. Part b shows a number line spanning 10 orders of magnitude, from 10 ⁻³ to 10 ⁷ . Part c shows a graph whose horizontal scale spans 10 orders of magnitude, from 10 ⁻³ to 10 ⁷ , and whose vertical scale extends from 0 to 10.

Fig. 2-1 . (a) A number line spanning three orders of magnitude. (b) A number line spanning 10 orders of magnitude. (c) A coordinate system whose horizontal scale spans 10 orders of magnitude and whose vertical scale extends from 0 to 10.

If you’re astute, you’ll notice that while the 0-to-10 scale is the easiest to envision directly, it covers more orders of magnitude than any of the others: infinitely many. This is so because no matter how many times you cut a nonzero number to 1/10 its original size, you can never reach zero.

Practice problems for these concepts can be found at: Scientific Notation for Physics Practice Test