Page 21 - Schaum's Outline of Theory and Problems of Applied Physics

6 USEFUL MATH [CHAP. 1

10 −4 = 0.0001 = 1 with decimal point moved 4 places to the left
10 −5 = 0.00001 = 1 with decimal point moved 5 places to the left
10 −6 = 0.000001 = 1 with decimal point moved 6 places to the left
0
10 = 1 = 1 with decimal point moved 0 places
1
10 = 10 = 1 with decimal point moved 1 place to the right
2
10 = 100 = 1 with decimal point moved 2 places to the right
3
10 = 1000 = 1 with decimal point moved 3 places to the right
4
10 = 10,000 = 1 with decimal point moved 4 places to the right
5
10 = 100,000 = 1 with decimal point moved 5 places to the right
6
10 = 1,000,000 = 1 with decimal point moved 6 places to the right
When numbers written in powers-of-10 notation are to be added or subtracted, they must all be expressed
in terms of the same power of 10:
3
3
2
3
3 × 10 + 4 × 10 = 0.3 × 10 + 4 × 10 = 4.3 × 10 3
It does not matter which power of 10 is used, as long as it is the same for all the numbers. Thus we get the same
answer as above if we proceed instead as follows:
2
3
2
2
2
3 × 10 + 4 × 10 = 3 × 10 + 40 × 10 = 43 × 10 = 4.3 × 10 3
Since a step is saved if the power of 10 used is that of the larger number, it makes sense to do this.
To subtract one number from another, the same procedure is followed. If the number being subtracted is the
larger of the two, the answer will be negative in sign, just as 3 − 5 =−2.
To multiply two powers of 10, add their exponents; to divide one power of 10 by another, subtract the
exponent of the latter from that of the former:
10 n n−m
n
m
n+m
(10 )(10 ) = 10 = 10
10 m
Reciprocals follow the pattern
1 −n
= 10
10 n
The rules of finding powers and roots of powers of 10 are
√
n 1/m
n m
n
(10 ) = 10 nm m 10 = (10 ) = 10 n/m
In taking the mth root, the power of 10 should be chosen to be a multiple of m. Thus
√ √ √ √
7
14
10 15 = ( 10)( 10 ) = 10 × 10 = 3.16 × 10 7
SOLVED PROBLEM 1.13
Examples of powers-of-10 notation:

20 = 2 × 10 = 2 × 10 1
3043 = 3.043 × 1000 = 3.043 × 10 3
8, 700,000 = 8.7 × 1,000,000 = 8.7 × 10 6
0.22 = 2.2 × 0.1 = 2.2 × 10 −1

0.000035 = 3.5 × 0.00001 = 3.5 × 10 −5