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10 × 10 × 10 × 10. Ten times itself four times is 10,000, so
4
10 is the scientific notation for 10,000. It is also equal to the EXAMPLE A.7
−4
number of zeros between the 1 and the decimal point; that is, What is 0.000732 in scientific notation? (Answer: 7.32 × 10 )
4
to write the longer form of 10 , you simply write 1, then move
the decimal point four places to the right; 10 to the fourth
power is 10,000. It was stated earlier that scientific notation provides
The power of ten table also shows that numbers smaller than 1 a compact way of dealing with very large or very small
have negative exponents. A negative exponent means a reciprocal: numbers, but it provides a simplified way to make calcula-
tions as well. There are a few mathematical rules that will
–1 _
1
10 = = 0.1 describe how the use of scientific notation simplifies these

10
calculations.
–2 _
1
10 = To multiply two scientific notation numbers, the coeffi-
= 0.01

100
cients are multiplied as usual, and the exponents are added alge-
3
2
1
–3
= 0.001
10 = _ braically. For example, to multiply (2 × 10 ) by (3 × 10 ), first
1,000 separate the coefficients from the exponents,
–4
To write the longer form of 10 , you simply write 1 and then (2 × 3) × (10 × 10 )
3
2
move the decimal point four places to the left ; 10 to the negative
fourth power is 0.0001. then multiply the coefficients and add the exponents.
Scientific notation usually, but not always, is expressed as 6 × 10 (2 + 3) = 6 × 10 5
the product of two numbers: (1) a number between 1 and 10 2 3
that is called the coeffi cient and (2) a power of ten that is called Adding the exponents is possible because 10 × 10 means
the exponent. For example, the mass of the Sun that was given in the same thing as (10 × 10) × (10 × 10 × 10), which equals
5
long form earlier is expressed in scientific notation as (100) × (1,000), or 100,000, which is expressed as 10 in
scientific notation. Note that two negative exponents add alge-
30
–2
–3
–5
1.97 × 10 kg braically; for example 10 × 10 = 10 [(–2) + (–3)] = 10 . A
negative and a positive exponent also add algebraically, as in
and the radius of a hydrogen atom is 5 –3 [(+5) + (–3)] 2
10 × 10 = 10 = 10 .
–11
5.0 × 10 m If the result of a calculation involving two scientific no-
tation numbers does not have the conventional one digit to
In these expressions, the coefficients are 1.97 and 5.0, and the
the left of the decimal, move the decimal point so it does,
power of ten notations are the exponents. Note that in both of
changing the exponent according to which way and how
these examples, the exponent tells you where to place the deci-
much the decimal point is moved. Note that the exponent in-
mal point if you wish to write the number all the way out in
creases by one number for each decimal point moved to the
the long form. Sometimes scientific notation is written with-
left. Likewise, the exponent decreases by one number for each
out a coefficient, showing only the exponent. In these cases, 3
decimal point moved to the right. For example, 938. × 10
the coefficient of 1.0 is understood, that is, not stated. If you 5
becomes 9.38 × 10 when the decimal point is moved two
try to enter a scientific notation in your calculator, however,
places to the left.
you will need to enter the understood 1.0, or the calculator will
30
not be able to function correctly. Note also that 1.97 × 10 kg To divide two scientific notation numbers, the coefficients
31
29
and the expressions 0.197 × 10 kg and 19.7 × 10 kg are are divided as usual and the exponents are subtracted. For
6
2
example, to divide (6 × 10 ) by (3 × 10 ), first separate the
all correct expressions of the mass of the Sun. By convention,
coefficients from the exponents,
however, you will use the form that has one digit to the left of
2
the decimal. (6 ÷ 3) × (10 ÷ 10 )
6
then divide the coefficients and subtract the exponents.
2 × 10 (6 − 2) = 2 × 10 4
EXAMPLE A.6
Note that when you subtract a negative exponent, for ex-
What is 26,000,000 in scientific notation? [(3) − (−2)] (3 + 2) 5
ample, 10 , you change the sign and add, 10 = 10 .
SOLUTION
Count how many times you must shift the decimal point until one digit
remains to the left of the decimal point. For numbers larger than the
digit 1, the number of shifts tells you how much the exponent is in- EXAMPLE A.8
creased, so the answer is
Solve the following problem concerning scientific notation:
2.6 × 10 7 4 –6
(2 × 10 ) × (8 × 10 )
__

which means the coefficient 2.6 is multiplied by 10 seven times. 8 × 10 4
628 APPENDIX A Mathematical Review A-6