Ten, Powers of.
This adjunct to calculations has become almost indispensable in working
with units of the C. G. S. system. It consists in using some power of 10
as a multiplier which may be called the factor. The number multiplied
may be called the characteristic. The following are the general
principles.
The power of 10 is shown by an exponent which indicates the number of
ciphers in the multiplier. Thus 10^2 indicates 100; 10^3 indicates 1,000
and so on.
The exponent, if positive, denotes an integral number, as shown in the
preceding paragraph. The exponent, if negative, denotes the reciprocal
of the indicated power of 10. Thus 10^-2 indicates 1/100; 10^-3 indicates
1/1000 and so on.
The compound numbers based on these are reduced by multiplication or
division to simple expressions. Thus: 3.14 X 10^7 = 3.14 X 10,000,000 =
31,400,000. 3.14 X 10^-7 = 3.14/10,000,000 or 314/1000000000. Regard must
be paid to the decimal point as is done here.
To add two or more expressions in this notation if the exponents of the
factors are alike in all respects, add the characteristics and preserve
the same factor. Thus:
(51X 10^6) + (54 X 10^6) = 105 X 10^6.
(9.1 X 10^-9) + (8.7 X 10^-9) = 17.8 X 10^-9.
To subtract one such expression from another, subtract the
characteristics and preserve the same factor. Thus:
(54 X 10^6) - (51 X 10^6) = 3 X 10^6.
If the factors have different exponents of the same sign the factor or
factors of larger exponent must be reduced to the smaller exponent, by
factoring. The characteristic of the expression thus treated is
multiplied by the odd factor. This gives a new expression whose
characteristic is added to the other, and the factor of smaller exponent
is preserved for both,
Thus:
(5 X 10^7) + (5 X10^9) = (5 X 10^7) + (5 X 100 X 10^7) = 505 X 10^7.
The same applies to subtraction. Thus:
(5 X 10^9) - (5 X 10^7) = (5 X 100 X 10^7) - (5 X 10^7) = 495 X 10^7.
If the factors differ in sign, it is generally best to leave the
addition or subtraction to be simply expressed. However, by following
the above rule, it can be done. Thus:
Add
5 X 10^-2 and 5 X 10^3.
5 X 10^3 = 5 X 10^5 X 10^-2
(5 X 10^5 X 10^-2) + (5 X 10^-2) = 500005 X 10^-2
This may be reduced to a fraction 500000/100 = 5000.05.
To multiply add the exponents of the factors, for the new factor, and
multiply the characteristics for a new characteristic. The exponents
must be added algebraically; that is, if of different signs the
numerically smaller one is subtracted from the other one, and its sign
is given the new exponent.
Thus;
(25 X 10^6) X (9 X 10^8) = 225 X 10^14.
(29 X 10^ -8) X (11 X 10^7) = 319 X 10^-1
(9 X 10^8) X (98 X 10^2) = 882 X 10^1