1. 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
529
To divide, subtract (algebraically) the exponent of the divisor from that of the dividend for the exponent of the new factor, and divide the characteristics one by the other for the new characteristic. Algebraic subtraction is effected by changing the sign of the subtrahend, subtracting the numerically smaller number from the larger, and giving the result the sign of the larger number. (Thus to subtract 7 from 5 proceed thus; 5 - 7 = -2.)
Thus; (25 X 10^6) / (5 X 10^8) = 5 X 10^-2 (28 X 10^-8) / (5 X 10^3) = 5.6 X 10^-11
[Transcriber's note: I have replaced ordinary exponential notation by the more compact and simpler "programming" representation. The last two example would be: 25E6 / 5E8 = 5E-2 28E-8 / 5E3 = 5.6E-11 ]
Definitions of ten, powers of
