We have argued previously that science deals with quantitative concepts, laws, and theories. Thus it is important that we know how to deal with numbers. Many times in science, especially in astronomy, extremely large or extremely small numbers confronts us. For example, the approximate number of stars in our Galaxy is 400 billion, or 400,000,000,000. Numbers of this size make the arithmetical operations of multiplication and division extremely cumbersome. Therefore, it is convenient to use what is known as scientific notation, which involves powers of ten. We begin by explaining exponents and their manipulations.
If n (known as an exponent) is a positive integer, then xn is defined as the number 1 multiplied n times by x; for example for n = 3, x raised to the third power is
x3 = 1*10*10*10.
By convention any number raised to the zeroth power is equal to 1. If a is the number 10, then we may form a table of powers of 10 with their accepted prefixes and symbols. For example, 1 million light years (ly) is 1 megalight year (Mly).
Powers of Ten Word Number Power Prefix Symbol trillion 1,000,000,000,000 1012 tera T billion 1,000,000,000 109 giga G million 1,000,000 106 mega M thousand 1,000 103 kilo k hundred 100 102 hecto h ten 10 101 deca da unit 1 100 tenth 0.1 10-1 deci d hundredth 0.01 10-2 centi c thousandth 0.001 10-3 milli m millionth 0.000,001 10-6 micro m billionth 0.000,000,001 10-9 nano n trillionth 0.000,000,000,001 10-12 pico p
There are rules for multiplication, division, raising to a power, or extracting a root for numbers expressed in exponent form:
Multiplication: Multiplication is accomplished by adding exponents. For example, where the dot denotes multiplication,
10-3*102*104 = 10-3+2+4 = 103 = 1000.
Division: Division is accomplished by subtracting exponents. For example,
104/102 = 104-2 = 102 = 100.
Exponentiation: Raising a number to a power is accomplished by multiplying exponents. For example,
(102)5 = 102*5 = 1010 = 10,000,000,000
(10-4)2 = 10-4*2 = 10-8 = 0.000,000,01
(2 x 102)3 = 23 x 102*3 = 8 x 106 = 8,000,000.
Roots: Extracting a root is also accomplished by multiplying exponents. For example,
(106)1/2 = 106*1/2 = 103 = 1000
(10-12)1/3 = 10-12*1/3 = 10-4 = 0.0001.
With these rules in mind we can now move to define scientific notation and consider some examples of its use:
Scientific Notation: Scientific notation is the expression of any number as the product of a number between 1 and 10 times a power of 10.
As examples of scientific notation for expressing numbers, consider the following examples:
- 2,380,000,000 may be written as 2.38 x 109
- 86,496 may be written as 8.6496 x 104
- 0.0005492 may be written as 5.492 x 10-4
From these examples a general rule is apparent for determining the numerical value and algebraic sign of the power of 10 to be used in expressing the number in scientific notation:
Determining the Power of Ten: The number of places the decimal point is shifted indicates the numerical value of the power of 10; if the shift is to the left, the algebraic sign is positive, and if the shift is to the right, the algebraic sign is negative.
For example, in the number 86,496 above, the decimal is shifted four places to the left so that the power of 10 is 4 and its algebraic sign is positive, or 8.6496 x 104. For the number 0.0005492 above the decimal is shifted four places to the right so that the power of 10 is also 4 but its algebraic sign is negative, or 5.492 x 10-4.
Applications of Scientific Notation
As mentioned above, scientific notation finds its greatest utility in multiplication or division, such as in Newton's law of gravitation:
F = Gm1m2/d2or F = (6.667 x 10-8 cm3/g*s2)(5.977 x 1027 g)(9.0 x 104 g)/(6.371 x 108 cm)2 = (358.6 x 1023) cm3*g*s2/(40.59 x 1016) g*s2*cm2 = (3.586 x 1025) g*cm/(4.059 x 1017) s2 = 8.834 x 107 dynes.
When numbers expressed in scientific notation are to be added or subtracted, then the following basic rule must be applied:
Addition and Subtraction: In addition or subtraction the power of 10 must be the same for all numbers to be added or subtracted, which will also be the appropriate power of 10 of the answer.
For example, let us add the mass of the Sun and Earth, as we might do in Newton's modified form of Kepler's third law:
Msun + Mearth = (1.989 x 1033 g) + (5.977 x 1027 g) = (1.989 x 1033 g) + (0.000005977 x 1033 g) = 1.989005977 x 1033 g.
It should also be noted that the units for each quantity in the last two examples are manipulated algebraically just as are the numbers. Thus in multiplication and division the units are multiplied and divided in order to obtain resultant units for the answer. And in addition and subtraction, numbers may be added or subtracted if they have the same units. Hence an extremely valuable means of verifying the correctness of a series of algebraic operations is to carry out first the operations with the units only. The result must be the correct units for the desired answer; that is if the desired answer is a force, then the units will not be those of energy.