Scientific Notation
The large numbers that characterize many scientific problems become much easier to manage if you employ scientific notation. Scientific notation involves taking a large, bulky number and replacing it with an expression that involves two numbers that express the value of the large, bulky number but do so in a more compact way. Of the two numbers, the first usually consists of a rational number less than 10. For example, you might see 3.4, 4.5, or 9.2. The second consists of 10 with an exponent. For example, you might see 102, 1022, or 1022. The exponent can be negative or positive, depending on whether you are addressing a number less than 1 or a number greater than 1. An example of a number less than one is 0.0003.
The first part of an expression given in scientific notation is usually called the coefficient of the number. The second part offers 10 raised exponentially. Consider, for example, the distance to the sun in miles. Expressed without scientific notation, this number is usually rounded to 92.9 million miles, which you express in this way:
92, 900, 000
Expressed in terms of scientific notation, this number becomes
9.29 × 107
Table 3.2 provides a few other representative values expressed in scientific notation.
| Item | Scientific Notation | As a Number |
|---|---|---|
| Distance from sun to farthest galaxy | 1.49 × 1010 light years | 14,900,000,000 |
| Distance from sun to Andromeda | 2.14 × 106 light years | 2,140,000 |
| Distance to farthest object yet seen | 1.57 × 1010 light years | 15,700,000,000 |
| Age of the solar system | 4.6 × 109 years | 4,600,000,000 |
| Age of the universe | 1.65 × 1010 years | 16,500,000,000 |
| Mass of the earth | 6.6 × 1021 tons | |
| Speed of light in miles per second | 1.86 × 105 | 186,000 |
| 100 | 1.0 × 102 | 100 |
| 1/10,000 | 1.0 × 10-4 | 0.0001 |
| Mass of an electron | 9.1 × 10-31 kilograms | |
| Wavelength of a gamma ray | 3.0 × 10-13 centimeter | |
| Planck’s Constant | 6.626 × 10-34 joules |
To represent the mass of the earth, you shift the decimal point to the right. The figure of 6.6 × 1021 becomes
6,600,000,000,000,000,000,000
Very small numbers, such as Planck’s Constant, are represented with a negative exponent, and the effect is to shift the decimal point 34 places to the left. Represented literally, 6.626 × 10-34 becomes
0.0000000000000000000000000000000006626
Carrying out calculations using scientific notation involves performing the usual mathematical operations with the coefficients, and then using the practices that pertain to exponents to deal with the powers of 10. Consider, for example, the problem of how far light travels in a year. If you begin with the speed of light as shown in Table 3.2, it is 1.86 × 105. On the other hand, there are (generally speaking) 365 days per year. To calculate the number of seconds in a year, you can use the relationship of days to hours, hours to minutes, and minutes to seconds: 365 × 24 × 60 × 60 = 31, 536, 000 seconds per year.
Expressed scientifically, you have 3.1536 × 107 seconds. To calculate the distance light travels in years you can set up the following expression:
(1.86 × 105)(3.1536 × 107) =
(1.86 × 3.1536) × 105+7 = 5.865696 × 1012 miles per year