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Scientific notation is a compact way of writing very large or very small numbers. A number written in scientific notation is expressed as a product of two numbers. a * 10^b In this form, the first factor is greater than or equal to 1 and less than 10. In other words, it needs to be in the interval [1,10). The second factor is a power of 10 where b is an integer. For example, the number 4 million can be rewritten as the product of 4 and a multiple of 10. Then, the multiple of 10 is rewritten as a base 10 power. 4 000 000 = 4 * 1 000 000 = 4 * 10^6 Very small decimal numbers can also be written in scientific notation. Consider a number where there are many zeros before the significant figures. Take as 0.000342 as an example. 0.000342 = 3.42 ÷ 10 000 = 3.42 * 10^(- 4) In such cases, numbers are expressed as a division by a multiple of 10. Division by a multiple of 10 is equivalent to multiplication by a base 10 power with a negative exponent. Consider a few more examples of numbers written in scientific notation.
| Decimal Form | Written as a Product or Division Expression | Scientific Notation |
|---|---|---|
| 4505 | 4.505 * 1000 | 4.505 * 10^3 |
| 8 320 000 | 8.32 * 1 000 000 | 8.32 * 10^6 |
| 0.0005 | 5 ÷ 10 000 | 5 * 10^(-4) |
| 0.0521 | 5.21 ÷ 100 | 5.21 * 10^(-2) |
An intuitive method to rewrite a number into scientific notation is to count the number of places the decimal needs to move
. Consider a number greater than 10. The decimal would move from right to left to make the number less than 10 but still greater than 1. The number of places the decimal moved indicates the positive exponent to be used for the base 10 power.
Similarly, for numbers less than 1, such as 0.000022, the decimal will move from left to right to make the number greater than or equal to 1 and less than 10. In this case, the number of places moved indicates the negative exponent to be used for the base 10 power.
Scientific notation is not only a convenient way to express cumbersome numbers. It also eases the comparison of numerical order of magnitude. For example, it may be difficult to determine how much larger 237 000 000 is compared to 4 530 000. However, it is easier to see that 2.37 * 10^8 and 4.5 * 10^6 differ by a factor of about 10^2 = 100.