Overview
Scientific notation is a method of representing very large numbers with positive exponents and very small numbers with negative exponents. Although any number can be used as a base to an exponent, scientific notation uses the decimal system, base 10.
The Decimal System
The decimal system is based on powers of 10. The number 1 is 100 and the number 10 is 101, by definition. The number 100 is 102, 1000 is 103, and 10000 is 104. The pattern is that the number of zeroes after the 1 is equal to the number represented by the exponent. Numbers that are smaller than 1 can be represented by negative exponents. A decimal such as 0.1, or 1/10, is represented with a negative exponent, 10-1. Similarly, 0.01, or 1/100, is represented as 10-2, 0.001 as 10-3, and so on.
Figure 1: The decimal system is organized in powers of 10.
Scientific Notation
Scientific notation is an application of the rules about exponents, and are a way to express very large numbers (such as the distances to planets, stars, and galaxies) and very small numbers (such as the size of microbes, viruses, and atoms. Suppose the number is the distance from the earth to the sun, approximately 93 million miles or 93 ∙ 106. Using standard scientific notation, the number with significant digits is always expressed as a number between 1 and 10, or 9.3. Now 9.3 can be transformed to 93 by multiplying by 101, so that 9.3 ∙101∙106 is 9.3 ∙107, also known as 93,000,000.
Figure 2: An example of standard scientific notation.
Significant Digits
Significant digits for the purpose of scientific notation refer to the part of the number that is not included in the zeroes that follow it (in large numbers) or precede it (in small numbers). The significant digits in 93 million are the 9 and the 3. If the number were 92,900,000, the significant digits would be 929. If the number were 0.002135, the significant digits would be 2135.
Scientific Notation in Other Bases
Base 2, also known as the binary system, is used in computer applications, and can also be used in scientific notation. Engineering notation is a variation of scientific notation where the power of 10 that is used is a multiple of 1000. Suppose the number 83200 is expressed in engineering notation. It would be written as 83.2∙ 103, while in standard scientific notation it would be 8.32 ∙104.
Figure 3: Engineering notation is a variation of scientific notation.
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