Scientific Notation
In science, measurements are often very large or very small. Using scientific notation makes these large and small numbers easier to work with.
Using scientific notation requires an understanding of exponents and bases. When a number is expressed as a base and an exponent, the base is the number that is used as a factor. The exponent tells how many times the base is multiplied by itself. For example, the number 25 can be expressed as a base and an exponent in the following way:
In the example above, 5 is the base and 2 is the exponent. In scientific notation, the base is always the number 10. The exponent tells how many times the number 10 is multiplied by itself.
A number written in scientific notation is expressed as the product of two factors, a number between 1 and 10 and the number 10 with an exponent. For example, the number 51,000 can be expressed in scientific notation. To find the first factor, move the decimal to obtain a number between 1 and 10. In this case, the number is 5.1. The exponent can be determined by counting the number of places the decimal point was moved. The decimal point was moved four places to the left. So, 51,000 expressed in scientific notation is 5.1 × 104.
Numbers that are less than one can also be expressed in scientific notation. In the case of numbers less than one, the decimal point must be moved to the right to obtain a number between 1 and 10. For example, in the number 0.000098, the decimal point must move five places to the right to obtain the number 9.8. When the decimal point is moved to the right, the exponent is negative. So, 0.000098 expressed in scientific notation is 9.8 × 10−5.
Calculating With Scientific Notation
Numbers expressed in scientific notation can be used in calculations. When adding or subtracting numbers expressed in scientific notation, the first factors must be rewritten so the exponents are the same.
Example
Follow these steps to add (4.30 × 104) + (2.1 × 103).
Move the decimal point in one of the expressions so the exponents are the same.
(43.0 × 103) + (2.1 × 103)
Add the first factors, keeping the value of the exponents the same.
(43.0 × 103) + (2.1 × 103) = 45.1 × 103
Move the decimal point so the first factor is expressed as the product of a number between and 1 and 10 and an exponent with base 10.
45.1 × 103 = 4.51 × 104
When numbers expressed in scientific notation are multiplied, the exponents are added. When numbers expressed in scientific notation are divided, the exponents are subtracted.
Example
Use the following steps to determine the area of a rectangular field that has a length of 1.5 × 103 meters and a width of 3.2 × 102 meters.
Write down the expressions to be multiplied.
(1.5 × 103 m)(3.2 × 102 m)
Multiply the first factors, add the exponents, and multiply any units.
= (1.5 × 3.2)(103 + 2) m X m
= 4.8 × 105 m2
Table of Contents
- Formulas and Equations
- Applying Formulas and Equations
- Mean, Median, and Mode
- Estimation
- Using Measurements in Calculations
- Effects of Measurement Errors
- Accuracy
- Precision
- Comparing Accuracy and Precision
- Significant Figures
- Calculating With Significant Figures
- Scientific Notation
- Calculating With Scientific Notation
- Dimensional Analysis
- Applying Dimensional Analysis