Scientific notation
Scientific notation is a mathematical system used to express very large or very small numbers in a more concise and manageable format. It reformats numbers using exponents and decimal points, making calculations and representation easier. In this system, a number is expressed as a product of a number between 1 and 10 and a power of ten, which indicates how many places the decimal point is shifted. The exponent is positive for large numbers (when the decimal moves to the left) and negative for small numbers (when it moves to the right).
For instance, the number 425 is written in scientific notation as 4.25 x 10², while a very small number like 0.0000000000776 becomes 7.76 x 10⁻¹¹. Operations involving scientific notation, such as addition, subtraction, multiplication, and division, follow specific rules, particularly concerning the exponents. This notation is widely used in scientific and engineering contexts, allowing professionals to handle extreme values efficiently and accurately. Understanding how to convert numbers to and from scientific notation is crucial for working with these figures in various fields.
Scientific notation
Scientific notation is the way scientists express both very large and very small numbers. Numbers are reformatted using exponents and decimal points to make the numbers easier and more concise to write. An exponent is a symbol used in mathematics to express how many times a number is multiplied by itself. It is sometimes called "to the power." It is written in superscript, which means it is set in smaller font and placed to the top right of a number. For example, 10 with an exponent of 2 (or 10 to the second power) would be written as 102. A decimal point is a dot (or period) that separates a whole number from tenths, hundredths, thousandths, etc. For example, 10 and 23 hundredths would be written as 10.23.
Numbers in scientific notation follow this format: first nonzero digit of the number, decimal point, the rest of the digits of the number, times 10 to the appropriate power. To figure out the last step, the exponent, or power, must equal the number of places the decimal point is moved to express the original large or small number. If the decimal point must be moved to the right to express a large number, the exponent will be positive. If the decimal point must be moved to the left to express a small number, the exponent will be negative.
Converting to Scientific Notation
To write 425 in scientific notation, write the first digit (4), follow with a decimal point and the rest of the digits (.25), times by 10 to the appropriate power, or number of places the decimal point must be moved to express the original number (2 to the right, so the exponent is positive 2). The result is 4.25 x 102.
For 397,000, write the first digit (3), followed by a decimal point and the rest of the digits (.97), and times by 10 to the fifth power (105) to equal 3.97 x 105.
The digits express the number of significant figures in the number, while the exponent shows the place of the decimal point. For example, 397,000 has three significant figures (397) and three zeros—the zeros are there as a placeholder when converting to scientific notation. In its scientific notation, 3.97 x 105, the exponent (5) shows that the decimal is moved five places to get 397,000.
Writing a very small number in scientific notation is done a little differently. For example, for 0.0000000000776, write the first digit (7), follow with a decimal point and the rest of the digits (.76), count the number of places the decimal point was moved (11 places). Because the decimal point must be moved to the left to express the original number, the exponent will be negative. The result is 7.76 x 10-11.
Converting from Scientific Notation
To convert a number in scientific notation back to a regular number is very easy. Use the exponent to determine the number of decimal spaces. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left. For example, for 3.97 x 105, the exponent is 5 and positive, which means the decimal point will be moved five places to the right. Remember, zeros will be used as placeholders. The result is 397,000.
For 2.2 x 10-8, the exponent is 8 and negative, which means the decimal point will be moved eight places to the left and zeros will be used as placeholders. The result is 0.000000022.
Addition and Subtraction
When adding or subtracting numbers expressed in scientific notation, they must have the same base, and exponents, or powers, must be equal. All numbers in scientific notation have a base of 10, so only the exponents need to match.
If the exponents are not equal, then they must be converted. Increase the smaller exponent to match the larger exponent and move the decimal point the same number of places. Once the exponents match, add or subtract the coefficients. Then, convert to scientific notation if needed.
Addition example:
5.678 x 108 + 6.1234 x 105
108 – 5 = 3
The smaller exponent must be increased by 3.
6.1234 x 108 = 0.0061234 x 108
5.678 x 108 + 0.0061234 x 108
5.6841234 x 108
Subtraction example:
4.6789 x 103 – 2.1 x 10-2
103 – (–2) = 5
The smaller exponent must be increased by 5.
2.1 x 10-2 = 0.000021 x 103
4.6789 x 103 – 0.000021 x 103
4.678879 x 103
Multiplication and Division
To multiply and divide numbers in scientific notation, the bases must be the same. However, all bases are 10 in scientific notation, so nothing has to be done before solving the problem. Answers may need to be converted to scientific notation.
For multiplication problems, first multiply the coefficients. Round the number of significant figures. Next, add the exponents. Then, convert to scientific notation if needed.
Multiplication example:
(4.2 x 1010) x (5.205 x 102)
4.2 x 5.205 = 21.9
1010 x 102 = 1010+2 = 1012
(4.2 x 1010) x (5.205 x 102)
21.9 x 1012
Convert 21.9 x 1012 to scientific notation.
2.19 x 1013
For division problems, first divide the coefficients. Round the number of significant figures. Next, subtract the exponents. Then, convert to scientific notation if needed.
Division example:
(2.02 x 107) ÷ (5.55 x 104)
2.02 ÷ 5.55 = 0.364
107 – 104 = 103
(2.02 x 107) ÷ (5.55 x 104)
Convert 0.364 x 103 to scientific notation.
3.64 x 102
Bibliography
"Math Skills Review: Scientific Notation." Department of Chemistry, Texas A&M University. Texas A&M University. Web. 28 Dec. 2014. http://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html
"The Scientific Notation." Laboratory for Atmospheric and Space Physics. Regents of the University of Colorado. Web. 28 Dec. 2014. http://lasp.colorado.edu/~bagenal/MATH/math1.html
"Scientific Notation." SparkNotes. SparkNotes LLC. Web. 28 Dec. 2014. http://www.sparknotes.com/math/algebra1/scientificnotation/summary.html
Stapel, Elizabeth. "Exponents: Scientific Notation." Purplemath. Purplemath. Web. 28 Dec. 2014. http://www.purplemath.com/modules/exponent3.htm