If you have ever taken a Chemistry class, you may have encountered the following numbers:
There are approximately 602,214,179,300,000,000,000,000 molecules in one mole of any substance (this quantity is called Avogadro's number).
Avogadro constant - Wikipedia, the free encyclopedia
The mass of an electron is approximately 0.0000000000000000000000000009109382 grams.
Electron rest mass - Wikipedia, the free encyclopedia
Notice these two values are inconvenient to write in regular decimal notation because there are so many digits to write. When values are very large or very small it is often more manageable to write the numerals using scientific notation
. Scientific notation - Wikipedia, the free encyclopedia
Scientific notation is a representation that uses a decimal number times a power of ten. The decimal value is usually written with one digit, not zero, in front of the decimal point, and when multiplied together with a power of 10, the scientific notation exactly equals the value it represents. This is referred to as the normalized form for scientific notation.
Before when we studied scientific notation (Session 13), we only worked with whole numbers. This meant that we allowed more than one non-zero digit and all of the powers of 10 were positive powers. Now we extend that notation to include any decimal fraction.
We write the mass of an electron (given above) in scientific notation. First, when changing to scientific notation, we move the decimal point until the first non-zero digit in the numeral is in front of the decimal point. Then the power of ten represents the number of place values the decimal point is moved.
Example:
So, in scientific notation, the mass of an electron in grams is approximately 9.109382 × 10–28.
Notice that 10–28 = 0.0000000000000000000000000001 and if we multiply that by 9.109382 the result is 0.0000000000000000000000000009109382, which is the decimal value with which we started.
When trying to remember whether to use a positive or a negative exponent, or whether to move the decimal point right or left, it is much better to think about the meaning of the multiplication involved in the scientific notation. Multiplying by a positive power of 10 makes the number larger and multiplying by a negative power of 10 makes the number smaller.
4.56 × 105 makes the 4.56 greater in value by 5 place values so it equals 456,000.
1.2 × 10–4 makes the 1.2 less in value by 4 place values so it equals 0.00012.
Likewise, to write a number that is greater than one in scientific notation, the power of 10 will be positive (or possibly zero), and to write a number less than one in scientific notation, the power of 10 will be negative.
Formal Definition: To express a number in scientific notation (normalized form), we write it in the form a × 10k , where a is a decimal with 1 ≤ a < 10 and k is an integer.
Example: In scientific notation, Avogadro's number is approximately 6.022141793 × 1023.
Matching this with the formal definition, a is the decimal 6.022141793 and k is the exponent 23.
If we multiply this out, the positive twenty-third power moves the decimal point 23 place values right, making the decimal part of the number larger by that many place values.
This is exactly equal to the original value of 602,214,179,300,000,000,000,000.
|
Standard Numeral |
Verbal Form |
Scientific Notation |
|
5,400,000,000 |
||
|
six hundred thirty-one million |
||
|
seventy-one ten-thousandths |
||
|
0.0000012 |
||
|
1.7 × 1010 |
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Date last modified: July 22, 2010.
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