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Using Scientific Notation

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Physics deals with realms of space from the size of less than a proton to the size of the universe. As such, you end up dealing with some very large and very small numbers. Generally, only the first few of these numbers are significant. No one is going to (or able to) measure the width of the universe to the nearest millimeter.
NOTE: This portion of the article deals with manipulating exponential numbers (i.e. 105, 10-8, etc.) and it is assumed that the reader has a grasp of these mathematical concepts. Though the topic can be tricky for many students, it is beyond the scope of this article to address.

In order to manipulate these numbers easily, scientists use scientific notation. The significant figures are listed, then multiplied by ten to the necessary power. The speed of light is written as: [blackquote shade=no]2.997925 x 108 m/s

There are 7 significant figures and this is much better than writing 299,792,500 m/s. (NOTE: The speed of light is frequently written as 3.00 x 108 m/s, in which case there are only three significant figures. Again, this is a matter of what level of precision is necessary.)

This notation is very handy for multiplication. You follow the rules described earlier for multiplying the significant numbers, keeping the smallest number of significant figures, and then you multiply the magnitudes, which follows the additive rule of exponents. The following example should help you visualize it:

2.3 x 103 x 3.19 x 104 = 7.3 x 107

The product has only two significant figures and the order of magnitude is 107 because 103 x 104 = 107

Adding scientific notation can be very easy or very tricky, depending on the situation. If the terms are of the same order of magnitude (i.e. 4.3005 x 105 and 13.5 x 105), then you follow the addition rules discussed earlier, keeping the highest place value as your rounding location and keeping the magnitude the same, as in the following example:

4.3005 x 105 + 13.5 x 105 = 17.8 x 105

If the order of magnitude is different, however, you have to work a bit to get the magnitudes the same, as in the following example, where one term is on the magnitude of 105 and the other term is on the magnitude of 106:

4.8 x 105 + 9.2 x 106 = 4.8 x 105 + 92 x 105 = 97 x 105

or

4.8 x 105 + 9.2 x 106 = 0.48 x 106 + 9.2 x 106 = 9.7 x 106

Both of these solutions are the same, resulting in 9,700,000 as the answer.

Similarly, very small numbers are frequently written in scientific notation as well, though with a negative exponent on the magnitude instead of the positive exponent. The mass of an electron is:

9.10939 x 10-31 kg

This would be a zero, followed by a decimal point, followed by 30 zeries, then the series of 6 significant figures. No one wants to write that out, so scientific notation is our friend. All the rules outlined above are the same, regardless of whether the exponent is positive or negative.

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