Speed of Light
Date: 06/08/2001 at 10:24:23 From: Kyrja Subject: Scientific Constants (2.99*10^2=299) I went to a scientific constants page and clicked on the link for the speed of light. Here's what I got: c = 2.99792458*10^8 m s^-1. What's m s^-1? Why isn't it just m/s (meters per second) instead? What's the little "-1" at the end? Why do they have to write 2.99792458*10^8, instead of just plain old 299,792,458?
Date: 06/08/2001 at 16:12:18 From: Doctor Rick Subject: Re: Scientific Constants (2.99*10^2=299) Hi, Kyrja. Thanks for your question! Are you familiar with exponents (including negative exponents) and scientific notation? Here are some pages from our Archives in case you aren't so confident about them. Scientific Notation http://mathforum.org/dr.math/problems/hauke10.2.98.html Properties of Exponents http://mathforum.org/dr.math/problems/crystal2.01.22.01.html Rules for Significant Figures and Decimal Places http://mathforum.org/dr.math/problems/ashley03.23.99.html You can find more using our Dr. Math Search page to search for words or phrases (select "that exact phrase") like scientific notation or negative exponent. http://mathforum.org/mathgrepform.html The unit "m s^-1" means "meters times seconds to the negative 1 power." A number raised to the -1 power is 1 over the number: -1 2^-1 = 2 = 1/2 Therefore, s^-1 means the same as 1/s or "per second," and m s^-1 means the same as m/s. I suppose scientists prefer to use the negative exponent instead of division because they are used to using exponents in scientific notation, and because division can be confusing (people get the order of operations mixed up). It doesn't make much difference with m/s versus ms^-1, but there are more complicated units. For instance, the unit of force (the newton) is the same as a kilogram-meter per second per second. We write this as -2 kg m s Using the exponent makes it stand out that seconds are in the denominator of the unit (because the exponent is negative). And you'd need an exponent anyway for this unit, unless you wrote "kg m/s/s" which looks really ugly, in my opinion. Now for the other part of your question. Why isn't the number just written 299,792,458? You're asking what the advantage is of using scientific notation. One reason is that scientists don't like to have to count things. When you write it as 2.99792458*10^8, the most important thing about the number is easy to find: how many digits it has. The "most significant digit" is the first one on the left - it is more important than any other digit in the number, because it stands for a bigger amount ("hundred millions," in this case). But even more important than the most significant digit is the NUMBER of digits: if you left out a digit, for example, then you'd essentially divide the number by 10, which is a bigger error than if you replaced the most significant digit by something else. In scientific notation, as I said, you can tell right away that the number is in the hundred-millions range (10 to the 8th power). We don't really care about the name "hundred million"; the number 8 says it all. We say it's "of the order 10^8." There is another big advantage about scientific notation, and it is particularly important at the Web site you found. This advantage is that it is easy to tell how PRECISE the number is. See the page on significant digits that I listed above. The speed of light is a number that has been found by very careful experiments. It's a big deal when someone does an experiment that gives the speed of light with more precision than the last experiment. And the number you found tells us that we now know the speed of light to within 0.00000001. That's a millionth of a percent! Again, the name of this percentage isn't important, but we say we know the speed of light to "8 decimal places." (This time, unfortunately, we have to count.) Having said all that, I have to note that while the discussion about precision would apply to other constants you might have seen on that Web site, it's not really true of the speed of light. In fact, the speed of light is EXACTLY 299,792,458 meters per second. Didn't I just say that it is measured by experiment, and we can't measure anything exactly? Well, yes, but not exactly. The speed of light is now DEFINED to be 299,792,458 meters per second. When a new experiment comes along, it changes how well the length of a meter is known, because a meter is defined as the distance that light travels in vacuum in 1/299,792,458 second. Here is a Web site that explains this. Time Line for the Definition of the Meter http://www.mel.nist.gov/div821/museum/timeline.htm - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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