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/