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August 10th, 2011, 02:03 PM  #1 
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Arithmetic mean: Why use its computation as its definition?
Why do teachers/profs and textbooks define the arithmetic mean using its method of computation? For example, they don't do it for the sine function, which they define as the length of the opposite side divided by the length of the hypotenuse of a right triangle. Then they give a method to compute the sine. And they don't do it for any other math function. Using a method of computation as the definition also causes circular reasoning. 
August 10th, 2011, 02:16 PM  #2  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Arithmetic mean: Why use its computation as its definiti Quote:
The arithmetic mean of two numbers is their average. You compute this by summing them and then halving the result. The sine ratio of a base angle in a right triangle is the ratio of the opposite side to the hypotenuse. You calculate this by forming a ratio. can be defined as and then you calculate it term by term... Why are you asking this? Are we going to see links to external sites in your posts soon (< cynicism warranted)  
August 10th, 2011, 05:48 PM  #3  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond  Re: Arithmetic mean: Why use its computation as its definiti Quote:
Quote:
 
August 10th, 2011, 06:47 PM  #4  
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti
I'm asking because all the authorities of the arithmetic mean (or the "average") that I have researched provide the computation of the mean as its definition. As to your request to provide external sites, all the ones I've researched give the algorithm as the definition. For example, Encyclopedia Britannica says: Quote:
As for your example of e^x, Wikipedia says that "the exponential function e^x can be characterized in a variety of equivalent ways". In particular it may be defined by the power series (which you provided), but also by a continued fraction ... obtained via an identity of Euler. So, claiming that the algorithm is the definition forces us to claim that e^x has multiple definitions. This is simply not so. Only one definition exists, which can be characterized several ways. So, when I ask for the definition of the arithmetic mean, I want the definition from which the algorithm derives.  
August 10th, 2011, 07:57 PM  #5  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 11th, 2011, 01:42 AM  #6  
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 11th, 2011, 01:49 AM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond  Re: Arithmetic mean: Why use its computation as its definiti
It's a sum, and I don't see why you're making a distinction in the first place. In reference to your remark about the sine ratio, dividing the length of a side of a right angled triangle by the length of its hypotenuse is as much a computation as adding together n numbers and dividing the result by n.

August 11th, 2011, 02:46 AM  #8 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti
There's nothing wrong or logically fallacious with defining a function using an algorithmic description. How else do you define the Ackermann function, for example?

August 11th, 2011, 05:02 PM  #9  
Senior Member Joined: Jul 2011 Posts: 245 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 11th, 2011, 11:41 PM  #10  
Member Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
Why do you think that should exist a definition befores the formula? Do you have such definition?  

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