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June 8th, 2015, 12:38 AM  #1 
Newbie Joined: May 2015 From: Australia Posts: 10 Thanks: 0  Opposite numbers
Is $\displaystyle \frac{x}{1}$ the opposite of$\displaystyle \frac{1}{x}$?
Last edited by deepblue; June 8th, 2015 at 12:42 AM. 
June 8th, 2015, 05:41 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 
There is no mathematical concept of "opposite numbers" because there are many different ways of being "opposite". If you mean "multiplicative inverse" or "reciprocal" then, yes, the reciprocal, i.e. multiplicative inverse, of 1/x is x/1= x.

June 8th, 2015, 06:17 AM  #3 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,134 Thanks: 720 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
As Country Boy says, the word you want here is "reciprocal". The operation where you flip the numerator and denominator like that is called "taking the reciprocal".

June 8th, 2015, 10:36 PM  #4 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18 
As Country Boy said, it depends on how you define opposite. An elementary text book on math brings up the idea of opposite in the topic of integers. It says x and x are opposites of each other; defining the opposite of a number as a number which when added to the original number yields the sum 0. Perhaps we can extend the idea to fractions. Things to note: 1. An opposite of a number is defined over an operation (+, , /, *, etc) 2. A number and its opposite subject to a given operation must yield the identity element of that particular operation (addition identity = 0, multiplication identity = 1) So, we can answer your question in two ways. 1) If the operation under consideration is addition, the opposite of 1/x is (1/x) 2) If the operation under consideration is multiplication, the opposite of 1/x is x What do you think? 
June 9th, 2015, 06:52 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896  (As long as x is not equal to 0. 0 has no reciprocal.)

June 9th, 2015, 07:44 AM  #6 
Senior Member Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176  Here is a problem of terminology. Everyone may think that his song it is the only reality. In my childhood, the day before yesterday, I heard the song so: "Two opposite numbers differs only by sign." (Seems a little dry and rough, but it is not far to be intuitive). Two numbers are inverse numbers of each other if and only if their product is equal to 1 . $\displaystyle \dfrac{x}{1}$ and $\displaystyle \dfrac{1}{x}$ could be admitted as inverse numbers. Last edited by aurel5; June 9th, 2015 at 07:54 AM. 

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