April 18th, 2017, 12:58 AM  #61  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Quote:
Now for the next imprecision trap to beware of consider this. Do you consider 1 less than 0 and 0 less than 1 ? Folks often forget that numbers extend to negative but very large in magnitude. So is 1 smaller than 0?  
April 18th, 2017, 01:18 AM  #62  
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory  Quote:
Indeed, I have come to another revelation. Indeed changes have occurred in mathematics over time and it is sometimes hard to know (or keep track) with every inclusion the changes it brings to the anxioms. There was a time when man knew only natural numbers and fractions, then came negative numbers.. Numbers were quite easy to understand and manipulate..they grew more complex with introduction of irrational numbers imaginary numbers and so forth.. The time we arrived to the concept $\infty$ we hadn't realised we gave an added meaning to 0. But still we were able to continue on despite not knowing this. We talk of approaching 0 or $\infty$ in calculus and we are correct.  
April 18th, 2017, 01:20 AM  #63 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory  It depends with what you mean by smallness. In value or in magnitude

April 18th, 2017, 02:40 AM  #64  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Quote:
Silly arguments over sign conventions in electrical circuit theory is one place where this happens. Did you not also introduce the ordering axiom somewhere in discussions?  
April 20th, 2017, 04:36 AM  #65 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory  
April 20th, 2017, 05:18 AM  #66  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,151 Thanks: 2387 Math Focus: Mainly analysis and algebra  Quote:
Quote:
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Learning about numbers (and mathematics in general) is like finding your way through a maze. It is easy to start with, but you can easily get lost going down the wrong path. One should always be capable of "resetting" oneself onto the known path. For this, the map of previous knowledge can be very useful. Mathematicians through the ages have worked to determine the best path through the maze. They have explored most paths (right and wrong) and left us with a map. Pretending that ones knowledge of the first few steps into the maze is sufficient to know it all is clearly nonsense. The process of learning mathematics takes us from the maze entrance to the boundaries of the current map. We can explore without referring to the map on the way there. It is good practice, but it won't change the maze. And when experienced guides tell you that you are way off the path, they are probably right.  
April 21st, 2017, 05:16 AM  #67 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory 
I agree. First few into the maze and thinking you know it all is being delusional. All I do is present ideas and back them up. I have a tendency to think I am right but I accept when I am wrong. Not that am disrespectful, but in discussions I don't consider a person's level of experience. My focus is on the topic. It is not on who is right or who is wrong. It is on what is right and what isn't. The point was, when 0 is thought of as a number, then 1/0 is undefined because we are dividing 1 by nothing. It even doesn't make sense.Those that think so are correct. But when you think of 0 as a concept, of which it is, then 1/0 is infinity. Just harmonisation of the two points. I have also avoided using equal to because 1/0 is not equal to any value. 
April 21st, 2017, 06:13 AM  #68  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,151 Thanks: 2387 Math Focus: Mainly analysis and algebra  That's good. Quote:
Having said that, you are right that there are times when it can be useful to think of $\frac10$ as being $\infty$ even though there is no rigour to the idea. There also times when it is better to think of $\frac10$ as being $\infty$. In fact there are times when it is useful to think of $\infty$ as being the same as $\infty$ and the whole number system becomes a closed loop. Remember that none of this is in any meaningful way true, but they are tools that can prove useful in generating results even though the can't form part of any proof. Similarly it can be useful to think of $\frac1\infty$ as being zero. Mathematics has many instances such as this where we find that we can take shortcuts by imagining things that aren't true and applying results outside of their valid context. Calculus in particular makes widespread use of such ideas to simplify the process of getting to answers. Such treatments must be proved rigorously, of course. But it's like a walks of life, if you properly understand the rules, their justification and their context, you can sometimes break them safely.  
April 21st, 2017, 09:38 AM  #69 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory 
Yes. True context is important. Such cannot be used in algebra. Or if used in special cases, there are considerations to be taken.

April 21st, 2017, 10:00 AM  #70  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Quote:
 

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