Comments on: A Quirky Introduction To Number Systems http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/ Learning shouldn't hurt. Let's share the insights that made difficult ideas click. Sun, 14 Mar 2010 21:58:00 +0000 http://wordpress.org/?v=2.8.4 hourly 1 By: Søren Kyale http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/#comment-264227 Søren Kyale Fri, 04 Dec 2009 22:06:55 +0000 http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/#comment-264227 First, I must give κῦδος(kudos) to Arithmeticus Simplex for pointing out the practicality of operations with roman numerals (it's important to acknowledge that a Roman provided with the argument of comparison would conclude that her system was vastly superior...and a Kyalian well versed in balanced ternary would correctly conclude balanced ternary to be superior to both Roman and decimal notation). On division by zero. I very much favor Abraham Walker, his non-standard analysis.* *(H. Jerome Keisler provided open access to his book, Elementary Calculus: An Infinitesimal Approach, under a Creative Commons by-nc-sa license. http://www.math.wisc.edu/~keisler/calc.html The book provides a highly approachable explanation of non-standard analysis.) Non-standard analysis (will be a funny name, if at some point in the future, it becomes standard) defines a range of positive numbers that are greater than zero and less than any positive real number. The additive inverse gives a negative range. And the inverses give infinite results. I prefer this method for dealing with infinity because, at least to me, it does not seem reasonable to think of 1 divided by zero as positive or negative or even non zero. To me, defining division by zero as undefined is not a bug. I think the bug is believing that we know nothing well enough assume that dividing by it should be defined (of course, if we do some day come do know nothing, then we may feel free to divide by it). So for all practical intents and purposes, I like to use the inverse of positive and negative infinitesimal numbers to represent infinity because I know the sign of their inverses. (I don't like taking the integral between negative and positive infinity because I don't think it's reasonable to say that infinity has a sign or non zero value). On the other hand, I do think it may be reasonable to define zero divided by zero as 1, provided that the "1" thus generated is given a universe of sets not compared with other divisions of zero by zero. (Note: I am not familiar with set theory, so please correct and forgive any errors in my use of terminology. Specifically, I'm the concept of Universe that I learned from Lewis Carrol [Dodgson], his Symbolic Logic). That is 0/0 of set universe A = 1 of set universe B, but 1_a does not equal 1_b. And 0_b/0_b = 1_c which does not equal 1_a or 1_b, or rather that 1_a = 1_b is one possibility out of an infinite set of roughly equivalent possibilities and therefore infinitely improbable. First, I must give κῦδος(kudos) to Arithmeticus Simplex for pointing out the practicality of operations with roman numerals (it’s important to acknowledge that a Roman provided with the argument of comparison would conclude that her system was vastly superior…and a Kyalian well versed in balanced ternary would correctly conclude balanced ternary to be superior to both Roman and decimal notation).

On division by zero. I very much favor Abraham Walker, his non-standard analysis.*

*(H. Jerome Keisler provided open access to his book, Elementary Calculus: An Infinitesimal Approach, under a Creative Commons by-nc-sa license.
http://www.math.wisc.edu/~keisler/calc.html
The book provides a highly approachable explanation of non-standard analysis.)

Non-standard analysis (will be a funny name, if at some point in the future, it becomes standard) defines a range of positive numbers that are greater than zero and less than any positive real number. The additive inverse gives a negative range. And the inverses give infinite results.

I prefer this method for dealing with infinity because, at least to me, it does not seem reasonable to think of 1 divided by zero as positive or negative or even non zero.

To me, defining division by zero as undefined is not a bug. I think the bug is believing that we know nothing well enough assume that dividing by it should be defined (of course, if we do some day come do know nothing, then we may feel free to divide by it).

So for all practical intents and purposes, I like to use the inverse of positive and negative infinitesimal numbers to represent infinity because I know the sign of their inverses. (I don’t like taking the integral between negative and positive infinity because I don’t think it’s reasonable to say that infinity has a sign or non zero value).

On the other hand, I do think it may be reasonable to define zero divided by zero as 1, provided that the “1″ thus generated is given a universe of sets not compared with other divisions of zero by zero.

(Note: I am not familiar with set theory, so please correct and forgive any errors in my use of terminology. Specifically, I’m the concept of Universe that I learned from Lewis Carrol [Dodgson], his Symbolic Logic).

That is 0/0 of set universe A = 1 of set universe B, but 1_a does not equal 1_b. And 0_b/0_b = 1_c which does not equal 1_a or 1_b, or rather that 1_a = 1_b is one possibility out of an infinite set of roughly equivalent possibilities and therefore infinitely improbable.

]]> By: Kalid http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/#comment-252351 Kalid Sat, 12 Sep 2009 01:18:35 +0000 http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/#comment-252351 @Mack: Thanks for writing -- for these types of problems, what helps me is to have a mental model of what's going on (some type of analogy or intuitive understanding). Once I have that, I can start playing with that model to figure out what will happen in the problem. It's difficult to describe because it's different for everyone, but for the inequality / data sufficiency examples, I might think about a number line (going left to right) and as each new equation comes along, it "shades out" different parts of the line. I think everyone has some type of model happening in their head, but it can be hard to recognize/talk about because it's so innate. @Mack: Thanks for writing — for these types of problems, what helps me is to have a mental model of what’s going on (some type of analogy or intuitive understanding). Once I have that, I can start playing with that model to figure out what will happen in the problem.

It’s difficult to describe because it’s different for everyone, but for the inequality / data sufficiency examples, I might think about a number line (going left to right) and as each new equation comes along, it “shades out” different parts of the line. I think everyone has some type of model happening in their head, but it can be hard to recognize/talk about because it’s so innate.

]]> By: Mack http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/#comment-252344 Mack Fri, 11 Sep 2009 22:14:13 +0000 http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/#comment-252344 I learn by memorizing. I've always had a great memory. This allowed me to learn to read early as I remembered the sound of the words. Eventually I could discern what word should sound like based on what I already know.... I will be taking the GMAT for applications to business school that in January and I'm trying to find a way to solve the problem that irecognize I have which is that my mind finds it difficult to keep the relationships of mathmaticvariables straight in my mind. I constantly jumble it in my mind. I find it difficult to solve problems that are worded slightly differently fromthe ones I've done before. Data sufficiency questions involving inequalities with variables baffle me cause there are too many factors to consider. I have to think about whether x and y could be positive or negative, fraction or integer and where one relationship would yield one result and another relationship would yield another result, I find it hard to keep all that information straight. I start getting confused and losing track of the relationships. I'm great when it's memorization but ask me to think and I get confused. I assume it's because I don't fully understand relationships and patterns but I don't know how to start to see these patterns of which you speak on your post. I learn by memorizing. I’ve always had a great memory. This allowed me to learn to read early as I remembered the sound of the words. Eventually I could discern what word should sound like based on what I already know…. I will be taking the GMAT for applications to business school that in January and I’m trying to find a way to solve the problem that irecognize I have which is that my mind finds it difficult to keep the relationships of mathmaticvariables straight in my mind. I constantly jumble it in my mind. I find it difficult to solve problems that are worded slightly differently fromthe ones I’ve done before. Data sufficiency questions involving inequalities with variables baffle me cause there are too many factors to consider. I have to think about whether x and y could be positive or negative, fraction or integer and where one relationship would yield one result and another relationship would yield another result, I find it hard to keep all that information straight. I start getting confused and losing track of the relationships. I’m great when it’s memorization but ask me to think and I get confused. I assume it’s because I don’t fully understand relationships and patterns but I don’t know how to start to see these patterns of which you speak on your post.

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