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teaching:topics:number:making

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teaching:topics:number:making [2020/09/15 14:35]
simon
teaching:topics:number:making [2022/02/16 23:23] (current)
simon
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-======making new numbers======+<WRAP right>​[[teaching:​topics:​number:​axioms#​there are important non-numbers]]</​WRAP>​ 
 +===making new numbers=== 
 +//Sometimes you can. Sometimes you cannot.// 
 While exploring and developing our idea of [[teaching:​topics:​number:​axioms:#​what is a number|number]] we have invented new numbers that allow us to While exploring and developing our idea of [[teaching:​topics:​number:​axioms:#​what is a number|number]] we have invented new numbers that allow us to
 give answers to questions that we previously could not answer with a number. give answers to questions that we previously could not answer with a number.
- +== == 
-So, maybe you are a farmer and you want to divide your field into three equal parts, you measure distance+Maybe you are a farmer ​(thousands of years ago) and you want to divide your field into three equal parts, you measure distance
 by '​chains'​ in your part of the world. There is actually a chain kept at the surveyor'​s house which is by '​chains'​ in your part of the world. There is actually a chain kept at the surveyor'​s house which is
 carefully laid out to mark off distances. If we measured that chain today wr would say it is a bit over carefully laid out to mark off distances. If we measured that chain today wr would say it is a bit over
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 right after `1`, you do not have any numbers in between! right after `1`, you do not have any numbers in between!
  
-So you make up fractions. The length you are after is \(1 2\over3\) chains. The surveyor folds his chain into+So you make up fractions. The length you are after is \(1{2\over3}\) chains. The surveyor folds his chain into
 three equal parts, and marks the '​thirds'​. three equal parts, and marks the '​thirds'​.
  
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 We could find ways to do these with fractions, with negative numbers, with real numbers, even We could find ways to do these with fractions, with negative numbers, with real numbers, even
-with complex numbers ​(after we invented \(\ \rm i\ \) as the solution to `\ x^2+1=0` ). But we cannot achieve a good working definition of `infty` that lets us use it in our number algebra.+with Complex Numbers ​(after we invented \(\ \rm i\ \) as the solution to `\ x^2+1=0` ). But we cannot achieve a good working definition of `infty` that lets us use it in our number algebra.
  
 [[teaching:​topics:​calculus:​limits#​Infinitessimals]] are a related problem. [[teaching:​topics:​calculus:​limits#​Infinitessimals]] are a related problem.
  
 Consider the [[teaching:​topics:​calculus:​sequences#​paradox|wolf and the sprinter]], a challenge has kept us wondering for millenia. Consider the [[teaching:​topics:​calculus:​sequences#​paradox|wolf and the sprinter]], a challenge has kept us wondering for millenia.
teaching/topics/number/making.1600144510.txt.gz · Last modified: 2020/09/15 14:35 by simon