teaching:topics:number:making
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teaching:topics:number:making [2020/09/15 14:09] 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> |
| - | While exploring and developing our idea of number we have invented new numbers that allow us to | + | ===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 | ||
| 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 | ||
| - | `20`m long. Your field is `5` chains wide. You try dividing it into `2` chain wide sections. But they are | + | `20`m long. Your field is `5` chains wide. You try dividing it into sections `2` chains wide. But they are |
| too big, after you mark off two of them there is not enough left for a third one the same size. So you try | too big, after you mark off two of them there is not enough left for a third one the same size. So you try | ||
| sections `1` chain wide. But you have a big part of your field left over, they are too small. How do you | sections `1` chain wide. But you have a big part of your field left over, they are too small. How do you | ||
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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 2/3` 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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| But that comes along with the notion of exactly zero. | But that comes along with the notion of exactly zero. | ||
| - | How does that fit with your idea of dividing? | + | How does that fit with your idea of dividing? Can you make a fraction with that new number as the denominator? Can you divide by zero? The answer must be bigger than any other number. |
| - | + | ||
| - | Can you divide by zero? | + | |
| - | Maybe you should give that answer a name as well, and use that like a number. | + | Maybe you should give that answer a name, and use it like a number. |
| Let's try `infty`. | Let's try `infty`. | ||
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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.1600142942.txt.gz · Last modified: 2020/09/15 14:09 by simon
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