teaching:topics:number:axioms
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teaching:topics:number:axioms [2021/09/25 23:00] simon [some important ideas first ...] |
teaching:topics:number:axioms [2024/05/02 12:03] simon [there are important non-numbers] |
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| **=>[[axioms-formal#a more formal way to define numbers]]** | **=>[[axioms-formal#a more formal way to define numbers]]** | ||
| - | We use them along with arithmetic like `times` and `+` to help us describe and understand many properties of the world we observe around us. | + | <WRAP #rationale/>We use them along with arithmetic like `times` and `+` to help us describe and understand many properties of the world we observe around us. |
| An astonishing variety of very different quantities we measure behave like numbers and have important properties derived using arithmetic. | An astonishing variety of very different quantities we measure behave like numbers and have important properties derived using arithmetic. | ||
| In physics we measure distance, time, mass and electrical charge and from them calculate properties like position, area, volume, speed, acceleration, force, pressure, temperature, density and energy to build a model of the way physical things interact. | In physics we measure distance, time, mass and electrical charge and from them calculate properties like position, area, volume, speed, acceleration, force, pressure, temperature, density and energy to build a model of the way physical things interact. | ||
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| <wrap #Order_of_operations />//Order of operations// in arithmetic:\\ | <wrap #Order_of_operations />//Order of operations// in arithmetic:\\ | ||
| - | do the brackets first; then any powers; then multiply and divide; then finally plus and minus. | + | do the brackets first;\\ <wrap indent/>then any powers;\\ <wrap indent/><wrap indent/>then multiply and divide;\\ <wrap indent/><wrap indent/><wrap indent/>then finally plus and minus. |
| <wrap #distributive />**distributive** tells how `+` and `times` work together, `quad 2times(3+4)=(2times3)+(2times4)=14`. This looks much neater (and is easier to read and understand) in our algebra notation where we do not use the `times` symbol... | <wrap #distributive />**distributive** tells how `+` and `times` work together, `quad 2times(3+4)=(2times3)+(2times4)=14`. This looks much neater (and is easier to read and understand) in our algebra notation where we do not use the `times` symbol... | ||
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| ====there are important non-numbers==== | ====there are important non-numbers==== | ||
| When describing numbers [[#What is a number|above]] we saw that dividing by zero was not possible, it did not make sense. Yet in other cases like that we pressed on, we made [[teaching:topics:number:making#making new numbers|new numbers]] to answer the questions the old ones could not and so made our collections of numbers more complete. We cannot find any solution to | When describing numbers [[#What is a number|above]] we saw that dividing by zero was not possible, it did not make sense. Yet in other cases like that we pressed on, we made [[teaching:topics:number:making#making new numbers|new numbers]] to answer the questions the old ones could not and so made our collections of numbers more complete. We cannot find any solution to | ||
| - | \(\ x^2=-1\ \) in the //Real Numbers//, so we name a new number that is the solution and call it \(\ \rm i\). It is not a //Real Number// of course but we find that a whole new kind of number emerges. We call them //Complex Numbers// and find they do behave very nicely as numbers --- following all the rules we talked about here. We also discover many areas where they are very useful, we find many systems where values are naturally expressed as //Complex Numbers//. | + | \(\ x^2=-1\ \) in the //Real Numbers//, so we name a new number that is the solution and give it a name: \(\rm i\). It is not a //Real Number// of course but we find that a whole new kind of number emerges. We call them //Complex Numbers// and find they do behave very nicely as numbers --- following all the rules we talked about here. We also discover many areas where they are very useful, we find many systems where values are naturally expressed as //Complex Numbers//. |
| Perhaps, then, we should simply define a new number, call it \(\large\infty\) and extend our number system even further. The problem is that it does not play by the number rules. If we include it as a number we lose much of what makes numbers so useful and powerful, we cannot use it as a value within our algebra without extra care and work. | Perhaps, then, we should simply define a new number, call it \(\large\infty\) and extend our number system even further. The problem is that it does not play by the number rules. If we include it as a number we lose much of what makes numbers so useful and powerful, we cannot use it as a value within our algebra without extra care and work. | ||
teaching/topics/number/axioms.txt ยท Last modified: 2024/05/02 13:04 by simon
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