teaching:topics:number:axioms
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teaching:topics:number:axioms [2021/09/25 22:56] simon |
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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| <wrap #rational_number />The collection of numbers found by starting with [[#one]], then [[#addition|adding]], then [[#reciprocal|dividing]], then [[#subtraction|subtracting]] we | <wrap #rational_number />The collection of numbers found by starting with [[#one]], then [[#addition|adding]], then [[#reciprocal|dividing]], then [[#subtraction|subtracting]] we | ||
| - | call the **Rational Numbers**. The Hindu mathematicians put these ideas together in a formal, mathematical way during the time Europeans call the Dark Ages, from about 650AD. Their anaysis was soon translated into Arabic. In the 1500s the Persian and Islamic mathematics that followed was translated and published in Europe, including much of our basic //algebra// and the //algorithms// we use for arithmetic. Those words come from the Arabic title and Persian author of one of those books. The modern symbols for the ten digits are originally from the Hindu written script of that time. | + | call the **Rational Numbers**. The Hindu mathematicians put these ideas together in a formal, mathematical way during the time Europeans call the Dark Ages, from about 650AD. Their analysis was soon translated into Arabic. In the 1500s the Persian and Islamic mathematics that followed was translated and published in Europe, including much of our basic //algebra// and the //algorithms// we use for arithmetic. Those words come from the Arabic title and Persian author of one of those books. The modern symbols for the ten digits are originally from the Hindu written script of that time. |
| <wrap #algebras />This is a rather important idea, we only just touch on it a little in | <wrap #algebras />This is a rather important idea, we only just touch on it a little in | ||
| high school --- but mathematicians keep creating new kinds of entities like these to talk about new kinds of models of things that we observe in the world around us. We develop the **algebras** that work with these new collections ... what we are learning | high school --- but mathematicians keep creating new kinds of entities like these to talk about new kinds of models of things that we observe in the world around us. We develop the **algebras** that work with these new collections ... what we are learning | ||
| - | now is Number //[[teaching:topics:algebra:first#Algebra]]//, and we look at //set algebra// also, manipulating | + | now is //Number [[teaching:topics:algebra:first#Algebra]]//, and we look at //Set Algebra// also, manipulating |
| //sets// and their //elements// using operations including //union// and | //sets// and their //elements// using operations including //union// and | ||
| //intersection// with relations like //in//. At school we also explore | //intersection// with relations like //in//. At school we also explore | ||
| - | //vector algebra//. | + | //Vector Algebra//. |
| ====What is a number?==== | ====What is a number?==== | ||
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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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