teaching:topics:number:axioms
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teaching:topics:number:axioms [2021/09/25 22:56] simon |
teaching:topics:number:axioms [2024/03/14 13:03] (current) simon [some important ideas first ...] |
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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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