teaching:topics:number:axioms
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
teaching:topics:number:axioms [2021/09/25 20:43] simon [What is a number?] |
teaching:topics:number:axioms [2024/03/14 13:03] (current) simon [some important ideas first ...] |
||
|---|---|---|---|
| Line 9: | Line 9: | ||
| **=>[[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. | ||
| Line 44: | Line 44: | ||
| <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... | ||
| Line 56: | Line 56: | ||
| <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 //[[teaching:topics:algebra:first#algebra|Number 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?==== | ||
| Line 70: | Line 70: | ||
| ... now let's make some more! | ... now let's make some more! | ||
| - | <wrap #Counting_Number /><wrap #Natural_Number /><wrap #order /> | + | <wrap #counting_number/><wrap #natural_number/><wrap #order/> |
| ===First we learn how to count.=== | ===First we learn how to count.=== | ||
| <wrap #next />We give the name //two// to the number after //one//, //three// to the number after //two// and make up a way of giving names to every **next** number. For example after //ninety nine// we call the //next// number //one hundred//. So we end up with an endless list of number-names, in a fixed **order**. | <wrap #next />We give the name //two// to the number after //one//, //three// to the number after //two// and make up a way of giving names to every **next** number. For example after //ninety nine// we call the //next// number //one hundred//. So we end up with an endless list of number-names, in a fixed **order**. | ||
| Line 79: | Line 79: | ||
| * we could find all the **[[#counting number]]s** by starting with //one//, and adding. | * we could find all the **[[#counting number]]s** by starting with //one//, and adding. | ||
| * **addition**<wrap #addition /> __of counting numbers__ can be seen as repeated '//[[#next]]//' number operations | * **addition**<wrap #addition /> __of counting numbers__ can be seen as repeated '//[[#next]]//' number operations | ||
| - | * that is: adding counting numbers is done by counting along the sequence of number-names | + | * that is: adding them is done by counting along the sequence of number-names |
| * the first thing we learn (or define, or teach) about numbers is their names, in order | * the first thing we learn (or define, or teach) about numbers is their names, in order | ||
| * then<wrap #count /> we learn how to count a collection of things by matching each one to a number-name | * then<wrap #count /> we learn how to count a collection of things by matching each one to a number-name | ||
| Line 98: | Line 98: | ||
| * this is the motivation for the conventional [[#order of operations]] in arithmetic | * this is the motivation for the conventional [[#order of operations]] in arithmetic | ||
| * in arithmetic we use brackets to change (or emphasise) the order of operations | * in arithmetic we use brackets to change (or emphasise) the order of operations | ||
| - | * and why we write equations with `+` and `-` signs and //[[teaching:topics:algebra:first#term]]s// (rather than using `times` symbols) | + | * and why we write equations with `+` and `-` signs between //[[teaching:topics:algebra:first#term]]s// (rather than using `times` symbols) |
| * hence we 'collect like terms' by adding their number-parts | * hence we 'collect like terms' by adding their number-parts | ||
| * hence we multiply an expression by multiplying __every__ (top level) term of the expression. | * hence we multiply an expression by multiplying __every__ (top level) term of the expression. | ||
| - | * <wrap #reciprocal /><wrap #multiplicative_inverse />the **reciprocal**, or //multiplicative inverse//, of a number is the number you [[#multiplication|multiply]] it by to get //[[#one]]// | + | * <wrap #division/><wrap #reciprocal/><wrap #multiplicative_inverse/>the **reciprocal**, or //multiplicative inverse//, of a number is the number you [[#multiplication|multiply]] it by to get //[[#one]]// |
| * or, the same thing if you already can divide: //one// divided by a number is its reciprocal | * or, the same thing if you already can divide: //one// divided by a number is its reciprocal | ||
| - | * <wrap #division />we use this to do //division//, we **multiply by the reciprocal to divide**. | + | * we use this to do **division**, we **multiply by the reciprocal to divide**. |
| * we need **fractions** to make a set of numbers (excluding zero) **[[#closed]] under division** <WRAP right half>\\ <WRAP tip> | * we need **fractions** to make a set of numbers (excluding zero) **[[#closed]] under division** <WRAP right half>\\ <WRAP tip> | ||
| **Fractions** (that is: [[#division]] and ratio) transform our number system from counting distinct things into | **Fractions** (that is: [[#division]] and ratio) transform our number system from counting distinct things into | ||
| Line 112: | Line 112: | ||
| really //understand// that difference. | really //understand// that difference. | ||
| </WRAP></WRAP> | </WRAP></WRAP> | ||
| - | * <wrap #fraction />we can find all the //fractions// by dividing pairs of //[[#counting number]]s// (then including their [[#negative]]s) | + | * <wrap #fraction/>we can find all the //fractions// by dividing pairs of //[[#counting number]]s// (then including their [[#negative]]s) |
| * the fraction bar in algebra combines brackets and dividing | * the fraction bar in algebra combines brackets and dividing | ||
| - | * <wrap #zero />we decide that **zero** is a number (this might be very familiar, but it is not at all trivial or obvious!) | + | * <wrap #zero/><wrap #additive_identity/>we decide that **zero** is a number (this might be very familiar, but it is not at all trivial or obvious!) |
| * adding //zero// (the **additive identity**) to any number leaves it unchanged | * adding //zero// (the **additive identity**) to any number leaves it unchanged | ||
| - | * <wrap #negative />the **negative**, or //additive inverse//, of a number is the number you add to it to get zero | + | * <wrap #negative/><wrap #additive_inverse/>the **negative**, or //additive inverse//, of a number is the number you add to it to get zero |
| * or<wrap #subtraction />, put the other way round: a number subtracted from //zero// is its negative | * or<wrap #subtraction />, put the other way round: a number subtracted from //zero// is its negative | ||
| * thus we define **subtraction**: we **add the negative to subtract**. | * thus we define **subtraction**: we **add the negative to subtract**. | ||
| - | * so //zero// is any number subtracted from itself | + | * put another way: //zero// is any number subtracted from itself |
| * we need zero and [[#negative]] numbers to make our set of numbers **[[#closed]] under subtraction** | * we need zero and [[#negative]] numbers to make our set of numbers **[[#closed]] under subtraction** | ||
| - | * now we have a new set of numbers --- //zero//, the //Counting Numbers//, //fractions// and their //negatives// ---- we call them the **[[#Rational Number]]s** | + | * now we have our new set of numbers --- //zero//, the //Counting Numbers//, //fractions// and their //negatives// --- we call them the **[[#Rational Number]]s**. We can add, subtract, multiply and ([[#almost always]]) divide these numbers to get another as the result. |
| - | * [[#multiplication]] of any number by zero gives zero | + | * <wrap #almost_always />[[#multiplication]] of any number by zero gives zero |
| * which means that **zero does not have a [[#reciprocal]]** | * which means that **zero does not have a [[#reciprocal]]** | ||
| * so **we cannot [[#division|divide]] by zero!** | * so **we cannot [[#division|divide]] by zero!** | ||
| Line 130: | Line 130: | ||
| [[axioms-formal#some set notation, as used here|more formal, abstract representation here]]. | [[axioms-formal#some set notation, as used here|more formal, abstract representation here]]. | ||
| - | We use these rules to build our **[[teaching:topics:algebra:first|Number Algebra]]**. | + | We use these rules to build our **Number [[teaching:topics:algebra:first#Algebra]]**. |
| ====more numbers==== | ====more numbers==== | ||
teaching/topics/number/axioms.1632566620.txt.gz · Last modified: 2021/09/25 20:43 by simon
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
