11th Oct '16 12:27:06 PM

**Koveras** Is there an issue? Send a Message

**Changed line(s) 5,6 (click to see context) from:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' * ''c'') * (''a'' * ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. That isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

**to:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' * ''c'') ~~* ~~= (''a'' * ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. That isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

30th Oct '15 4:43:25 AM

**MrUnderhill** Is there an issue? Send a Message

**Changed line(s) 3,4 (click to see context) from:**

Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' -- that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' zero (0), such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But as mentioned before, 0 * anything = 0 (or alternately, 0 + 0 = 0 no matter how many times you repeat the addition), so no matter how many copies of zero are added, the result will always be what was initially there -- zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / (any infinite number) evaluates to an ''infinitesimal'', which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities -- 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

**to:**

Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' -- that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' zero (0), such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But ~~as mentioned before, 0 * anything = 0 (or alternately, 0 + 0 = 0 no matter how many times you repeat the addition), ~~[[DepartmentOfRedundancyDepartment zero plus zero is always equal to zero]], so no matter how many copies of zero are added, the result will always be what was initially there -- zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / (any infinite number) evaluates to an ''infinitesimal'', which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities -- 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

30th Oct '15 4:37:46 AM

**MrUnderhill** Is there an issue? Send a Message

**Changed line(s) 3,4 (click to see context) from:**

Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' -- that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' zero (0), such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But as mentioned before, 0 + anything = 0, so no matter how many copies of zero are added, the result will always be what was initially there -- zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / (any infinite number) evaluates to an ''infinitesimal'', which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities -- 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

**to:**

Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' -- that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' zero (0), such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But as mentioned before, 0 ~~+ ~~* anything = ~~0, ~~0 (or alternately, 0 + 0 = 0 no matter how many times you repeat the addition), so no matter how many copies of zero are added, the result will always be what was initially there -- zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / (any infinite number) evaluates to an ''infinitesimal'', which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities -- 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

10th Nov '13 2:36:17 AM

**NarrativeLock** Is there an issue? Send a Message

**Changed line(s) 5,8 (click to see context) from:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' * ''c'') * (''a'' * ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. This isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

The real projective line (real numbers with infinity thrown in) does allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's ''almost'' a field, but not quite.

The real projective line (real numbers with infinity thrown in) does allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's ''almost'' a field, but not quite.

**to:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' * ''c'') * (''a'' * ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. ~~This ~~That isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

The [[http://en.wikipedia.org/wiki/Real_projective_line real projective~~line ~~line]] (real numbers with infinity thrown in) ~~does ~~''does'' allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's ''almost'' a field, but not quite.

The [[http://en.wikipedia.org/wiki/Real_projective_line real projective

10th Nov '13 1:17:33 AM

**NarrativeLock** Is there an issue? Send a Message

**Changed line(s) 5,6 (click to see context) from:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. This isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

**to:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' ~~+ ~~* ''c'') * (''a'' ~~+ ~~* ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. This isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

10th Nov '13 1:16:31 AM

**NarrativeLock** Is there an issue? Send a Message

**Changed line(s) 5,6 (click to see context) from:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. This isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see and guess what? ''They still can't

**to:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. This isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still ~~can't ~~

can't properly handle a value describable in two symbols!''

**Changed line(s) 11 (click to see context) from:**

All of this leads to fun mathematics mind-benders where you can mathematically prove that 1=2 and all sorts of other nonsense by neatly leaving out the important bit about equations of the form x/x=1 ''when x does not equal 0''.

**to:**

All of this leads to fun mathematics mind-benders where you can mathematically prove that 1=2 and all sorts of other nonsense by neatly leaving out the important bit about equations of the form x/x=1 ''when x does not equal ~~0''.~~0''[[note]]example: 1*0 = 2*0 = 0, therefore (1*0)/0 = (2*0)/0 = 1*(0/0) = 2*(0/0) (associativity -- see above), therefore 1*1 = 2*1 = 1 = 2[[/note]].

10th Nov '13 1:10:52 AM

**NarrativeLock** Is there an issue? Send a Message

**Changed line(s) 3,6 (click to see context) from:**

Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' - that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' zero, such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But as mentioned before, 0 + anything = 0, so no matter how many copies of zero are added, the result will always be what was initially there - zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / any of these evaluates to an ''infinitesimal'' number, which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities - 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. It is not the only one, however, and nowhere ''near'' the largest - the largest (and most general) one is usually called the

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. It is not the only one, however, and nowhere ''near'' the largest - the largest (and most general) one is usually called the

**to:**

Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' ~~- ~~-- that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' ~~zero, ~~zero (0), such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But as mentioned before, 0 + anything = 0, so no matter how many copies of zero are added, the result will always be what was initially there ~~- ~~-- zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / ~~any of these ~~(any infinite number) evaluates to an ~~''infinitesimal'' number, ~~''infinitesimal'', which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities ~~- ~~-- 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them.~~It is ~~This isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ~~only one, ~~''only'' totally ordered field, however, and nowhere ''near'' the largest ~~- ~~-- the largest (and most general) one is usually called ~~the ~~

the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see and guess what? ''They still can't

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them.

10th Nov '13 12:53:04 AM

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**Changed line(s) 3,4 (click to see context) from:**

Also, there are various sets of numbers that include "infintesimals" which are smaller in magnitude than any finite number other than zero. It's commonly possible to divide by an infinitesimal, which would produce an infinite number. You still can't divide by zero.

**to:**

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' + ''c'') * (''a'' + ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. It is not the only one, however, and nowhere ''near'' the largest - the largest (and most general) one is usually called the

25th Dec '12 4:56:06 AM

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Full disclosure: Mathematics has more than one meaning of zero. While you can't divide by zero in real numbers, you can use a process called "limits" to do something similar. For example, when you take the limit of dividing a positive number ''n'' by "positive zero" (the limit of ''x'' as ''x'' approaches zero from the positive side), you get another limit called "positive infinity," which basically means that as the number you divide by gets smaller, so long as it's above zero, the result gets larger. Taking the limit of dividing the same number by "negative zero" produces "negative infinity", and dividing by "two-sided zero" produces "projective infinity". But the physical consequences of these infinities, on the other hand, lead to [[OurWormholesAreDifferent things like black holes]].

Also, there are various sets of numbers that include "infintesimals" which are smaller in magnitude than any finite number other than zero. It's commonly possible to divide by an infinitesimal, which would produce an infinite number. You still can't divide by zero.

The real projective line (real numbers with infinity thrown in) does allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's ''almost'' a field, but not quite.

Related to the above, there's also a rather weird kind of mathematical structure called a [[https://en.wikipedia.org/wiki/Wheel_theory wheel]] where division is always possible. Some of the rules of algebra have to be sacrificed for this; for example, x/x=1 is not always true, and nor are 0x=0 or x-x=0. But it is a real thing.

All of this leads to fun mathematics mind-benders where you can mathematically prove that 1=2 and all sorts of other nonsense by neatly leaving out the important bit about equations of the form x/x=1 ''when x does not equal 0''.

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Also, there are various sets of numbers that include "infintesimals" which are smaller in magnitude than any finite number other than zero. It's commonly possible to divide by an infinitesimal, which would produce an infinite number. You still can't divide by zero.

The real projective line (real numbers with infinity thrown in) does allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's ''almost'' a field, but not quite.

Related to the above, there's also a rather weird kind of mathematical structure called a [[https://en.wikipedia.org/wiki/Wheel_theory wheel]] where division is always possible. Some of the rules of algebra have to be sacrificed for this; for example, x/x=1 is not always true, and nor are 0x=0 or x-x=0. But it is a real thing.

All of this leads to fun mathematics mind-benders where you can mathematically prove that 1=2 and all sorts of other nonsense by neatly leaving out the important bit about equations of the form x/x=1 ''when x does not equal 0''.

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