History Analysis / DivideByZero

30th Oct '15 4:43:25 AM MrUnderhill
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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
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0 + anything = anything, not zero.
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
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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.
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.
10th Nov '13 1:17:33 AM NarrativeLock
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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
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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!''

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
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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
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
10th Nov '13 12:53:04 AM NarrativeLock
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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.
to:
Also, there are various sets of numbers 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 include "infintesimals" which are smaller in magnitude than any finite is, ''a'' / ''b'' is a number other than 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. It's commonly possible This, naturally, leads to divide by an infinitesimal, which would produce an 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 number. You still can't divide by zero. 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
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