The properties that define a group are:
1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer {e.g., 3 + 5 = 8}.
2. Associativity. The operation must be associative-when combining {by the operation} three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed. Addition, for instance, is associative: {5 + 7} + 13 = 25 and 5 + {7 + 13} = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first.
3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3.
4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + {-4} = 0 and {-4} + 4 = 0.
The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.
1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer {e.g., 3 + 5 = 8}.
2. Associativity. The operation must be associative-when combining {by the operation} three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed. Addition, for instance, is associative: {5 + 7} + 13 = 25 and 5 + {7 + 13} = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first.
3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3.
4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + {-4} = 0 and {-4} + 4 = 0.
The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.
( Mario Livio )
[ The Equation That Couldn't Be ]
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