Integers, rationals and reals
- The integers under addition form a group. The identity element of the group is
, and the additive inverse is just the usual negative. In fact, the group of integers is an Abelian group: addition is commutative for integers. - The rational numbers under addition form a group. The identity element of the group is , and the additive inverse is just the usual negative. This group is Abelian, and the integers form a subgroup.
- The real numbers also form a group under addition. The rational numbers form a subgroup of the group of real numbers, and the integers form a smaller subgroup.
- The nonzero rational numbers under multiplication form a group. The identity element for this group is
. This group is also Abelian.
More generally, given any field, the field is a group under addition, and the nonzero elements of the field form a group under multiplication.
Some non-examples of groups are:
- The natural numbers under addition: There is no additive identity and there are no additive inverses.
- The nonzero integers under multiplication: The nonzero integers under multiplication have a multiplicative identity (namely). Hence, they form a monoid. But not every nonzero integer has an integer as its multiplicative inverse. In fact, the only invertible elements are
.
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