Example · High school
Terms
| Term | This example | Why |
|---|
| group |
✓ |
The integers form a group under addition, because adding $0$ changes nothing, adding $-n$ undoes adding $n$, and addition is associative.
|
| ring |
✓ |
The integers, with addition and multiplication, form a ring, because they form a commutative group under addition, multiplication is associative and has the identity $1$, and multiplication distributes over addition.
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| field |
✗ |
The integers, with addition and multiplication, do not form a field, because most integers do not have multiplicative inverses. For example, $2$ has no multiplicative inverse in $\mathbb{Z}$ because $1/2$ is not an integer.
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