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The set of Real numbers is a group with the operation addition. This implies the following properties for all x, y and z that are real:
- Closure: x + y is real.
- Associativity: (x + y) + z = x + (y + z).
- Identity: There exists a Real number, called the additive identity and represented by 0, such that 0 + x = x = x + 0.
- Invertibility: There exists a Real number, denoted by "-x", such that x + (-x) = 0 = (-x) + x.
In addition, the set of Real numbers is a ring and this means that the set is:
- Commutative: x + y = y + x
There is a second operation defined on the set, multiplication, such that multiplication is distributive over addition.
- Distributivity: x*(y + z) = x*y + x*z.
Finally, the set of Reals is a field so that division by non-zero elements is also defined along with the properties of multiplicative identity and inverses.
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A real number is any continuous quantity which can be represented as a point on a one-dimensional line. Real numbers are used for measuring properties of objects and phenomena in the natural and social world.
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