What are the different properties of a rational number?

Answer 1

The set of rational numbers is a mathematical field. This requires that if x, y and z are any rational numbers then their properties are as follows:

• x + y is rational : [closure of addition];
  
• (x + y) + z = x + (y + z) : [addition is associative];
• there is a rational number, denoted by 0, such that x + 0 = x = 0 + x : [existence of additive identity];
• there is a rational number denoted by -x, such that x + (-x) = 0 = (-x) + x : [existence of additive inverse];
• x + y = y + x : [addition is commutative];
• x * y is rational : [closure of multiplication];
  
• (x * y) * z = x * (y * z) : [multiplication is associative];
• there is a rational number, denoted by 1, such that x * 1 = x = 1 * x : [existence of multiplicative identity];
• for every non-zero x, there is a rational number denoted by 1/x, such that x * (1/x) = 1 = (1/x) * x : [existence of multiplicative inverse];
• x * y = y * x : [multiplication is commutative];
• x * (y + z) = x * y + x * z : [multiplication is distributive over addition].