Additive identity = 0Multiplicative identity = 1.
For addition, 0 and for multiplication, 1.
1 is the identity for multiplication. 1*x = x = x*1 for all rational x.
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
Yes, with respect to multiplication but not with respect to addition.
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].
For addition, 0 and for multiplication, 1.
1 is the identity for multiplication. 1*x = x = x*1 for all rational x.
A rational number is not. But the set of ALL rational numbers is.
No. The set of rational numbers is closed under addition (and multiplication).
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
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].
Yes, with respect to multiplication but not with respect to addition.
The set of rational numbers is closed under all 4 basic operations.
Yes. In general, the set of rational numbers is closed under addition, subtraction, and multiplication; and the set of rational numbers without zero is closed under division.
You need the rules of multiplication as well as of addition. But multiplication of integers can be viewed as repeated addition. Thus, if p/q and r/s are two rational numbers then their sum is(p*s + q*r)/(q*s)
The way in which the binary functions, addition and multiplication, are defined on the set of rational numbers ensures that the set is closed under these two operations.
Yes, it applies to even multiplication of fractions and rational and irrational numbers.