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).
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.
No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.
Rational numbers are closed under multiplication, because if you multiply any rational number you will get a pattern. Rational numbers also have a pattern or terminatge, which is good to keep in mind.
yes
Rational numbers are numbers that can be expressed as a fraction a/b where a and b are both integers and b is not equal to zero. All integers n are rational numbers because they can be expressed as the fraction n/1. Rational numbers are closed under addition, subtraction, multiplication and division by a non-zero rational. To be closed under addition, subtraction, multiplication and division by a non-zero rational means that if you have two rational numbers, when you add, subtract, multiple or divide them, you will get another rational number. For example, take the rationals 1/3 and 4/3. When you add them together, you get another rational number, 5/3. Same with the other operations. 1/3 - 4/3 = -1 (remember integers are rational, too) (1/3) * (4/3) = 4/9 (1/3) / (4/3) = 1/4
Additive identity = 0Multiplicative identity = 1.
Since the set of rational numbers is closed under addition, any rational number added to 0.5 will total another rational number.
It follows from the closure of integers under addition and multiplication.Any rational number can be expressed as a ratio of two integers, where the second is not zero. So two rational numbers may be expressed as p/q and r/s.A common multiple of their denominators is qs. So the numbers could also have been expressed as ps/qs and qr/qs.Both these have the same denominator so their sum is (ps + qr)/qs.Now, because the set of integers is closed under multiplication, ps, qr and qs are integers and because the set of integers is closed under addition, ps + qr is an integer.Thus the sum has been expressed as a ratio of two integers, ps + qr, and qs and so it is a rational number.
Any addition, subtraction, multiplication, or division of rational numbers gives you a rational result. You can consider 8 over 9 as the division of 8 by 9, so the result is rational.
Division by a non-zero rational number is equivalent to multiplication by its reciprocal.
The set of irrational numbers is not closed under addition because there exist two irrational numbers whose sum is a rational number. For example, if we take the irrational numbers ( \sqrt{2} ) and ( -\sqrt{2} ), their sum is ( \sqrt{2} + (-\sqrt{2}) = 0 ), which is a rational number. This demonstrates that adding certain irrational numbers can result in a rational number, confirming that the set is not closed under addition.