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Q: Is a rational number closed for addition and for multiplication?

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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

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.

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

Division by a non-zero rational number is equivalent to multiplication by its reciprocal.

Other than multiplication by 0 or by its own reciprocal, it if often not possible. Try it with pi, if you think otherwise.

It cannot be closed under the four basic operations (addition, subtraction, multiplication, division) because it is indeed possible to come up with two negative irrational numbers such that their sum/difference/product/quotient is a rational number, indicating that the set is not closed. You will have to think of a different operation.

Yes, it does.

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.

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.

Multiplication is repeated adding. Addition is a number with another number combined for a total.

You can have counting number in multiplication and addition. All integers are in multiplication, addition and subtraction. All rational numbers are in all four. Real numbers, complex numbers and other larger sets are consistent with the four operations.

It the combination is multiplication and the rational number is 0, then the result is rational. Otherwise it is irrational.

Rational and irrational numbers are part of the set of real numbers. There are an infinite number of rational numbers and an infinite number of irrational numbers. But rational numbers are countable infinite, while irrational are uncountable. You can search for these terms for more information. Basically, countable means that you could arrange them in such a way as to count each and every one (though you'd never count them all since there is an infinite number of them). I guess another similarity is: the set of rational numbers is closed for addition and subtraction; the set of irrational numbers is closed for addition and subtraction.

The distributive property of multiplication over addition.

Let your sum be a + b = c, where "a" is irrational, "b" is rational, and "c" may be either (that's what we want to find out). In this case, c - b = a. If we assume that c is rational, you would have: a rational number minus a rational number is an irrational number, which can't be true (both addition and subtraction are closed in the set of rational numbers). Therefore, we have a contradiction with the assumption that "c" (the sum in the original equation) is rational.

No, it cannot. The product of a rational and irrational is always irrational. And half a number is equivalent to multiplication by 0.5

No. 2 + sqrt(5) and 2 - sqrt(2) are both irrational but their sum, 4 is rational.

The relevant property is the closure of the set of rational numbers under the operation of addition.

It is an irrational number.

If a set is closed under an operation. then the answer will be a part of that set. If you add, subtract or multiply any two rational numbers you get another national number. But when it comes to division, it is closed except for one number and that is ZERO. eg 3.56 (rational number) ÷ 0 = no answer. Since no answer is not a rational number, that rational numbers are not closed under the operation of division.