Math and Arithmetic

# Why are odd integers closed under multiplication but not under addition?

The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.

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## Related Questions

If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.

negetive integers are not closed under addition but positive integers are.

You don't say that "an integer is closed". It is the SET of integers which is closed UNDER A SPECIFIC OPERATION. For example, the SET of integers is closed under the operations of addition and multiplication. That means that an addition of two members of the set (two integers in this case) will again give you a member of the set (an integer in this case).

They are not the same!The set of integers is closed under multiplication but not under division.Multiplication is commutative, division is not.Multiplication is associative, division is not.

The set of integers is not closed under multiplication and so is not a field.

Yes they are closed under multiplication, addition, and subtraction.

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. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.

Any time you add integers, the sum will be another integer.

Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.

Add two positive integers and you ALWAYS have a positive integers. The positive integers are closed under addition.

No. 1 + 3 = 4, which is not odd. In fact, no pair of odds sums to an odd. So the set is not closed under addition.

They form a closed set under addition, subtraction or multiplication.

Because the set is not closed under addition. If x and y are odd, then x + y is not odd.

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.

Yes, the set of integers is closed under subtraction.

No. The set of rational numbers is closed under addition (and multiplication).

Positive integers are greater than zero, negative integers are less than zero. The set of positive integers is closed under multiplication (and form a group), the set of negative integers is not.

A radical integer is a number obtained by closing the integers under addition, multiplication, subtraction, and root extraction.

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