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Q: Which set of numbers forms a field with respect to the operations of addition and multiplication?

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Yes, with respect to multiplication but not with respect to addition.

The set of integers is closed with respect to multiplication and with respect to addition.

No. An addition operation need not even be defined.

Yes, they are.

Closure with respect to addition and multiplication. Cummutative, Associative properties of addition and of multiplication. Distributive property of multiplication over addition.

While natural numbers are closed with respect to addition and mulitplication, they are missing the additive identity (zero). Furthermore, they are not closed with respect to two of the fundamental operations of arithmetic: subtraction and division.

I am not sure there are any fundamental operations of integers. The fundamental operations of arithmetic are addition, subtraction, multiplication and division. However, the set of integers is not closed with respect to division: that is, the division of one integer by another does not necessarily result in an integer.

Change the order of the terms. If A + B = B + A then A and B are said to commute with respect to addition. Although this may seem trivially true in the case of numbers and addition as "normally" defined, it is not true with respect to many mathematical operations.

1 is a whole number. It is the identity element with respect to multiplication but not addition.

Yes. That means that the product of two whole numbers is defined, and that it is again a whole number.

Yes, the whole numbers are closed with respect to addition and multiplication (but not division).The term "whole numbers" is not always consistently defined, but is usually taken to mean either the positive integers or the non-negative integers (the positive integers and zero). In either of these cases, it also isn't closed with respect to subtraction. Some authors treat it as a synonym for "integers", in which case it is closed with respect to subtraction (but still not with respect to division).

The question cannot be answered sensibly for two main reasons.First, the question mentions "there opposites" but does not specify WHERE the "there" refers to. Second, opposites are defined in the context of some operation. The opposite of 3 with respect to addition is -3 while the opposite with respect to multiplication is 1/3. There are other operations which will give yet more "opposites". There is no way of determining which one you mean.

Different sets of numbers have different properties. For example,The set of counting numbers is closed under addition but not under subtraction.The set of integers is closed under addition, subtraction and multiplication but not under division.Rational numbers are closed under all four basic operations of arithmetic, but not for square roots.A set S is "closed" with respect to operation # if whenever x and y are any two elements of S, then x#y is also in S. y = 0 is excluded for division.So, the answer depends on what you mean by "number".

It follows from the fact that real numbers area group with respect to addition and the decimal representation of numbers.

That is because 1 is the identity element of numbers with respect to multiplication.

No, it is not.

When you combine any two numbers in a set the result is also in that set. e.g. The set of whole numbers is closed with respect to addition, subtraction and multiplication. i.e. when you add, subtract or multiply two numbers the answer will always be a whole number. But the set of whole numbers is NOT closed with respect to division as the answer is not always a whole number e.g. 7÷5=1.4 The answer is not a whole number.

First of all there is no such thing as the associative property in isolation. The associative property is defined in the context of a binary operation. A binary operation is a rule for combining two elements (numbers) where the result is another (not necessarily different) element. Common mathematical binary operations are addition, subtractions, multiplication and division and the associative property does not apply to subtraction or division. Having established that it only makes sense to talk about the associative property in the context of an operation, the associative property of real numbers, with respect to addition, states that, for any three real numbers, x, y and z, (x + y) + z = x + (y + z) That is to say, the order in which the OPERATIONS are carried out does not matter. As a result, either of the above sums can be written, without ambiguity, as x + y + z. Thus associativity is concerned with the order of the operations and not the order of the numbers. Also, note that the order of the elements on which the operator acts may be important. For example, with matrix multiplication, (X * Y) * Z = X * (Y * Z) but X * Y â‰ Y * X So matrix multiplication is associative but not commutative.

No. For a set to be closed with respect to an operation, the result of applying the operation to any elements of the set also must be in the set. The set of negative numbers is not closed under multiplication because, for example (-1)*(-2)=2. In that example, we multiplied two numbers that were in the set (negative numbers) and the product was not in the set (it is a positive number). On the other hand, the set of all negative numbers is closed under the operation of addition because the sum of any two negative numbers is a negatoive number.

It means that for certain operations, if the operation appears twice, it doesn't matter in which order you do the operation. For example, for the common addition of real numbers:(a + b) + c = a + (b + c) Here is an example with numbers: (1 + 5) + 11 = 1 + (5 + 11) The parentheses indicate which addition should be done first.

It defines 0 as the identity in the group of numbers with respect to addition.

It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.

Yes, it is.

The concept of an identity property in arithmetic is of a process that does not alter the identity of a number, so with respect to addition, the number zero has the identity property; you can add zero to a number and that number does not change. With multiplication, the number one has the identity property; you can multiply anything by one, and it doesn't change.

All the elements in a group must be invertible with respect to the operation. The element 0, which belongs to the set does not have an inverse wrt multiplication.