Question Video: Determining the Associativity and Commutativity of Division over the Real Numbers | Nagwa Question Video: Determining the Associativity and Commutativity of Division over the Real Numbers | Nagwa

Question Video: Determining the Associativity and Commutativity of Division over the Real Numbers Mathematics • Second Year of Preparatory School

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True or False: √21 ÷ (9 ÷ 3) = (√21 ÷ 9) ÷ 3. Is division associative or nonassociative in ℝ? True or False: √7 ÷ √2 = √2 ÷ √7. Is division commutative or noncommutative in ℝ?

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Video Transcript

True or False: The square root of 21 divided by nine divided by three is equal to the square root of 21 divided by nine divided by three. Is division associative or nonassociative in the set of real numbers? True or False: The square root of seven divided by the square root of two is equal to the square root of two divided by the square root of seven. Is division commutative or noncommutative in the set of real numbers?

In the first part of this question, we need to check whether a proposed equation holds true. We can see that the two sides of the proposed equation involve the division of the same three real numbers. However, we are evaluating the divisions in a different order. To check if the equation holds true, we need to check if both sides of the equation are equal. To do this, let’s start by trying to rewrite both sides of the equation in the same form.

We first note that we can evaluate nine divided by three as three. So, the left-hand side of the equation is equal to the square root of 21 divided by three. We can rewrite the right-hand side of the equation in the same form by first rewriting root 21 divided by nine as root 21 over nine. We can then recall that dividing by three is the same as multiplying by one-third. Therefore, the right-hand side of the proposed equation is equal to root 21 over nine multiplied by one-third, which we can calculate is equal to root 21 over 27. We can then note that dividing root 21 by three will be a bigger number than dividing root 21 by 27. So these expressions cannot be equal. Hence, the answer to the first part of the equation is false.

In the second part of the question, we want to determine whether or not division is an associative operation over the set of real numbers. We can recall that for an operation to be associative, we need to be able to evaluate the operations in any order. Therefore, we need to check if 𝑎 divided by 𝑏 divided by 𝑐 is equal to 𝑎 divided by 𝑏 divided by 𝑐 for any real numbers 𝑎, 𝑏, and 𝑐.

However, we have already shown that this is not the case in the first part of the question. We have shown that if 𝑎 is equal to root 21, 𝑏 is equal to nine, and 𝑐 is equal to three, then the order we evaluate the division will affect the value. Hence, we have shown that the division operation is nonassociative over the set of real numbers.

In the third part of this question, we need to check if a proposed equation holds true. We can see that in the proposed equation we are switching the order of the numbers in the division. To check if the equation holds true, we first note that the square root of seven is greater than the square root of two, and both are positive. Therefore, dividing root seven by root two must be greater than dividing root two by root seven. Hence, the two sides of the proposed equation are not equal. So the answer is false.

In the final part of this question, we need to determine whether or not division is a commutative operation over the set of real numbers. To do this, we can start by recalling that a commutative operation is one in which we can switch the order of the elements. Therefore, for division to be commutative over the set of real numbers we need 𝑎 divided by 𝑏 to be equal to 𝑏 divided by 𝑎 for all real numbers 𝑎 and 𝑏. However, we have already shown that this is not the case in the third part of this question, since root seven over root two is not equal to root two over root seven. Hence, we have shown that the division operation is noncommutative over the set of real numbers.

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