Why operation * not defined in Q

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In summary, the conversation discusses the issue of defining the operation l/m * k/n = (l+k)/(m2+n2) in Q, given that l,k € Z and m,n € Z\{0}. It is noted that the operation can produce two different results when negating l and m, leading to the conclusion that the operation is not well defined.
  • #1
pakkanen
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Homework Statement



Show that l/m * k/n = (l+k)/(m2+n2) can not be defined as an operation in Q when l,k € Z and m, n € Z\{0}

I do not know what is the issue here? Should I know something about Q that this not fulfilled by the operation *?
 
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  • #2
pakkanen said:

Homework Statement



Show that l/m * k/n = (l+k)/(m2+n2) can not be defined as an operation in Q when l,k € Z and m, n € Z\{0}

I do not know what is the issue here? Should I know something about Q that this not fulfilled by the operation *?

Hint: What happens if you negate [itex]l[/itex] and [itex]m[/itex]?
 
  • #3
Ok.. So the same operation can produce two different results??

So that l/m * k/n = (l+k)/(m2+n2) ≠ (-l+k)/((-m)2+n2) = -l/-m * k/n = l/m * k/n
 
  • #4
pakkanen said:
Ok.. So the same operation can produce two different results??

So that l/m * k/n = (l+k)/(m2+n2) ≠ (-l+k)/((-m)2+n2) = -l/-m * k/n = l/m * k/n
That's right, so the operation is not well defined.
 
  • #5
Thank you very much jbunniii! Helped me a lot. I think we'll meet again.
 

FAQ: Why operation * not defined in Q

1. Why is the operation * not defined in Q?

The operation * is not defined in Q, or the set of rational numbers, because it does not always result in a rational number. For example, multiplying 1/2 and 1/3 results in 1/6, which is a rational number. However, multiplying 1/2 and 1/5 results in 1/10, which is not a rational number. Therefore, the operation * is not defined in Q to maintain the property that all elements in Q are rational numbers.

2. Can I still use the operation * in Q?

No, you cannot use the operation * in Q because it is not defined. However, you can use other operations such as addition, subtraction, and division, which are all defined in Q and result in rational numbers.

3. What is the significance of Q not having the operation * defined?

The fact that the operation * is not defined in Q highlights the importance of understanding the properties and limitations of different mathematical sets. Q, or the set of rational numbers, only includes numbers that can be expressed as a ratio of two integers. This means that certain operations, like multiplication, may not always result in a rational number and therefore cannot be defined in Q.

4. Can other sets have the operation * defined?

Yes, other sets such as the set of real numbers (R) or the set of complex numbers (C) have the operation * defined. This is because these sets include all numbers, not just rational numbers, and therefore the result of the operation * will also be included in the set.

5. How can I perform multiplication in Q if the operation * is not defined?

You can still perform multiplication in Q by using other operations such as repeated addition or using a calculator. For example, to multiply 1/2 and 1/3, you can add 1/2 three times (1/2 + 1/2 + 1/2 = 3/6) or use a calculator to find the decimal equivalent (0.5 x 0.333... = 0.1666...). However, it is important to remember that the result may not always be a rational number.

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