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 *?
That's right, so the operation is not well defined.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
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.
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.
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.
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.
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.