Ivan and polina have each drawn a conclusion about multiplying negative fractions. ivan: the product of an even number of negative fractions is always positive. polina: the product of an odd number of negative fractions is always negative. whose conclusion is correct? ivan only polina only both ivan and polina neither ivan or polina

Both are correct. This is easy to notice with a method called game signs, which I will explain below:

Operations with a number of different signals have a negative result. Operations of numbers with equal signs have positive results.

As you can see this is a multiplication operation, soon the game of signs will be done with multiplication.

then by multiplying the first negative 1 with the negative second 1, gives a positive result because both have the same sign. But when this positive result is multiplied with the negative third one of the equation the final result will be a negative 1.

Then -1 x -1 x -1= -1

As already explained, numbers with the same signal give a positive result. Since the equation only has two numbers, the result will be:

-1x-1 = +1

vector v such that t( p) = p+ v