Well, I guess it's suppsoed to be some kind of joke or comment on maths.
The idea is that the equation x2 +1=0 has
1) no real number solution, since each x in R has non-negative square
2) solution i in complex numbers, C, and
3) solutions 2 and 3 in quotient group Z5.
The idea of C is that while there's no real number x such that x2 =-1, we can imagine there is one, so let's call it imaginary number i. Some people did this during and after renessaince as a tool to solve equations, but they didn't accept them as answers (if I remember correctly). Then they found out that these imaginary numbers can be used in a rigorous way: calculate like i is just a unknown variable, and at the end use knowledge that i2=-1. For example:
(x+2i)(2x2-i)=2x3-ix+4ix2-2i2 (pay attention to the last term)
=2x3+4ix2-ix -2(-1)
=2x3+4ix2-ix +2. Here we have used i2=-1.
You also have geometric interpretation of C when instead of thinking numbers as a straight, you think a plane: Complex number x+iy is a point on a plane, whose cordinates are (x,y):
So the numbers without imaginary part are only the x-axis, which corresponds the straight of real numbers. Complex numers are added then like vectors, and for the multiplication there's simple geometric interpertation too, but explaining it on this brief text would probably just get you confused.
The quotient group Z5 is just group {0,1,2,3,4} (total of five numbers), and you calculate there just like with natural numbers, but whenever you get something bigger than four, you substract so many fives that the number is in the group again.
For an example, 3+2=0 (calculate 3+2=5, and substract 5).
2+2+2=1 (calculate 2+2+2=6 and substract 5)
4*3=2 (calculate 4*3=12 and substract 5, you get 7, substract 5 again, and you get 2)
22 +1 =0 (calculate 2*2+1=5 and substract 5).
32 +1 =0 (calculate 3*3+1=10, substract 5, you get 5, and substract 5 again, and you get 0).
Quotient groups are pretty simple also, and are covered in elementary algebra books. This wikipage seems to be pretty easy to understand too (probably more so than my explanation, but since written I won't bother to erase it).
Ambiguity of mathematics is whole different thing though. The equation at the top is itself ambiguous as it doesn't say what kind of thing x is supposed to be, so it's no wonder that the answers are ambiguous too. It's like asking "who is the president?". You have to narrow it down, president of what country? If the country is US, it's Barack Obama. If Finland, Tarja Halonen. If Sweden, there's no answer. However that doesn't mean that the state offices were ambiguously held.
The whole thing relies on the high school -way of thinking that equations themselves are meaningful, but of course you have to specify to what group the unknown variable belongs. This on the other hand comes from the thought that maths deals only with numbers, and the word "number" itself is unambiguous. So this thing boils down to asking amiguous question, and when there's no answer deducing that maths is ambiguous. Little like child thinking that "Who?" is in itself meaningful question.
As often in these cases, it also contains error, since also x=-i solves the equation in C.
The idea is that the equation x2 +1=0 has
1) no real number solution, since each x in R has non-negative square
2) solution i in complex numbers, C, and
3) solutions 2 and 3 in quotient group Z5.
The idea of C is that while there's no real number x such that x2 =-1, we can imagine there is one, so let's call it imaginary number i. Some people did this during and after renessaince as a tool to solve equations, but they didn't accept them as answers (if I remember correctly). Then they found out that these imaginary numbers can be used in a rigorous way: calculate like i is just a unknown variable, and at the end use knowledge that i2=-1. For example:
(x+2i)(2x2-i)=2x3-ix+4ix2-2i2 (pay attention to the last term)
=2x3+4ix2-ix -2(-1)
=2x3+4ix2-ix +2. Here we have used i2=-1.
You also have geometric interpretation of C when instead of thinking numbers as a straight, you think a plane: Complex number x+iy is a point on a plane, whose cordinates are (x,y):
So the numbers without imaginary part are only the x-axis, which corresponds the straight of real numbers. Complex numers are added then like vectors, and for the multiplication there's simple geometric interpertation too, but explaining it on this brief text would probably just get you confused.
The quotient group Z5 is just group {0,1,2,3,4} (total of five numbers), and you calculate there just like with natural numbers, but whenever you get something bigger than four, you substract so many fives that the number is in the group again.
For an example, 3+2=0 (calculate 3+2=5, and substract 5).
2+2+2=1 (calculate 2+2+2=6 and substract 5)
4*3=2 (calculate 4*3=12 and substract 5, you get 7, substract 5 again, and you get 2)
22 +1 =0 (calculate 2*2+1=5 and substract 5).
32 +1 =0 (calculate 3*3+1=10, substract 5, you get 5, and substract 5 again, and you get 0).
Quotient groups are pretty simple also, and are covered in elementary algebra books. This wikipage seems to be pretty easy to understand too (probably more so than my explanation, but since written I won't bother to erase it).
Ambiguity of mathematics is whole different thing though. The equation at the top is itself ambiguous as it doesn't say what kind of thing x is supposed to be, so it's no wonder that the answers are ambiguous too. It's like asking "who is the president?". You have to narrow it down, president of what country? If the country is US, it's Barack Obama. If Finland, Tarja Halonen. If Sweden, there's no answer. However that doesn't mean that the state offices were ambiguously held.
The whole thing relies on the high school -way of thinking that equations themselves are meaningful, but of course you have to specify to what group the unknown variable belongs. This on the other hand comes from the thought that maths deals only with numbers, and the word "number" itself is unambiguous. So this thing boils down to asking amiguous question, and when there's no answer deducing that maths is ambiguous. Little like child thinking that "Who?" is in itself meaningful question.
As often in these cases, it also contains error, since also x=-i solves the equation in C.
)
