Problems_with_Zero

Problems with Zero Zero is a perfect number, but zero is also a dangerous number. Many things can horribly go wrong with zero and you must really careful when you handle it. There is something that you cannot do with zero : you can’t divide something by zero or you can’t have thing like zero to the power of zero. And I was asked for this all the time from my younger brother, “Why can’t I divide by zero, I want to divide by zero and it’s infinity.. bla bla bla..” I’ll show him why we can’t divide by zero and it’s not just infinity, it’s more complicated than that. Let’s take a look at multiply, if you want multiply 10 by 5, you can do it by add 10 to 10 five times. And division is opposite to multiplication, that’s mean when you divide 20 by 4, you subtract 4 from 20 0+10=10 20-4=16 10+10=20 16-4=12 5 times 20+10=30 5 times 12-4=8 30+10=40 8-4=4 40+10=50 4-4=0 But now if I divide by zero, that means I subtract something by zero over and over, let’s try 10 divided by 0 : 10-0=10 10-0=10 infinity 10-0=10 10-0=10 ……… So it takes infinity times to divide 10 by zero, but you cannot say something equals to infinity; it’s like you say 1+1 = Green. And now my brother argue that there’s nothing wrong writing 10=infinity. Here’s why, if 10=infinity and 20=infinity, does that mean 10=infinity= 20 and therefore I can say 1=2 ' That’s nonsense. And now it’s time to calculus get involved, if we take the limit of 1x as x goes very close to zero limx→01x=infinity In this case, we can basically say that 10=infinity, but we have 2 different answers for this problem. As x goes to zero from the right, we have positive infinity and negative infinity if x goes from the left. What about 00 (zero to the power of zero) ' The other thing that my brother gets very annoy about is zero to the power of zero, because he has been taught that anything to the power of zero is one; and zero to the power of anything is zero. Now he doesn’t know what happen when they collide. This time, we have : limx→0xx And we also need to approach from both directions, limx→0+xx=1 and limx→0-xx=1 Because the left and right hand limit is equal to 1, so we can say limx→0xx=1 . But wait, there’s more! We only looked at the real number line, what if we use the complex numbers' In that case, when x approaches zero, the function xx approaches zero So that we can’t say for sure whether 00 equals 0 or 1; and the most satisfaction answer for 00 is undefined. How about 00 ' This is a very interesting question. In fact, it can be any answer you want it to be. Let’s start with xy . Now let y=x so the previous equation becomes xx At 0 , limx→0-xx=limx→0+xx=1 We also have y=-x , so now the equation is -xx ; at 0, limx→0--xx=limx→0+-xx=-1 And now y=0 ; therefore xy=x0 . And this brings us back to the very first part of dividing by zero. As we talked about this the acceptable answers should be ±∞ If the function goes vertically, this means x=0; and again, the equation turns into 0y and that equals to zero. So the answer for equation xx depends on which angle you approach, and so that you can make any answer you want it to be. Sources AP Calculus book TI-84 Plus Emulator Wolfram Alpha