Mathematically, 00 is defined to be 1, but the result can't be deduced using the four basic operations. Is this the reason why a pocket calculator usually returns an error?
Mathematically, 00 is defined to be 1, but the result can't be deduced using the four basic operations. Is this the reason why a pocket calculator usually returns an error?
00 is undefined, not 1. It is sometimes 'assigned' a value of 1 for convenience.
From what I remember in math class:
Assume you have a number XY where Y <> 0.
Divide it by itself: (XY ) / (XY )
Since the base X is the same, you divide by subtracting the Y of the denominator from the Y of the numerator, giving us Y-Y=0.
Since any non-zero number divided by itself = 1, we therefore derive that X0 = 1.
Now assume X = 0.
Zero to any power is still zero, so (0Y ) = 0, and that leaves us with 0/0.
You can't divide by zero. Result is undefined.
Or something like that.
What's the difference between a definition and an assignment? Windows Calculator returns 1, as do higher programming languages.
You can also prove that lim(x->0)xx = 1.
My TI-89 in school always made a point of telling you that 00 was assumed to be 1, and that may make the displayed result inaccurate.
A similar case occurs when 1/0 is defined to be infinity. Infinity is the result of lim(x->0) 1/x . But you can really screw things up if you make the assumption. Combine it with 1/∞ = 0, and you can "prove" 0 = 1.
Yes, but lim(x->0)0x = 0. So defining the result of 00 as 1 makes the function 0x discontinuous.
Yes, for many reasons it is still a good idea in some cases to define 00 = 1, for the same reason that we often define sin(0)/0 = 1. But it requires some care, and I personally have no problem saying that we let it be undefined instead.
The problem with using limits to find the value in question is that limits are what it could be, not what it is. That is, limits are an infinite process that gets closer and closer to the value but never actually reaches it. Then, it is for the above reasons that it is undefined.