Proving a^0=1: Step-by-Step Guide

[Rada Demorn]

I always thought that a point is nil-dimensional.

Perhaps I didn't convey the meaning properly that a power is dimensionality... ( i hope you don't take all this so seriously, eh? )

 

[Rada Demorn]

If we have a cube of side 2, it's volume is 8, Or, if we have a cube of bricks, with each side 2 bricks long, we have 8 bricks.

For a square of side 2, it's area is 4. Or, a square of bricks needs 4 bricks.

For a line of length 2, it's length is 2. Or, a line of bricks needs 2 bricks.

For a point, it's size is 0. Or, if we have a zero-dimensional array of bricks, then there are no bricks.

By this intuitive reasoning, we should have $2^0 = 0$.

Very, very wrong.

2^4 is the Figurate Number represented by the vertices (points) of a hypercube ( Dimension 4 }

2^3 is the Figurate Number represented by the vertices (points) of a cube ( Dimension 3 )

2^2 ................ of a square.

2^1 ........... by the end points of a line. ( Dimension 1 )

And 2^0 is just the point or the atom itself of Nil-Dimension.

 

[mathwonk]

rada you are quite right that we need the condition f(0) ≠ 0, and this is guaranteed by my assumption that f(x+y) = f(x).f(y) for all x,y, and f(n) = a^n for n = a positive integer and a > 0, since then f(1) = a ≠ 0, and thus f(x) ≠ 0 for all x. (if f(x) = 0, for some x, then for all y, f(y-x + x) = f(y-x).f(x) = f(y-x).0 = 0. i.e. thus either f(x) = 0 for all x or f(x) never equals zero.)