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Topic: Equivalent Definition of Exponentiation on N
Replies: 85   Last Post: Dec 2, 2013 10:07 PM

 Messages: [ Previous | Next ]
 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: Equivalent Definition of Exponentiation on N
Posted: Nov 19, 2013 4:29 PM

Correction

Here is an informal development of exponentiation on N (including 0) using x^2 =x*x as a starting point.

1. For all x in N, we have x^2 = x*x.

2. For all x, y we have x^(y+1) = x^y * x.

The case for exponents greater than 2 follows directly from this these 2 rules. What about the exponents 0 and 1?

Two cases to consider: x=/=0, x=0

Case 1: x=/=0.

Applying (2), we work the pattern backwards to obtain:

x^2 = x^1 * x = x * x.

Then x^1 = x by right cancelability.

Working the pattern backwards once more, we have:

x^1 = x^0 * x = x = 1*x

Then x^0 = 1 by right cancelablity.

Case 2: x=0

We cannot "work the pattern backwards," because we cannot use right cancelability as above.

From (2), we do have:

0^1 = 0^0 * 0 = 0

where 0^0 can have any value.

Again, we have x^1 = x, but there does not appear to be any restriction on 0^0. Any value will work.

Therefore, we can define ^ as a binary function on N such that:

1. x^2 = x*x for all x in N
2. x^(y+1) = x^y * x for all x in N

This definition leaves 0^0 undefined in the sense that 0^0 is assumed to be a natural number, but no specific value is assigned to it.

From this definition, we can derive:

1. x^1 = x for all x in N
2. x^0 = 1 for all x in N, x=/= 0
3. all the usual Laws of Exponents (see my previous thread).

Dan
Visit my new math blog at http://www.dcproof.wordpress.com

Date Subject Author
11/13/13 Dan Christensen
11/13/13 William Elliot
11/13/13 Dan Christensen
11/15/13 Shmuel (Seymour J.) Metz
11/13/13 Robin Chapman
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Jussi Piitulainen
11/14/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Dan Christensen
11/14/13 Bart Goddard
11/16/13 gnasher729
11/14/13 Bart Goddard
11/14/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Dan Christensen
11/14/13 quasi
11/14/13 Dan Christensen
11/14/13 quasi
11/14/13 Dan Christensen
11/14/13 Brian Q. Hutchings
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/16/13 Bart Goddard
11/16/13 Dan Christensen
11/16/13 Bart Goddard
11/16/13 fom
11/15/13 Dan Christensen
11/14/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Peter Percival
11/14/13 Robin Chapman
11/14/13 Peter Percival
11/15/13 Robin Chapman
11/15/13 Peter Percival
11/17/13 Marshall
11/14/13 Bart Goddard
11/14/13 Dan Christensen
11/14/13 Bart Goddard
11/14/13 fom
11/13/13 David C. Ullrich
11/13/13 Dan Christensen
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/15/13 Shmuel (Seymour J.) Metz
11/15/13 Jussi Piitulainen
11/25/13 Shmuel (Seymour J.) Metz
11/26/13 Jussi Piitulainen
11/16/13 gnasher729
11/17/13 Dan Christensen
11/17/13 Peter Percival
11/17/13 Dan Christensen
11/17/13 Peter Percival
11/17/13 Dan Christensen
11/19/13 Dan Christensen
11/19/13 LudovicoVan
11/19/13 Dan Christensen
11/29/13 Dan Christensen
12/2/13 Brian Q. Hutchings