(Number)^0 = 1

mathxyz · 2024-04-30 09:49:40

Let a = any number

Prove that a^0 = 1, if a does not equal 0.

Last edited by mathxyz (2024-04-30 09:53:53)

Bob · 2024-04-30 19:59:49

Read the (number)^1 answer first.

Using rule one a^n x a^0 = a^(n+0) = a^n so a^0 acts like it is 1 so it is defined to be 1.

Bob

mathxyz · 2024-05-01 04:53:04

Bob wrote:

Read the (number)^1 answer first.
Using rule one a^n x a^0 = a^(n+0) = a^n so a^0 acts like it is 1 so it is defined to be 1.
Bob

Ok. I totally will compare notes for a^1 = a and a^0 = 1..

KerimF · 2024-05-02 09:26:08

[a*a*a*a... m times / a*a*a*a... n times] = a^m/a^n = a^(m-n)
Similary, for any value of x we can write:
1 = a^x/a^x = a^(x-x) = a^0

Phrzby Phil · 2024-05-02 15:06:08

In my opinion this is not per se a "proof," but rather several explanations for why a^0=1 (for a not= 0) is a reasonable definition.

mathxyz · 2024-05-02 18:20:20

Phrzby Phil wrote:

In my opinion this is not per se a "proof," but rather several explanations for why a^0=1 (for a not= 0) is a reasonable definition.

Can you then show the actual mathematical proof?

Bob · 2024-05-03 05:18:32

In order to make a proof you need to start with the axioms for the behaviour of indices. As these rules are fairly obvious for powers that are positive integers I prefer to show students those first and then develop general properties from there.

I'd be interested in seeing an actual proof

Bob

mathxyz · 2024-05-03 08:04:00

Bob wrote:

In order to make a proof you need to start with the axioms for the behaviour of indices. As these rules are fairly obvious for powers that are positive integers I prefer to show students those first and then develop general properties from there.

I'd be interested in seeing an actual proof

Bob

You are correct. I like to know the WHY of things. Anybody can memorize a bunch of formulas but knowing how to derive them is actual learning.

Phrzby Phil · 2024-05-03 11:47:28

A definition does not have a proof.

What it does need is to be consistent with other statements in a given theory, and to be useful in the various formulas.

Bob's and KerimF's explanations show how it fits with the rules of arithmetic, as odd as it seems for a^0 = 1.

Last edited by Phrzby Phil (2024-05-04 07:37:58)

mathxyz · 2024-05-04 00:50:02

Phrzby Phil wrote:

A definition dos not have a proof.
What it does need is to be consistent with other statements in a given theory, and to be useful in the various formulas.
Bob's and KerimF's explanations show how it fits with the rules of arithmetic, as odd as it seems for a^0 = 1.

I will return to this topic at another time.

mathxyz · 2024-05-08 10:27:17

Phrzby Phil wrote:

A definition does not have a proof.
What it does need is to be consistent with other statements in a given theory, and to be useful in the various formulas.
Bob's and KerimF's explanations show how it fits with the rules of arithmetic, as odd as it seems for a^0 = 1.

https://m.youtube.com/watch?v=kPTp82EGjv8&pp=ygUNUHJvdmUgYV4wID0gMQ%3D%3D

mathxyz · 2024-05-08 10:32:31

Bob wrote:

In order to make a proof you need to start with the axioms for the behaviour of indices. As these rules are fairly obvious for powers that are positive integers I prefer to show students those first and then develop general properties from there.

I'd be interested in seeing an actual proof

Bob

https://m.youtube.com/watch?v=kPTp82EGjv8&pp=ygUNUHJvdmUgYV4wID0gMQ%3D%3D

Math Is Fun Forum

#1 2024-04-30 09:49:40

(Number)^0 = 1

#2 2024-04-30 19:59:49

Re: (Number)^0 = 1

#3 2024-05-01 04:53:04

Re: (Number)^0 = 1

#4 2024-05-02 09:26:08

Re: (Number)^0 = 1

#5 2024-05-02 15:06:08

Re: (Number)^0 = 1

#6 2024-05-02 18:20:20

Re: (Number)^0 = 1

#7 2024-05-03 05:18:32

Re: (Number)^0 = 1

#8 2024-05-03 08:04:00

Re: (Number)^0 = 1

#9 2024-05-03 11:47:28

Re: (Number)^0 = 1

#10 2024-05-04 00:50:02

Re: (Number)^0 = 1

#11 2024-05-08 10:27:17

Re: (Number)^0 = 1

#12 2024-05-08 10:32:31

Re: (Number)^0 = 1

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