Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

(Number)^0 = 1

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

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: (Number)^0 = 1

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

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^0 = 1

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

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 248

Re: (Number)^0 = 1

[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

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Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
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Phrzby Phil

Member

From: Richmond, VA

Registered: 2022-03-29

Posts: 50

Re: (Number)^0 = 1

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.

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World Peace Thru Frisbee

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^0 = 1

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

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: (Number)^0 = 1

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^0 = 1

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

Member

From: Richmond, VA

Registered: 2022-03-29

Posts: 50

Re: (Number)^0 = 1

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)

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World Peace Thru Frisbee

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^0 = 1

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

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^0 = 1

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

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^0 = 1

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