negative exponents?

[Savage]

Hi, had a question, currently writing a book on negative exponents and I am stuck.

I define exponents as whole, positive numbers

With this definition I have been able to prove the addition theorem of exponents as well as:

a^b divided by a^c = a^(b-c) when b is greater than c

a^b divided by a^c = 1 / [ a^(c-b) ] when c is greater than b

Now, I am tempted to define negative exponents, but I want this book to give a non-bs thorough look at why negative exponents should be allowed and if so, how they can be derived.

Right now, I am tempted to just define them and show how the underlined portion illustrates that, but is there a more elementary way?

[Aves]

Is there a reason for this limitation? I don't see why exponents should be limited to whole, positive numbers.
 

[Savage]

Because I begin by explaining to people that an exponent is a list of how many times we want to multiply the same thing over and over again, so we can condense it. It is like counting, back in the day, before even negative numbers were ever thought of, everything was "nothing or greater"

To allow negative exponents goes against this ideal.

[flyingcat]

Correct me if I'm wrong, but can't you say negative exponents are "dividing" by that many times?

[Savage]

Correct me if I'm wrong, but can't you say negative exponents are "dividing" by that many times?

I want to...I guess what I am getting at is this:

Take fractional exponents (non-integers) for example, I can define those pretty well because in the book I take a look at what the heck we mean when we take a part of an integer (for example, 1/10 = cut something into 10 equal sized pieces, give me 1 piece, then relate this to exponents).

I guess, how would you take normal negative numbers and illustrate examples to help understand what a negative exponent is.

[Aves]

I remember that they used debt to explain negative numbers in grade school; maybe you could go off of that?

I have three apples in my pocket but I owe Bob five; if I give Bob my three apples, I still owe him two apples. -- To demonstrate 3-5 = -2

[Savage]

I remember that they used debt to explain negative numbers in grade school; maybe you could go off of that?

I have three apples in my pocket but I owe Bob five; if I give Bob my three apples, I still owe him two apples. -- To demonstrate 3-5 = -2

Do you have any suggestions for this:

I ultimately would want to justify that 1 / [a^(c-b)]    =    a^(b-c) when c is bigger.

Example, 2 divided by 2^2 = 1/2 = 2^(2-1) on the bottom, but how would I legitimately justify  2^(1-2) on the top from that result?

[dragonsdemesne]

Well, since a negative exponent is equal to the inverse of the positive exponent, i.e. a^(-b) = 1/(a^b), you could perhaps explain it in terms of the inverse somehow.  If you are writing a book for publication, as opposed to a school assignment or something, you probably shouldn't define an exponent as a positive whole number, because exponents can not only be negative, but can be fractional and/or complex as well.  It would, to use an analogy we all understand here, be like saying Elements is a game with four different elements, earth, air, fire, and water, and then be stuck at explaining the presence of the 8 remaining elements, as well as 'other' cards.  Also, I think that in your two stated definitions above, that those would be true whether or not b>c.

[Savage]

Yeah, I thought about it last night. I'll probably just define negative numbers instead of attempting to derive them. My whole idea behind restricting "whole and positive" was to not give so much information at once, but instead make things simple and slowly remove the restrictions one by one as long as define what we mean.

[Arum]

I second the idea of using negative exponents as inverse positive exponents. That makes it easier, and you can say that negative exponents are just a notation used for 1/a^x.