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Re: On Multiplicative Inverse Notation
Posted:
May 21, 2000 10:48 PM
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Pagadala wrote:
> The minus sign is prefixed to "a" but not to the additive identity(0)
> in case of additive inverse, while the multiplicative identity (1)
> gets
> the minus sign in case of multiplicative inverse.
Because they are different things. a^-1 = 1/a. Even if you ignore the
fact that you're strictly dealing with exponentiation, a superscripted
-1 following an entity is a common way to indicate the inverse; it's
done with all sorts of mathematical objects, including functions and
matrices.
> Also, the definition of multiplicative inverse uses exponential
> notation, while the definition of additive inverse uses only additive
> operators (+ and -).
It is just a notation. That a^-1 suggests to you exponentation is good,
because the symbol is often chosen delierately (with exponentiation a^-1
= 1/a [for all a != 0, of course]), so as to be consistent with
exponentiation rules (which may not have even been introduced yet).
It's just a symbol, just as a prefixed - for the additive inverse is (-a
= 0 - a, but -a is a distinct notation).
> The more appropriate way to define it would have been -
> ".... denoted by 1/a"
> or ".... denoted by 1^(-a)" (?!)
>
> I'm sure you can't digest the second version.
No, because it's wrong. 1^-a = 1/1^a = 1, not the multiplicative
inverse of a. Additive inverse is indicated by a prefixed dash.
Multiplicative inverse is indicated by a postfixed, superscripted -1.
THey are notations which are unrelated to subtraction and
exponentiation, except in that they are chosen to be consistent with it
once those are added to one's repertoire. "-a is the additive inverse
of a" does not involve subtraction at any point, but obviously -a (the
additive inverse) = 0 - a (an operation involving subtraction and the
additive identity).
In fact, subtraction itself is often defined in terms of the additive
inverse: a - b == a + (-b).
> Then, i feel, we shouldn't define it in exponential form.
It's not defined in terms of exponentiation. It is defined in terms of
a superscripted -1. It resembles exponentiation, but that is as a
bonus, not because the notation itself involves exponentiation.
--
Erik Max Francis / max@alcyone.com / http://www.alcyone.com/max/
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