Confusion with definition and notation of reciprocal.

[infranatural]

Hello everyone,

I have some conceptual issues with aforementioned definitions.

How is exactly multiplicative inverse defined? Say, for a rational, nonzero number a/b, its reciprocal is b/a. Is there a certain operation that transforms a/b to b/a?

Also, the notation for multiplicative inverse of any real number (except zero) x is 1/x. Is 1/x a unique symbol or one that indicates operation of division of 1 by x?
For example, if x=2/3, should i see its inverse as 1/x=3/2, or as an operation of division, that is 1/x=1/(2/3)? I know that in the end the answer is the same, but what i'd like to know is if division is included in the "process" of obtaining that inverse or is it by definition that we just "flip" the numbers.

 

[The Jericho]

Hey,

a^-b

(a "raised to" -b)

Or do you mean an alternative way to this too?

 

[infranatural]

Hey,

a^-b

(a "raised to" -b)

Or do you mean an alternative way to this too?

No, no, a/b, a rational number, where a is some nonzero integer, and b is a natural number. No exponentiation here.

 

[HallsofIvy]

If a is any non-zero number then its reciprocal is defined as the number, b, such that ab= 1.

"Is 1/x a unique symbol or one that indicates operation of division of 1 by x?" Yes, it indicate division of 1 by x. If ab= 1, and a is not 0, we can divide both sides by a to get b= 1/a.

Your question seems to be more about notation than mathematics.

 

[CellCoree]

Greetings,

I understand your confusion with the definition and notation of reciprocal. Let me provide some clarification on this topic.

To start, the multiplicative inverse of a number is defined as the number that, when multiplied by the original number, gives a product of 1. In the case of a rational number a/b, its multiplicative inverse would be b/a. This can also be thought of as "flipping" the numerator and denominator. There is not a specific operation that transforms a/b to b/a, it is simply a matter of understanding the concept of a multiplicative inverse.

Regarding the notation, 1/x is a unique symbol that represents the multiplicative inverse of x. It is not indicating the operation of division, but rather it is a shorthand way of writing the inverse. So for your example, if x=2/3, its inverse would be written as 1/x=3/2, not as 1/x=1/(2/3). However, both expressions ultimately represent the same number, just written in a different way.

I hope this clears up any confusion you may have had. It is important to understand the concept of a multiplicative inverse and how it relates to division, but in terms of notation, it is simply a matter of convention. Keep exploring and questioning, that is the essence of science!