Definition of additive inverse and operation of negation.

[Cicnar]

Hello everyone.

I am slightly confused by these ideas so i would like your help. How is additive inverse defined? Is unary negation an operation in its own right just like those more familiar, like addition, multiplication? Or something else?

 

[micromass]

Hello everyone.

I am slightly confused by these ideas so i would like your help. How is additive inverse defined? Is unary negation an operation in its own right just like those more familiar, like addition, multiplication? Or something else?

It would help us if you would say in which context you ask this. Are you studying things like group theory right now?? Are you talking about the additive inverse in the integers, the reals??

Let's assume that we're talking about [itex]\mathbb{Z}[/itex]. One of the "axioms" for the integers says that for each integer [itex]x\in \mathbb{Z}[/itex], there exists a [itex]y\in \mathbb{Z}[/itex] such that x+y=0.

Now, it can be proven that this y is unique. Indeed, let z be another number such that x+z=0, then y=0+y=(x+z)=(y+x)+z=0+z=z.
This unique y is called the additive inverse of x and is denoted by -x. So for each x, there exists a unique number -x such that x+(-x)=0.

This of course gives a unary operation [itex]\mathbb{Z}\rightarrow \mathbb{Z}:x\rightarrow -x[/itex]. This is indeed an operation in its own right, but it is an operation that can be defined by just using the operation +. Nevertheless, it is something important to look at the inverse as an operation itself.

 

[Cicnar]

Actually, I'm learning high school algebra. But the lack of rigor, if you wish, irritates me. Because of that i was trying to find more info on the internet and by doing so, I've stumbled upon such things like rings, group theory, etc. Now, i don't understand those, though i think i have some intuitive idea what they might represent, but what i found satisfying is the treatment of arithmetic operations as functions (transformations).

Negation is not mentioned at this level, rather, it is "replaced" by the idea of multiplying by -1. Although the results are the same, something was "wrong" to me here, especially to let many students think that additive inverse is defined using properties of another operation (multiplication).
Following the same rationale I presume that multiplicative inverse is also a unary operation? Also, there is operation that maps each element of a set to itself (+a), right? Does it have a name?

One more thing; the minus sign - it has three different meanings, true? It is used to indicate subtraction (binary op.), negation (unary) and one more where it does not represent a transformation (operation) but it is actually a part of symbolical representation of an element, like -3 (negative three), or -a in a+(-a)=0. Is that correct?

I read your proof, makes sense, though i think you dropped a y in one place, y=0+y=Y+(x+z)=(y+x)+z=0+z=z (that capital y).

 

[jbriggs444]

One more thing; the minus sign - it has three different meanings, true? It is used to indicate subtraction (binary op.), negation (unary) and one more where it does not represent a transformation (operation) but it is actually a part of symbolical representation of an element, like -3 (negative three), or -a in a+(-a)=0. Is that correct?

Yes, that is correct.

If you were writing a parsing algorithm for a programming language you would want to distinguish between the use of a minus sign for subtraction, for unary negation or as part of a numeric literal.

 

[Cicnar]

So, real numbers x and -x are equally "worthy" in a symbolical sense, yes? For example, if i have some real world problem that is transcribed into a linear equation, i can denote my unknown quantity with a -x (rather than x) and solve it for -x (not x)? I am asking this because in K12 math, x has "priority" over -x, though, i think, in a mathematical sense, they are on a same level, that is, -x is not some number that you get by applying some operation on x to get -x.

I guess only reason why x is preferred is because we don't have to write that extra minus sign so often (which, in this case, is a way of distinction of symbols, not a unary or binary operation) and we use notation -x only when we need to point out that we are talking about a number opposite of x.

For example, the simple formula for area of a triangle, b*h/2=A where b is base and h is height, A is area. If i rewrote that as (-b)(-h)/2=-A and defined -b to be the base, -h is heigh, and -A is area then i haven't done anything wrong, right? Of course, by defining so, the first, "usual" formula loses its validity, but i think you see what i am asking here.

 

[jbriggs444]

So, real numbers x and -x are equally "worthy" in a symbolical sense, yes? For example, if i have some real world problem that is transcribed into a linear equation, i can denote my unknown quantity with a -x (rather than x) and solve it for -x (not x)? I am asking this because in K12 math, x has "priority" over -x, though, i think, in a mathematical sense, they are on a same level, that is, -x is not some number that you get by applying some operation on x to get -x.

I may not be understanding you here. My apologies if the following does not address your concerns.

I come from a computer science background. And we would say that:

1234.0 is a numeric literal. It specifies a number explicitly.
-1234.0 is also a numeric literal. It specifies a number explicitly.

x is a variable name. In computer science, we would call the "x" an "identifier".

-x is an expression. It denotes the action of taking the additive inverse of the value of the variable named x.

We do not use "-x" as a variable name (aka identifier) because that would be confusing. It would be indistinguishable from the notation "-x" denoting the action of negating a variable named x.


Note that there is a syntactic ambiguity between

-1234.0 as a numeric literal and
-1234.0 as an expression denoting the negation of the numeric literal 1234.0

Although parsers can be anal about such things, in practice this ambiguity is harmless. Either way the value of the term is -1234.0

 

[rcgldr]

Note that there is a syntactic ambiguity between
-1234.0 as a numeric literal and
-1234.0 as an expression denoting the negation of the numeric literal 1234.0

APL is a programming language that uses different syntaxes for this: ¯1234.0 is a numeric literal while -1234.0 is unary negation of the numeric literal 1234.0. I don't know if there are other programming languages that do this.

In finite field math, it is common to only have positive integers, such as {0, 1, 2, 3, 4} for a finite field defined as math modulo 5. In this case -3 always means negation. The additive inverse of 3 is 2, since (3 + 2)%5 == 0 (where % means modulo), so -3%5 == 2. The multiplicative inverse for a is defined as b where (a b)%5 == 1. So (÷1)%5 == 1, (÷2)%5 == 3, (÷3)%5 == 2, (÷4)%5 == 4. All non zero numbers can be considered as powers of 2: (2⁰)%5 == 1 , (2¹)%5 == 2, (2²)%5 == 4, (2³)%5 == 3. If addition or subraction of logarithms is used tor multiply or divide modulo 5, the addition or subraction of logarithms is done modulo 4.

 

[Studiot]

You have asked this question in the maths section about algebra and it is a good one which has received a good answer.

However in the physical world we need to be careful about properties we take for granted in algebra.

For instance we like (a+b) = (b+a) or a = a + (b) + (-b), but in there real world there are simple circumstances where this is not so.

For instance start at P and walk 1000m (b) due south then 1000m due east (a) to Q
Now start again at P and walk 1000m due east then 1000m due south. You will not have arrived at Q.

(a+b) ≠ (b+a)

Again start at P and walk 1000m due east to R (c)
Now start again and walk 1000m due south (d) then walk 1000m due east (c) then walk 1000m due north (-d). You will not have arrived at R

c ≠ c + (d) + (-d)

 

[rcgldr]

For instance start at P and walk 1000m (b) due south then 1000m due east (a) to Q Now start again at P and walk 1000m due east then 1000m due south. You will not have arrived at Q.

That is because the Earth is spherical. Start at 500m north of the equator on an idealized spherical Earth and you will end up at the same spot. If you change the statement to walking .009° (degrees) east and .009° south, the order won't matter with an idealized spherical world.

On a flat plane, vector addition is communitave a + b = b + a.

 

[Cicnar]

Thanks everyone for your effort.

Jbriggs, i see a relation, but i can’t really comment that since my knowledge of computer science is next to none.

Studiot, am i missing something here or is everything that you just said...wrong? Either way, this has nothing to do with what i asked.

EDIT
My apologies Studiot, i was very much wrong.

Anyway, the more i think about it, the more it makes sense.
Number a is on a number line. Also, by definition, number -a is on a number line.
Therefore, minus sign here is a part of symbolical notation (a way to distinct from number a), not an indicator of negation. The fact that we actually come to the number -a by negating number a is a different story. At the end, it boils down to the usual, lazy, old habit that mathematicians have: we use the same notation for distinction and for an operation because we can; it does no harm and it saves a bit of work. Very similar reasoning works for multiplicative inverses, x and 1/x. At least i think so.

 

[Studiot]

For someone just learning high school algebra you show remarkable perception, and suggested you thought your studies were a bit staid.

I thought I would extend your horizon.

Of course my example is correct - it just depends upon your system of algebra - don't forget that the a, b , c etc stand for something.

 

[Cicnar]

Thank you Studiot, both for the kind words and the advice.

Now, can anyone of you guys confirm this? Am i on the right track?

 

[Studiot]

Now, can anyone of you guys confirm this? Am i on the right track?

Micromass indicated in his reply that the answer to your original questions depends upon context.

There are number systems (and other mathematical object) for which we can give meaning to the idea of the binary operations of addition and subtraction.

However the operation of negation of a single number may not make sense in such a number system.

With complex numbers (beyond high school arithmetic) for instance the concept of positive or negative has no meaning. However these numbers have the operation of conjugation to compensate.

You may have done some work with the idea of greater than and less than with real numbers.

This is possible because the real numbers are what is called 'well ordered'. That is we can arrange them in succesive order we call increasing.

However this, again, is not possible with complex numbers.

There are even weirder mathematical objects awaiting your interest.