# Term for: “equal and opposite” and “sum to zero” (physics)

I'm looking for a word or term that encompasses both of the following conditions:

• equal and opposite
• AND
• sum to zero

I am looking specifically at a term for a pairing relationship in the context of physics.

Here are some examples:

NUMBER

• ( +5 ) <> ( -5 )
• equal magnitude
• opposite sign
• ( +5 ) ADD ( -5 ) sums to zero

VECTOR / FORCE

• ( magnitude 5 angle 0 degrees ) <> ( magnitude 5 angle 180 degrees )
• equal magnitude
• opposite direction
• ( magnitude 5 angle 0 degrees ) ADD ( magnitude 5 angle 180 degrees ) sums to zero

ELECTRIC CHARGE

• ( 2+ ) <> ( 2- )
• equal magnitude
• opposite charge sign
• ( 2+ ) ADD ( 2- ) sums to zero

Thus far, I've found the terms:

equilibrant : a force capable of balancing another force and producing equilibrium

~ but this is for forces only

anti-parallel : In a Euclidean space, two directed line segments, often called vectors in applied mathematics, are antiparallel, if they are supported by parallel lines and have opposite directions

~ but this is for vectors only and does not strictly mean equal

What may help to convey exactly what I am looking for is to consider the pairing relationship from the "opposite end" - let me express my examples again from this point of view:

NUMBER

• we start with number ( 0 )
• we split ( 0 ) into
• ( +5 ) <> ( -5 )

VECTOR / FORCE

• we start with vector/force ( magnitude 0 )
• we split ( magnitude 0 ) into
• ( magnitude 5 angle 0 degrees ) <> ( magnitude 5 angle 180 degrees )

ELECTRIC CHARGE

• we start with charge ( 0 )
• we split charge ( 0 ) into
• ( 2+ ) <> ( 2- )

## 4 Answers

Additive inverse.

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

Vectors -

All the following examples are in fact abelian groups:

Complex numbers: −(a + bi)  =  (−a) + (−b)i. On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above).

Addition of real- and complex-valued functions: here, the additive inverse > of a function f is the function −f defined by (−f )(x) = − f (x) , for all x, such that f + (−f ) = o , the zero function ( o(x) = 0 for all x ).

More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):

Sequences, matrices and nets are also special kinds of functions.

In a vector space the additive inverse −v is often called the opposite vector of v; it has the same magnitude as the original and opposite direction. Additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is point reflection in the origin. Vectors in exactly opposite directions (multiplied to negative numbers) are sometimes referred to as antiparallel.

vector space-valued functions (not necessarily linear),

In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 + x ≡ 0 (mod 11).

From Wikipedia - Additive inverse

• Thanks. Inverse was one of the terms I did explore, but initially I wasn't completely comfortable with it being a mathematical term, and also having to differentiate between additive inverse and multiplicative inverse. That being said, coupling inverse with @Will Kunkel's suggestion of symmetry has brought me to a good answer. – TaoRich Jul 29 '16 at 8:00

How about just saying "balanced"? From Oxford Dictionaries:

A condition in which different elements are equal or in the correct proportions.

If you'd like something that conveys the idea that they are equivalent more directly, I'd suggest symmetric:

Made up of exactly similar parts facing each other or around an axis; showing symmetry.

Or in this case, you might even want to say antisymmetric:

Unaltered in magnitude but changed in sign by exchange of two variables or by a particular symmetry operation.

• That does make sense, but it doesn't convey all of the meaning I am looking for. The term "balanced" is not limited to a pairing relationship - one could have 4 forces that are balanced, without any direct equal and opposite pairing of any 2 forces. – TaoRich Jul 28 '16 at 14:44
• Perhaps "symmetrical" might be closer, then? – Will Kunkel Jul 28 '16 at 14:47
• Your edit was helpful - symmetry looks like the right place to start. Wiktionary gives this definition: Exact correspondence on either side of a dividing line, plane, center or axis. Exploring the classes of symmetry a bit deeper leads me to reflection symmetry. I'll give this some more thought and will try to refine something like: symmetric pair reflected about zero – TaoRich Jul 29 '16 at 7:43
• And just to comment on antisymmetric - the definition you quote looks like a great fit for my needs, but when I search around I see that the term is used formally with quite a specific meaning in the area of wavefunctions. That will cause some confusion or ambiguity in my writing. – TaoRich Jul 29 '16 at 7:47

This is the best I can think of: mutually counterbalancing

mutual: directed by each toward the other

counterbalance: to have an effect that is opposite but equal to (something) : to balance (something) by being opposite

• Thanks ... counterbalance was also one of the words I did explore previously, but again (in my opinion) it falls short of the generalisation I am looking for - to my mind counterbalance comes with an implicit association with forces. Also, although it does convey equal and opposite, it doesn't quite convey the fact that if we add the pair, then the pair disappears. Yes, there is the concept of equilibrium but I also want a more forceful mental association with annihilation. I feel that inverse captures that better. – TaoRich Jul 30 '16 at 12:55

Thanks to all for the suggestions - you have led me to a good answer so I thought I would post it up as a response to my own question.

I did some searching around these two suggestions:

• @Phil Sweet : inverse
• @Will Kunkel : symmetry

I found this:

• The definition of the inversion symmetry operator is that it transforms a vector into a different vector of same magnitude but antiparallel orientation.

( Nice to see the word antiparallel resurfacing )

So going back to my original question and examples - I'll go with this statement:

" The following can all be classed as inversion symmetry pairs "

NUMBER

• ( +5 ) <> ( -5 )

VECTOR / FORCE

• ( magnitude 5 angle 0 degrees ) <> ( magnitude 5 angle 180 degrees )

ELECTRIC CHARGE

• ( 2+ ) <> ( 2- )

I may refine the terms a bit more, but I'm pretty comfortable with that.

• I'm marking my own answer here as correct - simply to close the question. I'm happy with where I have got to by following the suggestions from the other answers. Thanks all. – TaoRich Jul 30 '16 at 12:40