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Additive Inverse Calculator


Additive-Inverse Calculator

Definition of “ additive-inverse” : The additive-inverse of a number or an expression is another number or expression such that when they are added together, the sum should be equal to zero.
For example, 1 is the additive-inverse of -1. That is because (1) + (-1) = 1 – 1 = 0. Similarly the additive-inverse of x+2 would be –(x+2). 
That is because (x+2) + (-(x+2)) = x + 2 + (-x – 2) = x + 2 – x – 2 = x – x + 2 – 2 = 0 + 0 = 0. 
Additive-inverse is also called the opposite. 
That means, the additive-inverse of a number n is also called the opposite of the number n.

Formula for Calculating Additive-Inverse:

The formula for finding the additive-inverse of a number or expression is as follows:
Ia = -(a), where Ia is the additive-inverse of a and a is any number or expression.
In general terms additive-inverse of any number a is also written as –a.

Proof of Formula of Additive-Inverse:

Suppose there are two numbers a and b that are additive-inverse of each other. 
Then by the definition of additive-inverse that we stated above, we can say that,
a + b = 0 
Subtracting a from both sides we have:
b = 0 – a
Since the zero (0) on the right side has no value we omit it. 
So that now we have:
b = -a
But we know that b is the additive-inverse of a (Ia)
So we can say that : Ia = -a or Ia = - (a)
Hence proved.

Additive-Inverse Property:

If a set of numbers has elements such that each element has its additive-inverse, then that particular set of numbers is said to obey additive-inverse property. 
For example, the set of integers. If we pick an integer, say 5, then its additive-inverse would be -5. 
Therefore the additive-inverse of the integer 5 and -5 exist. 
Similarly additive-inverses of all integers would exist. 
However if we consider the set of natural numbers, we find that the natural number 5 does not have an additive-inverse in the set of natural numbers. 
Similarly none of the natural numbers have additive-inverse in the set of natural numbers. 
Therefore the set of natural numbers does not obey additive-inverse property. 
Except for natural numbers all the other number sets obey additive-inverse property, such as integers, rationals, irrationals, reals, complex, etc. 
 

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From Wikipedia

Additive inverse

In mathematics, the additive inverse, or opposite, of a numbera is the number that, when added to a, yields zero. The additive inverse of F is denoted −F.

For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.

In other words, the additive inverse of a number is the number's negative. For example, the additive inverse of 8 is −8, the additive inverse of 10002 is −10002 and the additive inverse of x² is −(x²).

The additive inverse of a number is defined as its inverse element under the binary operation of addition. It can be calculated using multiplication by −1; that is, −n = −1 × n.

Integers, rational numbers, real numbers, and complex number all have additive inverses, as they contain negative as well as positive numbers. Natural numbers, cardinal numbers, and ordinal numbers, on the other hand, do not have additive inverses within their respective sets. Thus, for example, we can say that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.

General definition

The notation '+' is reserved for commutative binary operations, i.e. such that x + y = y + x, for all x,y. If such an operation admits an identity elemento (such that x + o (= o + x) = x for all x), then this element is unique (o' = o' + o = o). If then, for a given x, there exists x' such that x + x' (= x' + x) = o, then x' is called an additive inverse of x.

If '+' is associative ((x+y)+z = x+(y+z) for all x,y,z), then an additive inverse is unique

( x" = x" + o = x" + (x + x') = (x" + x) + x' = o + x' = x' )

– y instead of x + (–y).

For example, since addition of real numbers is associative, each real number has a unique additive inverse.

Other examples

All the following examples are in fact abelian groups:

  • addition of real 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):
* complex valued functions,
* vector space valued functions (not necessarily linear),
  • sequences, matrices and nets are also special kinds of functions.
  • In a vector space additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is inversion in the origin.
  • In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a+x â‰¡ 0 (mod n). This additive inverse does always exist. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3+x â‰¡ 0 (mod 11).


From Yahoo Answers

Question:

Answers:additive inverse of any number 'a' simply means a number 'b' such that a+b=0, that is b = -a So, if the complex number is 'p + iq', its additive inverse is: - p - iq If you need to know abt multiplicative inverse, google on it, i am attaching a good link i know.

Question:What is the additive inverse of 3? what is the additive inverse of -3?

Answers:An additive inverse is the number when added will equal zero, thus -3 is the additive inverse of 3, and 3 is the additive inverse of -3. Best of luck! :)

Question:I have a few questions i am wondering about... What is the sum of a number and its additive inverse? What is the product of a number and its mulitplicative inverse? please just answer i am really wondering what it is!

Answers:1. Zero; that is the definition of additive inverse. For any number x, the additive number of x is -x, so that x + (-x) = 0. 2. One; that is the definition of multiplicative inverse. For any number x, the multiplicative inverse of x is 1/x, so that (x)(1/x)= 1.

Question:i just need to know these two! you'll get best answer! 2.5 -4 3/5 thanks

Answers:Well, go back to the definition: What do you add to 2.5 to get zero? x + 2.5 = 0 What do you multiply by to get one? x * 2.5 = 1 Solve those, and there's your first pair of answers. Do the same for the second one. If you have a calculator, there's an even easier way: the +/- button does additive inverse; the 1/x button does multiplicative inverse.

From Youtube

Additive Inverses :This short video describes and depicts additive inverses.

Additive Inverse :WEBSITE: www.teachertube.com Short little video giving the definition of additive inverse