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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)
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.