Additive Inverse
Additive inverse is what number you add to a given number to make the sum zero. For example, if we take the number 3 and add 3 to it, the result is zero. Hence, the additive inverse of 3 is 3. We come across such situations in our dailylife where we nullify the value of a quantity by taking its additive inverse. The right amount of fire and ice will cancel out each other and the result will be zero. Let us learn the additive inverse property of real and complex numbers.
What is Additive Inverse?
The additive inverse of a number is its opposite number. If a number is added to its additive inverse, the sum of both the numbers becomes zero. The simple rule is to change the positive number to a negative number and vice versa. 7+ (7) =0. Thus 7 is the additive inverse of 7 and 7 is the additive inverse of 7.
Additive Inverse Property
When the sum of two real numbers is zero, then each real number is said to be the additive inverse of the other. R+(R) = 0, where R is a Real number. R and R are the additive inverses of each other. For example: ¾+ (¾) = 0. Here ¾ is the additive inverse of ¾and ¾ is the additive inverse of ¾.
Let’s say you have a bucket of water at room temperature. You add a liter of hot water to it which makes the overall temperature of the bucket rise by a certain amount. Now, add another liter of cold water to it. The contrasting temperatures of water added to the bucket will cancel out each other, and the result will be a bucket of water at room temperature. The same rule applies while finding the additive inverse of a number. The additive inverse property holds good for both real numbers and complex numbers.
Additive Inverse of Real Numbers
The given number can be a whole number, a natural number, an integer, a fraction, a decimal, or any real number. The additive inverse is just the negative of the given number.
Example  Given Number  Additive Inverse 

Natural Numbers  12  12 
Whole Numbers  6  6 
Integers  12  12 
Fractions  4/5  4/5 
Decimals  2.5  2.5 
Additive Inverse of Complex Numbers
The algebraic property of complex numbers states the existence of additive inverse. Given any complex number z ∈ C, there is a unique complex number, denoted by −z, such that z+ (−z)=0. Moreover, if z = (x,y) with x,y ∈ R, then −z = (−x,−y).
Let Z = x + iy be the given complex number. Then its inverse is Z = xiy. For example, the additive inverse of i1 = (i1) = i +1
Additive Inverse Formula
The general formula for the additive inverse of a number can be given in the form of the number itself. Any number when added to its negative will cancel out each other and give the overall sum as zero. We require to find the negative of that given number N. In other words, we need to find –1 × (N). Hence, we can say that :
Additive Inverse of N = –1 × (N)
Additive Inverse in Algebraic Expressions
The property of additive inverse terms can be extended to algebraic expressions. Following the same rule as stated above, the additive inverse of an algebraic inverse is one that makes the sum of all the terms zero. Note: The additive inverse of the expression is (expression). The additive inverse of x^{2 }+ 1 is  (x^{2 }+ 1) = x^{2 } 1.
For example, Jimmy had two apples and three peaches. The total number of apples and peaches he had can be represented as 2x + 3y, where 'x' represents the apples and 'y' represents the peaches. He distributed the fruits among his five other friends. Now, the expression can be written as:  x  x  y  y  y, which shows how all the fruits he had are distributed or given away. Therefore, the additive inverse of 2x + 3y is 2x  3y, making the sum of all the elements zero.
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Additive Inverse Examples

Example 1: What is the additive inverse of 6/14?
Solution:
We know that the given number and its additive inverse = 0
Let x be the additive inverse.
6/14 + x = 0
x = 6/14
Answer: The additive inverse.of  6/14 is 6/14.

Example 2: What is the additive inverse of the expression 13x + 5y  9z?
To find the additive inverse of the given expression, we need to find the additive inverse of the whole expression.
The additive inverse is calculated by multiplying the whole equation by 1.
1(13x + 5y  9z) = 13x  5y + 9z
Answer: The additive inverse of the given expression is –13x – 5y + 9z.

Example 3: Find the sum of the complex number 1 + i√3 and its additive inverse.
Solution: Given z = 1 + i√3
The additive inverse of z = z
z= (1 + i√3) = 1 i√3
1 + i√31 i√3 = 0
Answer: The sum of the complex number 1 + i√3 and its additive inverse is 0.
FAQs on Additive Inverse
What is the Additive Inverse?
Additive inverse is what you add to a number to make the sum zero. For example, the additive inverse of 4 is 4 because their sum is zero. When two numbers are added together to get 0, then we say both the numbers are additive inverses of each other.
How Do You Find The Additive Inverse of a Given Number?
Just change the sign of the given number and we get its additive inverse. For example, the additive inverse of 8 is 8. The additive inverse of 1/8 is 1/8. The additive inverse of 35 is 35.
What is the Additive Inverse Formula?
The additive inverse formula is 1 × R, where R is any real number.
What is the Additive Inverse of zero?
Since zero does not have a positive or negative sign associated with it, the additive inverse of zero is zero.
Is Additive Inverse Same as Additive Identity?
No, additive inverse and the additive identity are not the same. The additive inverse of a given number is got by just reversing its sign. Together the given number and its additive inverse give 0. Whereas, the additive identity of any given number is 0, as the additiion of the given number and zero gives the same number.
The additive inverse of 4 is 4.( 4  4 = 0)
The additive identity of 4 is 0. (4 + 0 = 4)
What is the Difference Between Additive Inverse and Multiplicative Inverse?
Additive inverse is what you add to a number to make the sum zero, whereas, the multiplicative inverse is the reciprocal of the given number, which when multiplied together, gives the product as 1.
Does 0 Have an Additive Inverse?
Since zero does not have a positive or negative sign associated with it, the additive inverse of zero is zero.
What is the Additive Inverse of 8?
The additive inverse of 8 is 8. This can be verified as: 8 +( 8 )= 0.
What is the Additive Inverse of 12?
The additive inverse of 12 is 12. This can be verified by: 12 +( 12) = 0.
What is the Additive Inverse of 2/9?
The additive inverse of 2/9 is 2/9. This can be verified by: 2/9 +(2/9) = 0.
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