Additive Inverse: Formula, Properties, Fraction & Complex Numbers

Additive inverse is the number, which when added to a number, gives a sum of zero. For example, if we take ‘4’ and add ‘-4’ to it, the net result is 0. Therefore, the additive inverse of 4 is -4. By the same dictum, the additive inverse of -4 is -(-4) = 4. Additive inverse is also known as the opposite number, a sign change or negation. To find out the additive inverse of a number we simply have to change the sign of it or multiply it with -1. The properties of the additive inverse are applicable to all the real numbers as well as the complex or imaginary numbers.

Key Takeaways- Additive inverse, complex numbers, multiplicative inverse.

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What is Additive Inverse?

An Additive Inverse of a number can be defined as the value, which on adding to the original number gives a net value of zero. Simply put, the additive inverse of a number will be its opposite number. Suppose, if ‘p’ is an original number then its additive inverse will be ‘-p’, such that p+ (-p)=0. 

Additive Inverse

We can determine the additive inverse for the real numbers as well as complex numbers. The simple rule to find an additive inverse is to change a positive number to the negative number of the same value or vice-versa.

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General Formula of Additive Inverse

To find an additive inverse we have to find the opposite number of the number given. The general formula of the additive inverse is as follows.

Additive Inverse of a number = (-1) x number

For example, if we want to find the additive inverse of the number 50, we will simply multiply (-1) with it.

Additive inverse of 50 = (-1) x 50 = -50

If we want to find out the additive inverse of -60, we can apply the same rule.

Additive inverse of -60 = (-1) x (-60) = 60

Also Read: Permutations and Combinations 

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Properties of Additive Inverse

Here are some properties of Additive Inverse, based on the negation of the original number. If p is the original number, then its additive inverse will be -p, we will see the properties of -p as follows.

• −(−p) = p
• (-p)2 = p2
• −(p + q) = (−p) + (−q)
• −(p – q) = q − p
• p − (−q) = p + q
• (−p) × q = p × (−q) = −(p × q)
• (−p) × (−q) = p × q

These additive inverse properties hold good for the real numbers as well as complex numbers.

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Additive Inverse of Real Numbers

The given number can be a whole number, a natural number, a fraction, a decimal. An integer or any other real number. The additive inverse of that real number is just the negative of the given number. The above-mentioned properties also apply.

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Additive Inverse of Complex Numbers

The complex numbers are the combinations of the real numbers and imaginary numbers, described in the form of,

 A+iB 

where, 

A → Real Number 

iB → Imaginary Part

The algebraic property of complex numbers shows the existence of their additive inverse.

For example, if any given complex number z? C, there will be a unique complex number, maybe denoted by -z, such that z + (-z) = 0. Moreover, if z = (p,q) and p,q ? R, then -z= (-p,-q).

Let z = p + iq be the given complex number. Then the inverse of it will be -z = -p - iq.

Thus, the additive inverse will be,

 z + (-z) = p + iq - p - iq = 0

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Additive Inverse of a Fraction

The general rule of the additive inverse applies to fractions as well. For example, the additive inverse of a fraction p/q is –p/q and vice-versa. It is because,

 p/q + (-p/q) = 0

So, the additive inverse of a positive fraction will be the same fraction with a negative sign, while for a negative fraction, its additive inverse will be the same fraction without a negative sign i.e., a positive sign.

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Additive Inverse and Multiplicative Inverse

In mathematics, there are two properties of numbers: additive inverse and multiplicative inverse properties related to addition and multiplication operation respectively. For a number p, -p is the additive inverse and 1/p is the multiplicative inverse. The comparison between the two is tabulated below.

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Things To Remember

• The additive inverse of a number is a value that gives a net outcome of zero when added to the original number.
• The additive inverse of a number will be the opposite number of it which simply means it will have the opposite sign.
• The formula to find the Additive Inverse of a number ‘p’ will be (-1) x p.
• The general principles of additive inverse apply to both real numbers and complex numbers.
• Additive inverse and multiplicative inverse are two different things. In the case of Additive Inverse, we find the opposite number of an original number by changing its sign such that the sum total of two is always zero. On the other hand, the multiplicative inverse, we take the reciprocal of the number.

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