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What comes to your mind when you think of the word ‘inverse’? Inverse means something opposite. So, what could be the meaning of the word ‘additive inverse’?
The term additive inverse is related to the term additive identity. The additive inverse of a number is a number that when added to the number gives an additive identity.
For example, additive inverse of $5$ is $-5$, since $5 + \left(-5 \right) = 0$.
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 $0$ (zero).
The simple rule to get an additive inverse of a number is to change the sign of a number, i.e.,
- change the positive number to a negative number
- change the negative number to a positive number
For example, the additive inverse of $8$ is $-8$ and the additive inverse of $-6$ is $6$.
Observe that $8 + \left(-8 \right) = 0$ and $-6 + 6 = 0$. For more on addition and subtraction check Addition and Subtraction of Integers.
Note: $0$ is an additive identity of a number.
Additive Inverse Property
The additive inverse property of numbers states that when a number is added to its additive inverse, the sum obtained is an additive identity, i.e., zero ($0$).
Mathematically it is represented as: For any number $a$, if $a + b = 0$, then $b$ is the additive inverse of $a$.
For example, $9 + \left(-9\right) = 0$, therefore, $-9$ is additive inverse of $9$.
Let’s find out which sets of numbers have an additive inverse.
Additive Inverse of Natural Numbers
A set of natural numbers is a set of numbers that starts with $1$ and moves on to $2$, $3$, $4$, and so on. That is $N = \{1, 2, 3, 4, …\}$.
You have seen above that for any positive number, its additive inverse is a negative number. Since the set of natural numbers does not have any negative numbers, therefore, the numbers in the set of natural numbers do not have an additive inverse.
For a set of natural numbers, the properties – closure property, commutative property, associative property, distributive property, and multiplicative identity property hold, but additive identity property, additive inverse property, and multiplicative inverse property do not hold.
Additive Inverse of Whole Numbers
A set of whole numbers is a set of numbers that starts with $0$ and moves on to $1$, $2$, $3$, and so on. That is $W = \{0, 1, 2, 3, …\}$.
You have seen above that for any positive number, its additive inverse is a negative number. Since the set of whole numbers does not have any negative numbers, therefore, the numbers in the set of whole numbers do not have an additive inverse.
For a set of whole numbers, the properties – closure property, commutative property, associative property, distributive property, additive identity property, and multiplicative identity property hold but, additive inverse property, and multiplicative inverse property do not hold.
Additive Inverse of Integers
A set of integers is a set of numbers that contains all the natural numbers $1$, $2$, $3$, … and their corresponding negative values $-1$, $-2$, $-3$, …, along with $0$, ie., $Z = \{…, -3, -2, -1, 0, 1, 2, 3, … \}$.
You have seen above that for any positive number, its additive inverse is a negative number. And in a set of integers, for every positive integer, there is a corresponding negative number and for every negative number and for every negative number, its additive inverse is a positive number, and there is a corresponding positive number, therefore, the set of integers has an additive inverse for every number.
For a set of whole numbers, the properties – closure property, commutative property, associative property, distributive property, additive identity property, multiplicative identity property, and additive inverse property hold but, the multiplicative inverse property does not hold.
Additive Inverse of Rational Numbers
Rational numbers are the numbers that can be expressed in the form $\frac {p}{q}$, where both $p$ and $q$ are integers and $q \ne 0$.
As the set of rational numbers contains both negative and positive numbers and as seen above that for any positive number, its additive inverse is a negative number and for every negative number, its additive inverse is a positive number, therefore the set of rational numbers has an additive inverse for every number.
For a set of rational numbers, the properties – closure property, commutative property, associative property, distributive property, additive identity property, multiplicative identity property, additive inverse property, and multiplicative inverse property hold.
Additive Inverse of Irrational Numbers
Irrational numbers is the numbers that when expressed in decimal form have non-terminating and non-recurring decimal places.
The additive inverse of any irrational number will be a number with an opposite sign, therefore the set of an irrational number will have an additive inverse for every irrational number.
For a set of irrational numbers, the properties – commutative property, associative property, distributive property, additive inverse property, and multiplicative inverse property holds, whereas, closure property, additive identity property, and multiplicative identity property do not hold.
Additive Inverse of Real Number
A set of real numbers is a set consisting of all the sets – natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Therefore, the set of real numbers will have an additive inverse for every real number.
For a set of real numbers, all the properties – closure property, commutative property, associative property, distributive property, additive identity property, multiplicative identity property, additive inverse property, and multiplicative inverse property holds.
Difference Between Additive Inverse and Multiplicative Inverse
Following are the differences between additive inverse and multiplicative inverse
| Additive Inverse | Multiplicative Inverse |
| To find the additive inverse of a number, you have to just change the sign of the number | To find the multiplicative inverse of a number, you have to take its reciprocal |
| The formula for finding the additive inverse of a number $a$ is $-a$ | The formula for finding the multiplicative inverse of a number $a$ is $\frac {1|{a}$ |
| When an additive inverse is added to a number the result is $0$ | When a multiplicative inverse is multiplied to a number the result is $-1$ |
Conclusion
An additive inverse of a number is a number that when added with the number results in an additive identity. Some sets have additive inverse for the numbers whereas some sets do not have additive inverse for the numbers.
Practice Problems
- Which of the following sets have additive inverse for the numbers?
Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers
- Which of the following sets do not have additive inverse for the numbers?
Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers
- Find the additive inverse of
- $19$
- $-25$
- $56.8$
- $-0.004$
- $\frac {3}{4}$
- $-\frac {45}{21}$
- $2 \frac {1}{5}$
- $\sqrt{2}$
- $9 + \sqrt{3}$
- $-12 + \sqrt{5}$
Recommended Reading
- Associative Property – Meaning & Examples
- Distributive Property – Meaning & Examples
- Commutative Property – Definition & Examples
- Closure Property(Definition & Examples)
- Additive Identity of Decimal Numbers(Definition & Examples)
- Multiplicative Identity of Decimal Numbers(Definition & Examples)
- Multiplicative Inverse of Decimal Numbers – Definition & Property (With Examples)
FAQs
What is the 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 $0$ (zero). $0$ is also called the additive identity of a number.
How do you find the additive inverse of a number?
To find the additive inverse of a number, you have to just change the sign of a number. The additive inverse of a positive number is a negative number and that of a negative number is a positive number.
For example, additive inverse of $47$ is $-47$ and additive inverse of $-91$ is $91$.
Also, the additive inverse of $\frac {2}{11}$ is $-\frac {2}{11}$ and additive inverse of $-\frac {7}{15}$ is $\frac {7}{15}$.
What is the additive inverse formula?
For any number $a$, the additive inverse is $-1 \times a$.
For example, additive inverse of $4$ is $-1 \times 4 = -4$ and additive inverse of $-12$ is $-1 \times \left(-12 \right) = 12$.
What is the additive inverse of zero?
Since zero ($0$) does not have any sign, neither positive nor negative, therefore, the additive identity of $0$ is $0$.
Note: You do not have to add anything to $0$ to get $0$ and nothing means $0$.
What is the additive inverse property?
The additive inverse property states that the sum of a number and its additive inverse is always $0$. For example, $7 + \left( -7 \right) = 0$.
Is additive inverse the same as additive identity?
No, the additive inverse is not the same as the additive identity. The additive identity for every number is $0$, whereas the additive inverse is a number with an opposite sign.
The additive identity for all the numbers $5$, $-8$ and $\frac {2}{3}$ is $0$, whereas the additive inverse of $5$ is $-5$, $-8$ is $8$ and that of $\frac {2}{3}$ is $-\frac {2}{3}$.
What is the difference between additive inverse and multiplicative inverse?
The additive inverse is what we add to a number to make the sum zero ($0$), whereas, the multiplicative inverse is the reciprocal of the given number, which when multiplied together, gives the product as $1$.
