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In mathematics, you perform different types of operations. These operations result in different types of outputs depending on the operator used with the numbers. Sometimes the result remains the same or some specific value irrespective of the numbers chosen.
One of such numbers or elements is known as additive identities. In this article, you’ll learn about these special numbers – additive identity.
What is Additive Identity?
The additive identity is related to the process of addition and as the word ‘identity’ signifies, additive identity is a number that when added to a number gives the output that is the same as that of the number itself.
In the case of an addition, such a number is ‘$0$’ (zero). You might have noticed that when $0$ is added to any number, the result is always the same number.
For example, $12 + 0 = 0 + 12 = 12$. You see here that when $0$ is added to $12$, the result is the number itself i.e., $12$. Let’s consider one more example. $-395 + 0 = 0 + \left(-395 \right) = -395$. Here also the result remains the same after adding $0$. It means for $-395$ also, the identity element is $0$.
In general, we can say that for any number $a$, $0$ is called an additive identity, as $a + 0 = 0 + a = a$.
Additive identity exists for most the set of numbers. Still, there are few sets of numbers, where additive identity does not exist. Let’s explore which of the set of numbers has the additive identity and for which the additive identity does not exist.
Additive Identity 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 $N = \{1, 2, 3, 4, …\}$.
Here, you’ve noticed that there is no number $0$ in the set of natural numbers. Thus, the set of natural numbers does not have an additive identity.
For a set of natural numbers, the properties – closure property, commutative property, associative property, distributive property, and multiplicative identity property hold, whereas the additive identity property does not hold.
Additive Identity 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 $W = \{0, 1, 2, 3, …\}$.
Here, you’ve noticed that the first number is $0$ which is the additive identity. Thus, the set of whole numbers has an additive identity.
For a set of whole numbers, the properties – closure property, commutative property, associative property, distributive property, multiplicative identity property, and additive identity property holds.
Additive Identity 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 $-1$, $-2$, $-3$, …, alongwith $0$, ie., $Z = \{…, -3, -2, -1, 0, 1, 2, 3, … \}$.
You’ve noticed that the number $0$ exists in the set of integers $Z$, thus, the set of integers has an additive identity.
For a set of integers, the properties – closure property, commutative property, associative property, distributive property, multiplicative identity property, and additive identity property holds.
Additive Identity 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 $q \ne 0$.
Also, every whole number such as $0$, $1$, $2$, … can be expressed as $\frac {0}{1}$, $\frac {1}{1}$, $\frac {2}{1}, …$, therefore, $0$ is a rational number and is an additive identity, and hence, the set of rational numbers have an additive identity.
For a set of rational numbers, the properties – closure property, commutative property, associative property, distributive property, multiplicative identity property, and additive identity property holds.
Additive Identity of Irrational Numbers
Irrational numbers are the numbers that when expressed in decimal form have a non-terminating and non-recurring decimal place.
$0$ the additive identity is a terminating decimal number $\left(0.0\right)$, therefore, $0$ is not an irrational number and hence, a set of irrational numbers do not have an additive identity.
For a set of irrational numbers, the properties – commutative property, associative property, and distributive property holds, but closure property, additive identity property, and multiplicative identity property does not hold.
| Number Category | Does Additive Identity Exist? |
| Natural Numbers | No |
| Whole Numbers | Yes |
| Integers | Yes |
| Rational Numbers | Yes |
| Irrational Numbers | No |
| Real Numbers | Yes |
Difference Between Additive Identity and Multiplicative Identity
These are the differences between additive identity and multiplicative identity of numbers.
| Additive Identity | Multiplicative Identity |
| Used in the addition operation | Used in the multiplication operation |
| $0$ is the additive identity | $1$ is the multiplicative identity |
| Expressed as $a + 0 = 0 + a = a$ | Expressed as $a \times 1 = 1 \times a = a$ |
| Example: $56 + 0 = 0 + 56 = 56$ | Example: $39 \times 1 = 1 \times 39 = 39$ |
identity.
Practice Problems
1. Which of the following is the additive identity for numbers?
$1$
$0$
$-1$
None of these
2. Which of the following sets have an additive identity?
Natural numbers
Rational numbers
Whole numbers
Irrational numbers
Integers
Real numbers
3. Which of the following sets do not have an additive identity?
Natural numbers
Rational numbers
Whole numbers
Irrational numbers
Integers
Real numbers
4. The additive identity which when added to a number gives the same result
- True
- False
5. The additive identity which when multiplied by a number gives the same result
- True
- False
6. The additive identity which when divided by a number gives the same result
- True
- False
FAQs
What is additive identity?
An additive identity is a number that when added to any number gives the same number. The number $0$ is called the additive identity of numbers, since for any number $a$, $a + 0 = 0 + a = a$.
Which number in math is called the additive identity?
$0$ is called the additive identity of numbers, since for any number $a$, $a + 0 = 0 + a = a$.
Are additive identity and multiplicative identity the same?
No, additive identity and multiplicative identity are two different concepts. Additive identity is for addition operation, whereas multiplicative identity is for multiplication operation.
The additive identity is $0$, whereas the multiplicative identity is $1$.
The additive identity when added to any number gives the same result, whereas the multiplicative identity when multiplied by any number gives the same result.
Do all sets of numbers have an additive identity?
No, all sets of numbers do not have an additive identity. The sets having additive identity are whole numbers, integers, rational numbers, and real numbers. The sets that do not have additive identity are natural numbers and irrational numbers.
Conclusion
An additive identity is a number that when added to a number gives the same result. $0$ is the additive identity of numbers used in math. Some sets have an additive identity, whereas there are some sets that do not have an additive identity.
