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In mathematics, you use different types of numbers and operators in your calculations. The operators in mathematical expressions along with the numbers result in different types of results. You might have noticed that many times an operation performed on a certain number gives the same result. Such types of numbers(or elements) are known as identities.
All operators are associated with their own identities. The identity associated with multiplication is known as multiplicative identity. Let’s understand what is a multiplicative identity and what are its properties.
What is Multiplicative Identity?
The multiplicative identity is related to the process of multiplication and as the word ‘identity’ signifies, multiplicative identity is a number that when multiplied by a number gives an output that is the same as that of the number itself.
In the case of multiplication, such a number is ‘$1$’ (one). You might have noticed that when $1$ is multiplied by any number, the result is always the same number.
For example, $19 \times 1 = 1 \times 19 = 19$. You see here that when $1$ is multiplied by $19$, the result is the number itself i.e., $19$. Let’s consider one more example. $-27 \times 1 = 1 \times \left(-27 \right) = -27$. Here also the result remains the same after multiplying by $1$. It means for $-27$ also, the identity element is $1$.
In general, we can say that for any number $a$, $1$ is called a multiplicative identity, as $a \times 1 = 1 \times a = a$.
Multiplicative identity exists for most of the set of numbers. Still, there are few sets of numbers, where multiplicative identity does not exist. Let’s explore which of the set of numbers has the multiplicative identity and for which the multiplicative identity does not exist.
Note: $-1$ is not a multiplicative identity.
Multiplicative 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 $N = \{1, 2, 3, 4, …\}$.
Since the first element of the set of natural numbers is $1$, we see that the multiplicative identity exists in the case of natural numbers. Thus, the set of natural numbers has a multiplicative 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.
Multiplicative 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 $1$ exists in the set of whole numbers which is a multiplicative identity. Thus, the set of whole numbers has a multiplicative 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.
Multiplicative 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 $-1$, $-2$, $-3$, …, along with $0$, ie., $Z = \{…, -3, -2, -1, 0, 1, 2, 3, … \}$.
You’ve noticed that the number $1$ exists in the set of integers $Z$, thus, the set of integers has a multiplicative identity.
For a set of integers, the properties – closure property, commutative property, associative property, distributive property, multiplicative identity property, and additive identity property holds.
Multiplicative 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, $1$ is a rational number and is a multiplicative identity, and hence, the set of rational numbers have a multiplicative 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.
Multiplicative Identity of Irrational Numbers
Irrational numbers are numbers that when expressed in decimal form have non-terminating and non-recurring decimal places.
$1$ the multiplicative identity is a terminating decimal number $\left(1.0\right)$, therefore, $1$ is not an irrational number and hence, a set of irrational numbers does not have a multiplicative 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 Multiplicative Identity Exist? |
| Natural Numbers | Yes |
| 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: $39 \times 1 = 1 \times 39 = 39$ |
Practice Problems
- Which of the following is the multiplicative identity for numbers?
- $1$
- $-1$
- $0$
- None of these
- Which of the following sets have a multiplicative identity?
- Natural Numbers
- Whole Numbers
- Integers
- Rational Numbers
- Irrational Numbers
- Real Numbers
- Which of the following sets does not have a multiplicative inverse?
- Natural Numbers
- Whole Numbers
- Integers
- Rational Numbers
- Irrational Numbers
- Real Numbers
- The multiplicative identity which when added to a number gives the same result
- True
- False
- The multiplicative identity which when subtracted from a number gives the same result
- True
- False
- The multiplicative identity which when multiplied by a number gives the same result
- True
- False
FAQs
What is multiplicative identity?
A multiplicative identity is a number that when multiplied by any number gives the same number. The number $1$ is called the multiplicative identity of numbers, since for any number $a$, $a \times 0 = 0 \times a = a$.
Which number in math is called the multiplicative identity?
$1$ is called the multiplicative identity of numbers, since for any number $a$, $a \times 1 = 1 \times a = a$.
Is $-1$ also a multiplicative identity of numbers?
No, $-1$ is not a multiplicative identity of numbers, since when a number is multiplied by $-1$, its sign changes and hence no longer remains the same number.
Are multiplicative identity and additive identity the same?
No, multiplicative identity and additive identity are two different concepts. The multiplicative identity is for multiplication operations, whereas additive identity is for the addition operation.
The multiplicative identity is $1$, whereas the additive identity is $0$.
The multiplicative identity when multiplied by any number gives the same result, whereas the additive identity when added to any number gives the same result.
Do all sets of numbers have a multiplicative identity?
No, all sets of numbers do not have a multiplicative identity. The sets having multiplicative identity are natural numbers, whole numbers, integers, rational numbers, and real numbers. The set that does not have a multiplicative identity is irrational numbers.
Conclusion
A multiplicative identity is a number that when multiplied with a number gives the same result. $1$ is the multiplicative identity of numbers used in math. Some sets have a multiplicative identity, whereas there are some sets that do not have a multiplicative identity.
