# Consider each set of numbers and determine if the set has an additive identity, additive inverse, multiplicative identity, or a multiplicative inverse. Explain your reasoning for each. a. the set of natural numbers.

Consider each set of numbers and determine if the set has an additive identity, additive inverse, multiplicative identity, or a multiplicative inverse. Explain your reasoning for each. a. the set of natural numbers.
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The set of natural numbers is $\left\{1,2,3,4,\dots \right\}$.
The additive identity of a number, aa, is always 0 since $a+0=0+a=a$. Since 0 is not included in the set of the natural numbers, then the set does not have an additive identity.
The additive inverse of a number aa is −a since $a+\left(-a\right)=-a+a=0$. The set of natural numbers does not include any negative numbers so for all natural numbers a, the additive inverse of −a would not be a natural number. The set then does not have an additive inverse.
The multiplicative identity of a number aa is always 1 since $a\cdot 1=1\cdot a=a$. Since 1 is included in the set of the natural numbers, then the set does have a multiplicative identity does have a multiplicative identity.
The multiplicative inverse of a number aa is always $\frac{1}{a}$ since $a\cdot 1a=1$. If aa is a natural number not equal to 1, then $\frac{1}{a}$ is not a natural number. For example, $a=2$ would have a multiplicative inverse of $\frac{1}{a}=\frac{1}{2}$ which is not a natural number. The set of natural numbers then does not have a multiplicative inverse. ​