 # 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. Chardonnay Felix 2021-01-16 Answered
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. ​