Properties of numbers — distributive, identity and inverse

The additive identity is the number that leaves any other number unchanged when added to it. We need a number $x$ such that $a+x=a$ for any number $a$.

Adding 0 to any number leaves it unchanged: $38+0=38$, $-5+0=-5$, $100+0=100$. This works for every number.

The additive identity is 0. It is the unique number that satisfies $a+0=a$ for all values of $a$.

We need the number that keeps any number unchanged under addition. If adding it to 5 still gives 5, it's the additive identity.

$5+1=6\ne 5$. Adding 1 changes the number, so 1 is not the additive identity.

$5+(-1)=4\ne 5$ and $5+10=15\ne 5$. Both -1 and 10 change the number, so neither is the additive identity.

$5+0=5$ ✓. 0 leaves any number unchanged under addition, confirming it is the additive identity.