Be on the lookout for the following special names for zero and one!
SPECIAL PROPERTIES OF $\,0\,$ and $\,1$

For all real numbers
[beautiful math coming... please be patient]
$\,x\,$,
$\,x + (x) = 0\,$.
A number added to its opposite always gives zero.
The opposite of a number is also called the additive inverse.

For all real numbers
[beautiful math coming... please be patient]
$\,x\,$,
$\,x\cdot 0 = 0\,$.
Any number multiplied by zero gives zero.

For all nonzero real numbers
[beautiful math coming... please be patient]
$\,x\,$,
$\,\displaystyle\frac{x}{x} = x\cdot\frac{1}{x} = 1\,$.
A nonzero number divided by itself (or multiplied by its reciprocal) always gives one.
The number
[beautiful math coming... please be patient]
$\,\frac{1}{x}\,$
is called the reciprocal of $\,x\,$ or the
multiplicative inverse of $\,x\,$.
Multiplying a number by its reciprocal gives the number $\,1\,$.
Every nonzero number has a reciprocal; zero does not have a reciprocal.
EXAMPLES:
Decide if the given number is
[beautiful math coming... please be patient]
$\,0\,$, $\,1\,$, or a different number:
[beautiful math coming... please be patient]
$\displaystyle\frac13 + (\frac 13)$ 
Answer: $\,0\,$ 
[beautiful math coming... please be patient]
$\displaystyle 0\cdot\frac 13$ 
Answer: $\,0\,$ 
[beautiful math coming... please be patient]
$\displaystyle 2\cdot \frac{1}2$

Answer: $\,1\,$ 
[beautiful math coming... please be patient]
$\displaystyle\frac{1/7}{1/7}$ 
Answer: $\,1\,$ 
[beautiful math coming... please be patient]
$\displaystyle 3\bigl(\frac13\bigr)$

Answer: not $\,0\,$, and not $\,1$ 