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5. The role of 0 and 1

Zero \((0)\) is a rational number, and the sum of any rational number with zero\((0)\) results in the same rational number.
Thus, ab+0=0+ab=ab, for every rational number ab.
 
\(0\) is called the additive identity for rationals.
Example:
(i) Consider the rational number 35.
 
 Then, we have 35+0=0+35=35.
 
35+0=35+05=3+05=35 and similarly, 0+35=05+35=0+35=35.
 
Therefore, 35+0=0+35=35.
 
(ii) Consider the rational number −23.
 
Then, we have −23+0=0+−23=−23.
 
−23+0=−23+03=−2+03=−23 and similarly, 0+−23=03+−23=0+−23=−23.
 
Therefore, −23+0=0+−23=−23.
One \((1)\) is a rational number and the product of any rational number with one \((1)\) result in the same rational number.
For any rational number ab, we have ab×1=1×ab=ab.
 
\(1\) is called the multiplicative identity for rationals.
Example:
(i) Consider the rational number 34.
 
Then, we have 34×1=1×34=34.
 
34×1=34×14=3×14=34 and 1×34=14×34=1×34=34.
 
Therefore, 34×1=1×34=34.
 
(ii) Consider the rational −913.
 
Then, we have −913×1=1×−913=−913
 
−913×1=−913×113=−9×113=−913 and 1×−913=113×−913=1×−913=−913.
 
Therefore, −913×1=1×−913=−913.
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