Every rational number has a reciprocal.
True or false. Give reasons.
Every rational number has a reciprocal.
True or false. Give reasons.
False,
For every non-zero rational number a/b there exists a rational number b/a such that
a/b × b/a = 1 = b/a × a/b
The rational number b/a is called the multiplicative inverse or reciprocal of a/b and is denoted by (a/b)-1.
The reciprocal of 12 is 1/12
The reciprocal of 5/16 is 16/5.
The reciprocal of 3/4 is 4/3 i.e., (3/4)^-1 = 4/3.
The reciprocal of -5/12 is 12/-5 i.e., (-5/12)^-1 = 12/-5.
The reciprocal of (-14)/17 is 17/-14 i.e., (-17)/14.
The reciprocal of -8 is 1/-8 i.e., (-1)/8.
The reciprocal of -5 is 1/-5, since -5 × 1/-5 = -5/1 × 1/-5 = -5 × 1/-5 × 1 = 1.
.Note: The reciprocal of 1 is 1 and the reciprocal of -1 is -1. 1 and -1 are the only rational numbers which are their own reciprocals. No other rational number is its own reciprocal. We know that there is no rational number which when multiplied with 0, gives 1. Therefore, rational number 0 has no reciprocal or multiplicative inverse.
False,
For every non-zero rational number a/b there exists a rational number b/a such that
a/b × b/a = 1 = b/a × a/b
The rational number b/a is called the multiplicative inverse or reciprocal of a/b and is denoted by (a/b)-1.
The reciprocal of 12 is 1/12
The reciprocal of 5/16 is 16/5.
The reciprocal of 3/4 is 4/3 i.e., (3/4)^-1 = 4/3.
The reciprocal of -5/12 is 12/-5 i.e., (-5/12)^-1 = 12/-5.
The reciprocal of (-14)/17 is 17/-14 i.e., (-17)/14.
The reciprocal of -8 is 1/-8 i.e., (-1)/8.
The reciprocal of -5 is 1/-5, since -5 × 1/-5 = -5/1 × 1/-5 = -5 × 1/-5 × 1 = 1.
.Note: The reciprocal of 1 is 1 and the reciprocal of -1 is -1. 1 and -1 are the only rational numbers which are their own reciprocals. No other rational number is its own reciprocal.We know that there is no rational number which when multiplied with 0, gives 1. Therefore, rational number 0 has no reciprocal or multiplicative inverse.
