Reciprocals

In mathematics, in case of numbers we say that reciprocals and in case of trigonometry we say it as inverse, but both the meaning are same here. We know that, any natural number can be expressed in the form of fraction.

For example: 4 can be written as 4/1, where 4 is the numerator and 1 is the denominator.

Thus, we can say that,

Fractions are written in the form of ratio of two numbers. The upper number is called as numerator and the number below is called as denominator.

For example: a/b is the fraction, in which a is numerator and b is the denominator.

Reciprocals:

Hence, the reciprocal of the number is obtained by interchanging or reversing the numbers from numerator and denominator. That means, the number in numerator goes to denominator and the number in denominator goes to numerator. And the final number we will get is the reciprocal.

For example:

The reciprocal of x is 1/x, where x≠0.

Thus, reciprocal of a number is simply interchanging the numbers in numerator and denominator.

For example:

  • The reciprocal of 2 is ½, because 2 can be written as 2/1 and hence its reciprocal is ½.
  • Reciprocal of 5 is 1/5, because 5 can be written as 5/1 and hence its reciprocal is 1/5.
  • Reciprocal of 2/3 is 3/2.

Note:

  • We cannot find the reciprocal of zero. Because reciprocal of zero is not defined.

Hence, 1/0 → undefined

  • The product of a number and its reciprocal always gives the value 1 i.e. unity.

Hence, reciprocal of a number is its multiplicative inverse.

For example:

  • The reciprocal of 6 is 1/6.

Hence, product of number and its reciprocal = 6*1/6 = 1

  • Also, the reciprocal of 2/3 is 3/2.

Hence, product of number and its reciprocal = 2/3*3/2 = 1

Reciprocals of proper fractions:

Proper fractions are the fractions in which numerator is less than denominator

For example:

  • 5/7 is the proper fraction, in which numerator is 5 and denominator is 7.

Thus, reciprocal of proper fraction 5/7 is 7/5, and 7/5 is the improper fraction as numerator is greater than denominator.

And also, 5/7*7/5= 1

  • Again, 11/12 is the proper fraction and its reciprocal is 12/11, which is the improper fraction. And also, 11/12*12/11 = 1.
  • Thus, we can say that, the reciprocal of proper fraction is the improper fraction.

Reciprocals of improper fractions:

Improper fractions are those fractions in which numerator is greater than or equal to the denominator.

For example:

  • 8/7 is the improper fraction as numerator is greater than denominator. The reciprocal of improper fraction 8/7 is 7/8 and which is the proper fraction as numerator is less than the denominator.

And, 8/7*7/8 = 1

  • Also, the reciprocal of improper fraction 6/5 is 5/6, and 5/6 is the proper fraction as numerator is less than the denominator.

And 6/5*7/5 = 1

  • Thus, we can say that, the reciprocal of the improper fraction is the proper fraction.

Reciprocal of mixed fraction:

  • Mixed fractions are those which are written in the form of whole number and the proper fraction both together.

For example:

2 2/3, 4 5/6, 3 6/7 all are the mixed fractions and then can simplified to improper fractions.

  • Thus, to find the reciprocal of mixed fraction, first we have to convert mixed fraction into improper fraction and then we have to find its reciprocal.

For example:

  • 2 2/3 is the mixed fraction, then its improper fraction form is as below.

2 2/3 = (3*2 +2)/ 3 = 8/3, thus we get the improper fraction form of mixed fraction.

Hence, the reciprocal of 8/3 is 3/8.

  • Similarly, the mixed fraction 4 5/6 is converted into improper fraction as below.

4 5/6 = (6*4 + 5)/6 = 29/6 is the improper fraction form.

Thus, the reciprocal of 29/6 is 6/29.

Reciprocal of decimal number:

Decimal numbers are those which are having whole number part and decimal part also.

For example: 0.5, 0.33 are the decimal forms.

  • To find the reciprocal of decimal we first convert the decimal into fraction and then we find its reciprocal.

Let, 0.5 is the decimal number then 0.5 = ½ is the proper fraction form of 0.5.

Thus, reciprocal of 0.5 i.e. ½ is 2.

And 2*1/2 = 2*(0.5) = 1

 

  • Similarly, we can find the reciprocal of 0.3.

0.3 = 1/3 is the proper fraction form of 0.3

Hence, the reciprocal of 0.3 i.e. 1/3 is 3.

And 3*1/3 = 3*0.3 = 1

Reciprocal of a negative number:

The reciprocal of negative number is calculated similarly as the above process only negative sign we have to give it, which remains as it is.

For example:

  • The reciprocal of -5 is 1/-5 or – (1/5).
  • The reciprocal of -2/3 is – (3/2).
  • The reciprocal of -5/6 is – (6/5).
  • The reciprocal of -0.5 = – (1/2) is -2.

Updated: July 14, 2021 — 11:52 pm

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