How to convert non-terminating and repeating decimals to fractions
Noteworthy Notes:
1) All non-terminating and non-repeating decimals are Irrational numbers.
2) All non-terminating and repeating decimals are Rational numbers.
Non-terminating and repeating decimals are also called recurring decimals. Sometimes these are expressed by a line over the repeating digits.
For examples :
2.7777…… = 2.7
1.05693693693693….. = 1.05693
Students sometimes find very tricky to work with them. As they are rational numbers , then by definition they can be expressed in the form of p/q.
We will find out how to do that in this post.
Remember , that our target is to eliminate the digits after the decimal. Just notice how…
Suppose we are given to convert the following two decimals to their fractional equivalent:
ex-1) 0.6666….. or 0.6 ex-2) 1.05693693693693….. or 1.05693
Just follow the steps below with the given two examples:
Step-1: Write an equation, x = the given decimal.
| Ex-1: x = 0.6666….. | Ex-2: x = 1.05693693693693….. |
Step-2: Count the number of digits after decimal, the repeated pattern starts. Suppose it is equal to y. Then, multiply x with 10y .
| Ex-1 : x = 0.6666….. Here the repeated pattern is “6”, which gets repeated just after the decimal. Therefore here, y = 0 and 10o = 1 Hence, 1 × x = 1 × 0.66666….. = 0.66666…. | Ex-2 : x = 1.05693693693693….. Here the repeated pattern is “693”, which gets repeated after two digits (0 and 5) of the decimal. Therefore here, y = 2 and 102 = 100 Hence, 100 × x = 100 × 1.05693693693693….. = 105. 693693693….. |
Step-3: Count the number of digits in the repeated pattern, let this be, z . Now multiply the given decimal, x by 10y+z .
| Ex-1 : x = 0.6666….. 1 × x = 0.6666… Here the number of digits in the repeated pattern is “1”. Therefore here, z = 1==> y+z= 0+1 = 1 so , 1o1 = 10 Hence, 10 × x = 10× 0.66666….. = 6.666666… | Ex-2 : x = 1.05693693693693….. 100x = 105.693693693693….. Here the number of digits in the repeated pattern is “3”. Therefore here, z = 3==> y+z= 2+3 = 5 so , 1o5 = 100000 Hence, 100000 × x = 100000× 1.05693693693693….. = 105693.693693693….. |
Step-4: Subtract the equation of step-2 from the equation of step-3 and solve for x , to get the value in the form of p/q .
| Ex-1 : x = 0.6666….. equation of step-2 : x = 0.6666… equation of step-3: 10x = 6.666666… Calculate differences of the LHS and RHS of the two equations and solve for x : 10x – x = 6.6666… – 0.6666…. 9x = 6 x = 6/9 or reduce further to get x = 2/3 | Ex-2 : x = 1.05693693693693….. equation of step-2 : 100x = 105.693693693693….. equation of step-3: 100000x = 105693.693693693….. Calculate differences of the LHS and RHS of the two equations and solve for x : 100000x -100 x =105693.693693693….. – 105.693693693693….. 99900x = 105588 x = 105588/99900 or reduce further to get x =52794/49950 =26397 /24975 |
Now , take out your calculator to reverse-calculate and check if you answer is correct :
| Ex-1 : x =6/9 = 2/3 = 0.666…. = 0.6 | Ex-2 : x =105588/99900 = 52794/49950 =26397 /24975 = 1.05693693693…. = 1.05693 |
Initially , the steps may seem to be daunting but if you follow and practice converting some repeating decimal to their fractional equivalents , you will understand that it’s all about eliminating the digits after the decimal and finding p/q.
Good Luck !!
The Reader’s challenge :
Convert the following decimals to their fractional equivalent and comment with your answers:
1) 10.0474747….. or 10.047
2) 0.212121…. or 0.21