A decimal number consists of a whole-number part and a fractional part separated by a decimal point. Sometimes, the digits after the decimal point end after a certain place, and sometimes they continue forever. Based on how the digits continue after the decimal point, decimals are classified into two types:
Terminating Decimals
Non-Terminating Decimals (Repeating and Non-Repeating)
Terminating Decimal
A terminating decimal is a decimal number that has a limited number of digits after the decimal point and can be expressed as a fraction in the form p/q, where q ≠ 0.
Example: 23.5 is a terminating decimal as it has a finite number of digits after the decimal point and can be written as 47/2 in p/q form.
Representation of Terminating Decimals on Number Line
A terminating decimal ends after a few digits, so it is easy to show it on the number line. To represent such numbers, we break the number line into equal parts and mark the point.
Steps:
Find between which two whole numbers the decimal lies.
Divide the interval into equal parts based on decimal place like for tenth ( 10 equal parts) and for hundredths ( 100 equal part).
Count the required number of parts from the left.
Put a dot at that position.
Now, let us look at the space between 3.7 and 3.8. We can divide this interval into 10 equal parts. Each small part will represent 0.01. The first mark to the right of 3.7 will be 3.71, then 3.72, 3.73, 3.74, and so on.
Now, we want to locate 3.735. This number lies between 3.73 and 3.74. Again, divide the space between 3.73 and 3.74 into 10 equal parts. The first mark to the right of 3.73 will be 3.731, then 3.732, 3.733, 3.734, and so on. Somewhere between these marks, we will find 3.735.
Non-Terminating Decimal
A non-terminating decimal is a decimal number that has infinite digits after the decimal point and does not come to an end.
A non-terminating decimal is classified into two types:
Non-Terminating Recurring Decimals
Non-Terminating Non-Recurring Decimals
Non-Terminating Recurring Decimals
A non-terminating recurring decimal is a decimal that continues forever but repeats in a fixed pattern and can be written in p/q form.
Example:
1/7 = 0.\overline{142857}
The block 142857 repeats after every 6 digits. Since the decimal continues forever with a fixed repeating pattern, it is a non-terminating recurring decimal.
Example 1 : Express 2.757575... in p/q form.
Solution:
Given decimal is 2.666666...
let x = 2.75757575...
Here, we can observe that the numbers 75 are repeating. Since we have two digits repeating, let us multiply x with 100.
100x = 275.75757575...
100x = 273.75757575... + 2.75757575...
We know that x = 2.75757575...
So, 100x = 273+ x
100x − x = 273
99x = 273
x = 273/99 = 91/33
Hence, 2.75757575... = 91/33.
Non-Terminating Non-Recurring Decimals
A non-recurring decimal is a non-terminating decimal in which the digits do not repeat after the decimal point without terminating. A non-terminating recurring decimal cannot be expressed in p/q form.
1.23489..., 25.755780..., 345.532901..., etc. are some examples of non-terminating, non-recurring decimals.
Representation of Non-Terminating Decimals on Number Line
Non -Terminating are numbers which continue without ending and can be show their approximate position on the number line. We cannot mark the exact point because the decimal never ends, but we can get closer and closer to the number by taking more decimal places.
To locate 6.47777… on the number line (up to four decimal places), we consider 6.4777. The number lies between 6 and 7, so we first divide this interval into 10 equal parts. Since 6.4777 is between 6.4 and 6.5, we zoom into that part and again divide it into 10 equal sections. Next, we see that it lies between 6.47 and 6.48, so we divide this smaller region into 10 parts. Finally, we look between 6.477 and 6.478 and divide it again. We find that 6.47777… is closer to 6.4777 than 6.4778. By dividing the number line into smaller and smaller parts, we can show the approximate position of any non-terminating decimal.
Solved Examples
Example 1: Identify the terminating and non-terminating decimals among the numbers given below.
a) 6.678 b) 1.3333333... c) 5.214178134... d) 0.15
Solution:
a) 6.678 is a terminating decimal because it has a finite number of digits after the decimal point. The number of digits after the decimal point is 3.
b) 1.3333333... is a non-terminating recurring decimal, because here the number 3 is repeating without terminating.
c) 5.214178134... is a non-terminating, non-recurring decimal, because the digits after the decimal are not repeating and are not terminating.
d) 0.15 is a terminating decimal because it has a finite number of digits after the decimal point. The number of digits after the decimal point is 2.
Example 2: Determine whether 2.895 is a terminating or non-terminating decimal.
Solution:
Given decimal is 2.895,
Which can be expressed as 2895/1000.
2.895 is a terminating decimal because it has a finite number of digits after the decimal point.
It terminates after three decimal places and can be expressed in the p/q form.
Example 3: Prove that 1.333333... can be expressed in p/q form where p and q are integers and q ≠ 0.
Solution:
The given decimal is 1.333333...
let x = 1.33333333... = 1.\bar{3}
Here, we can observe that the number 3 is repeating. Since we have one digit repeating, let us multiply x with 10.
10x = 13.3333333...
10x = 12 + 1.333333...
We know that x = 1.333333...
So, 10x = 12 + x
10x − x = 12
9x = 12
x = 12/9 = 4/3.
Hence, 1.\bar{3} = \frac{4}{3}
Example 4: Is 11/12 a terminating or repeating decimal?
Solution:
The given number is 11/12 whose decimal value is 0.91666666... We can observe that the number 6 is repeating and is not terminating. So, 11/12 is a terminating decimal.
