Terminating Decimal
- What is a Terminating Decimal?
- Definition of Terminating Decimal
- Identifying the Terminating Decimal
- Non-Terminating and Other Types of Decimals
- Terminating Decimal Characteristics
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is a Terminating Decimal?
A terminating decimal is called a decimal whose repeating digit is zero. In other words, a terminating decimal is a decimal fraction whose decimal places are limited and whose decimal representation stops at some point.
For example, `0.25` is a terminating decimal; if we write a few more decimal places, we get `0.25000`. We can see its repeating digits are `0`. So, it is a terminating decimal.
`3.125` is a terminating decimal fraction that ends after three decimal places. As we increase the digits after `0.125`, the digits will be `0`, which are repeating and insignificant.
A decimal is also a number that consists of a whole number and a fractional part. These decimal fractions are written in the form of `5.645`, where `5` is the whole number and `0.645` is the fractional part. Decimals are mainly of two types:
- Terminating decimals: They have a finite number of digits after the decimal, and their repeating digits are `0`.
- Non-terminating decimals: These decimals go on indefinitely after the decimal point with or without repeating a pattern.
You can understand this in a situation where you had `$10` and a coin worth `25` cents. So, after counting the total money in dollars, you will end in with `$10.25`. This `10.25` is a terminating decimal.
Definition of Terminating Decimal
Terminating decimals are decimals that have a finite number of digits after the decimal.
Consider `9.56` with the following image to understand the terminating decimal.
It is important to note here that the rational numbers are terminating decimals. In other words, we can say that terminating decimals are rational numbers, i.e., they can be written in the form of fractions.
For example, consider the following.
- `5.5=55/10=11/2`
- `3.75=375/100=15/4`
Identifying the Terminating Decimal
A terminating decimal can be identified by the following checkpoints
- The number of digits after the decimal point is finite, i.e., the number of digits ends after some counting.
- If you encounter a fraction, then check for its equivalent rational number.
- Check for the denominator ending in `10, 100, 1000`, etc.
- If, after simplification, the denominator of rational number ends in the form of `2^a xx 5^b` or `2^a` or `5^b`, where `a, b` are positive integers, then it will be a terminating decimal number.
Terminating Decimal Number Theorem
If a rational number is of the form `p/q`, then `p/q` can be a terminating decimal if and only if `q` contains `2’s` and/or `5’s` in its prime factorization.
Example: `15/16=15/2^4`, therefore it must be a terminating decimal, which is `0.9375`.
Non-Terminating and Other Types of Decimals
A non-terminating decimal number is defined as a decimal number in which the digits after the decimal point go on indefinitely without ending or terminating. It also means that the number of decimal places continues to infinity. The digits after the decimal can be repeating, non-repeating, or mixed, which classifies them into three categories.
- Non-repeating, non terminating decimals (irrational numbers): These numbers are the representations of irrational numbers such as the square root of `2`, pi, and the golden ratio, etc. These fractions are non terminating, non-repeating means that the digits after decimal do not repeat for example `pi=3.14159265359`....
- Repeating Decimals or Recurring Decimals: These are the non terminating decimals, but the digits after the decimal repeat themselves in a pattern. The repeating part is indicated by the bar over the repeating digits. For example, `1/3=0.333333..… = 0.\bar3`
- Mixed decimals with repeating patterns: This pattern can be understood by the digits after the decimal that is repeating as well as non-repeating. For example `0.12748347444445555`…..
Terminating Decimal Characteristics
- Conversion to fractions: Terminating decimals can be converted to fractions by dividing them by the denominators of a power of `10`. For example, `0.35` can be written as, `0.35=35/100`.
- Whole numbers as terminating decimals: A whole number can be represented as a terminating decimal. For example, `7` can be written as `7.00`, which is a terminating decimal with two decimal places.
- Rational numbers: Terminating decimals can be categorized as a subset of rational numbers This means that the terminating decimals can be expressed as integer ratios. For example, `0.8` can be written as `4/5`.
- Decimal place value: The digits after the decimal point correspond to the decreasing power of ten. This means that the first decimal place represents the tenth place, the second represents the hundredth place, and so on.
Solved Examples
Example `1`: Convert the following terminating decimal into a fraction:
`0.635`
Solution:
Here, `0.635` can be written as `635/1000`.
After simplifying it we get `127/200`.
Example `2`: Convert `0.0625` into a fraction.
Solution:
Here, `0.0625` can be written as `625/10000`.
After simplifying it we get `1/16`.
Example `3`: Is `10/6` terminating or non-terminating?
Solution:
Here, `10/6=5/3`
After dividing `5` by `3`, we get `1.6666`….. . So, the digits after decimal go on repeating up to infinity. Therefore, `10/6` is a non-terminating decimal.
Example `4`: Which one of the following is the terminating decimal `sqrt 2/1,5/20, 10/3, and 7/3`?
Solution:
This can be solved by simplifying the fractions as follows.
`(sqrt2)/(1)=sqrt2=1.414…..=>` Non terminating decimal
`(5)/(20)=0.25=>` Terminating decimal
`(10)/(3)=3.33333…..=>` Non terminating decimal
`(7)/(3)=2.3333….=>` Non terminating decimal
Example `5`: Can you identify that `4.675` is a terminating decimal?
Solution:
Here, the decimal is given as `4.675`.
To identify the terminating decimal, divide and multiply `4.675` by `1000`
`4.675/1000 xx 1000=4675/1000`
Since the above decimal can be expressed in the form of `p/q`, it is a terminating decimal.
Practice Problems
Q`1`. Identify the terminating decimal.
- `-(sqrt10)/(1)`
- `-pi`
- `-1/3`
- `-(7)/(56)`
Answer: d
Q`2`. Which of the following is a terminating decimal?
- `3.33333……`
- `0.6785`
- `0.666666…..`
- `3.14159265359……`
Answer: b
Q`3`. _______ is a non-terminating repeating decimal.
- `3.14159265359`
- `2.7275`
- `4.1789`
- `3.433333……`
Answer: d
Q`4`. If a rectangle is defined by its `2.675` cm of length and `4.37` cm of breadth. Select its area from the following:
- Non-terminating
- Terminating
- Recurring Decimals
- Mixed decimals
Answer: b
Q`5`. State true or false for the following statement.
“Digits after a decimal determine whether the decimal is terminating or non-terminating”.
- True
- False
Answer: a
Frequently Asked Questions
Q`1`. Can the terminating decimals be expressed in simple fraction form?
Answer: Yes, the terminating decimal can be expressed in fraction form.
Q`2`. Is there any other method to determine whether the given fraction is terminating or non-terminating?
Answer: Although the division method can determine the fraction, the prime factorization theorem can be used to find its nature by factoring the denominator into `2^a xx 5^b` where `a, b` are positive integers.
Q`3`. Is `44/80` a terminating decimal?
Answer: `44/80=11/20=0.55`. Therefore, it is a terminating decimal.
Q`4`. Is `pi` a terminating decimal?
Answer: `pi` `= 22/7` `= 3.14159265359` and so on, so pi is a non-terminating decimal. It is a non-recurring and non-repeating decimal. It is an irrational number.
Q`5`. Are the terminating decimals always rational?
Answer: Not all rational numbers are always considered terminating decimals. The rational numbers with prime factors other than `2 and 5` in the denominators will result in non-terminating decimals.