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Terminating Decimal

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What is Terminating Math Definition?

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Before learning about Terminating Decimal Meaning, let us learn about decimals. The decimal numeral scheme is used to represent both integer and non-integer numbers. It is Hindu–Arabic numeral system's expansion to non-integer numbers. Decimal notation refers to the way numbers are represented in the decimal system. A decimal is a fraction that has been composed in a particular manner. Instead of writing 1/2, you should use the decimal 0.5 to represent the fraction, with the zero in the one’s place and the five in the tenths place. 

Terminating Number


What is a Decimal?

Decimal derives from the Latin word Decimus, which means tenth, and is derived from the root word Decem, which means eight. As a consequence, the decimal system has a base of 10 and is often referred to as a base-10 system. A number in the metric system may also be referred to as decimal. Or used as an adjective, decimal refers to a counting scheme. 

Decimal Number

Below are the two forms of decimals:

  • Terminating decimals (or) Non-recurring decimals

  • Non Terminating Decimal (or) Recurring decimals

Let us understand about the difference between Terminating And Non Terminating Decimals, Terminating Decimal And Repeating Decimal

Terminating Decimal

Define Terminating Decimal

Terminating Decimal Definition is a decimal number with a finite number of digits after the decimal point. A Terminating Decimal Is, such as 5.65, can be interpreted as the repeated decimal 5.650000000000000..., but the number is usually labeled as terminating when the repeating digit is nil. Both ending decimals are rational numbers that can be represented as reduced fractions with no prime number variables other than two or five in the denominators.


Non Terminating Decimal 

What is Non Terminating Meaning?

A non-terminating, non-repeating decimal is a decimal number that lasts indefinitely with no repeating digits. This sort of decimal can't be expressed as a fraction, because it's an irrational number. When expressing a fraction in decimal form, we get any remainder when we split it. If the dividing method does not result in a remainder equal to zero, the decimal is referred to as a non-Terminating Meaning In Maths. In certain cases, a digit or a set of digits in the decimal portion repeats itself. Non-terminating repeating decimals, also known as pure repeated decimals, are a type of non-terminating repeating decimal. A bar is mounted on the replicated portion to represent these decimal numbers. 

Non Terminating Repeating Decimals Are Rational and they can be written as p/q, where q is not equal to zero.

Non Terminating Decimal


Repeating Decimal

A decimal representation of a number of periodic digits and an infinitely repeated element that is not zero is known as a repeating decimal. A number can be seen to be rational if and only if its decimal representation repeats or terminates. The repetend or reptend is a digit series that can be replicated forever. Since the zeros can be omitted and the decimal terminates before these zeros, this decimal representation is considered a terminating decimal rather than a repeated decimal where the repetend is a zero. Any terminating decimal representation can be written as a decimal fraction with a power of ten as the denominator.

Repeating Decimal


How to Convert Non-terminating Repeating Decimal to Fraction Problem?

Simply follow the five steps below when translating non-Terminating And Repeating Decimals to a fraction:

Step 1: Make x to be the repeating decimal you're turning to a fraction.

Step 2: Look for the repeating digit in the repeating decimal (s).

Step 3: To the left of the decimal point, put the repeated digit(s).

Step 4: To the right of the decimal point, put the repeated digit(s).

Step #5: Subtract the left sides of the two equations using the two equations you find in steps 3 and 4. Subtract the right sides of the two equations after that. Simply ensure that the variance from both sides is positive when you deduct.


Use of Decimal

When it comes to money, dealing with decimal numbers is unavoidable. In a number of cases, such as when translating paise to rupee. It is not necessary for the length of an item to be a multiple of the specified graduation when calculating its length. When using a metre scale to measure the length of a table, the length may not be a whole number; instead, it may fall within two graduations on the metre scale. The decimal numbers are used in these cases. For representing weight, we use decimal numbers. When buying a watermelon, for example, it may not necessarily weigh in whole numbers; it may weigh less than 2 kg but more than 1 kg. In such cases, the shopkeeper must determine how much a watermelon can cost depending on its weight.


Theorem

  1. Let x be a rational number with a decimal expansion that ends. Then prime factorization of q is of the form 2\[^{n}\]5\[^{m}\], where n and m are non-negative integers, and x can be expressed in the form p/q, where p and q are co-prime.

  2. Let x = p/q be a rational number with prime factorization 2\[^{n}\]5\[^{m}\] and non-negative integers n and m. As a consequence, x has a decimal extension that ends.

  3. Let x =p/q be a rational number with a q prime factorization that is not  2\[^{n}\]5\[^{m}\], and n and m be non-negative integers. Then x has a non-terminating and repeating decimal extension (recurring).


Solved Examples

1. Transform \[\frac{8}{11}\] to decimal.

Ans:

Solved Example 1


2. Identify \[\frac{8}{3}\] as terminating or non-terminating.

Ans:

Solved Example 2


3. Convert 0.7777777... to fraction.

Ans:

Let us start by equating 0.7777777… to x,

x = 0.777777...

10x = 7.777777

10x - x = 7.777777 - 0.7777777

9x = 7

Ans: x = \[\frac{7}{9}\]

FAQ (Frequently Asked Questions)

1. What is an Example of a Terminating Decimal?

Ans:

Meaning of terminating decimal are logical decimal numbers with a finite number of digits that can be expressed in decimal form. An ending decimal is readily identifiable. It's a terminating decimal if the number doesn't have a bar over it. A decimal with an end digit is known as a terminating decimal. It's a decimal number with a limited number of digits (or terms). 0.15, 0.86, and so on. The term "non-terminating decimal" refers to a decimal that does not have an end term. There are an infinite number of terms in it. The numbers that come to an end after a few repetitions after the decimal point are known as terminating decimals.

2. How Do You Say Whether a Number is Terminating or Not?

Ans:

A terminating decimal or a repeated decimal may be written for any logical number. Simply split the denominator by the numerator. You get a terminating decimal if you end up with a remainder of 0. Otherwise, the remainders would tend to repeat after a bit, resulting in a repeated decimal. Decimals that end and non-terminating but repeating decimals are all logical. These decimals can be written as a/b fractions. Consider the following example .375=3/8 Irrational decimals are non-terminating and non-repeating decimals, such as root 2, root 3, and so on. After the decimal point, the digits can carry on indefinitely.