Understanding Terminating Decimals

Oct 13, 2025 by wbrum.org 23 views

Introduction

In mathematics, decimals are a way to represent fractions with a denominator that is a power of 10. A terminating decimal is a decimal that has a finite number of digits after the decimal point. In this article, we will explore how to express a terminating decimal as a quotient of integers, and if possible, reduce it to its lowest terms.

Understanding Terminating Decimals

A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example, 0.5, 0.25, and 0.125 are all terminating decimals. On the other hand, decimals like 0.333... and 0.666... are repeating decimals, also known as non-terminating decimals.

Expressing Terminating Decimals as Quotients of Integers

To express a terminating decimal as a quotient of integers, we need to follow these steps:

Move the decimal point: Move the decimal point of the terminating decimal to the right until we have a whole number.
Multiply by 10: Multiply the whole number by 10 for each place value that we moved the decimal point.
Write as a fraction: Write the whole number as a fraction with the denominator equal to 10 raised to the power of the number of places we moved the decimal point.

Example 1: Expressing 0.9613 as a Quotient of Integers

Let's use the above steps to express 0.9613 as a quotient of integers.

Move the decimal point: Move the decimal point of 0.9613 to the right until we have a whole number. We get 9613.
Multiply by 10: Multiply 9613 by 10 for each place value that we moved the decimal point. We get 96130.
Write as a fraction: Write 96130 as a fraction with the denominator equal to 10 raised to the power of the number of places we moved the decimal point. We get 96130/10000.

Reducing to Lowest Terms

To reduce the fraction 96130/10000 to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

Finding the GCD

To find the GCD of 96130 and 10000, we can use the Euclidean algorithm.

Divide the larger number by the smaller number: Divide 96130 by 10000. We get 9 with a remainder of 6130.
Replace the larger number with the smaller number and the smaller number with the remainder: Replace 96130 with 10000 and 10000 with 6130.
Repeat the process: Divide 10000 by 6130. We get 1 with a remainder of 3870.
Repeat the process: Divide 6130 by 3870. We get 1 with a remainder of 2260.
Repeat the process: Divide 3870 by 2260. We get 1 with a remainder of 1610.
Repeat the process: Divide 2260 by 1610. We get 1 with a remainder of 650.
Repeat the process: Divide 1610 by 650. We get 2 with a remainder of 310.
Repeat the process: Divide 650 by 310. We get 2 with a remainder of 10.
Repeat the process: Divide 310 by 10. We get 31 with a remainder of 0.

The GCD

The GCD of 96130 and 10000 is 10.

Reducing the Fraction

To reduce the fraction 96130/10000 to its lowest terms, we divide both the numerator and the denominator by the GCD, which is 10.

The Reduced Fraction

The reduced fraction is 9613/1000.

Conclusion

In this article, we learned how to express a terminating decimal as a quotient of integers and reduce it to its lowest terms. We used the example of 0.9613 to illustrate the process. By following these steps, we can express any terminating decimal as a quotient of integers and reduce it to its lowest terms.

References

Euclid: The Elements
Gauss: Disquisitiones Arithmeticae

Appendix

Here are some additional resources that you may find helpful:

Decimal to Fraction Converter: A online tool that converts decimals to fractions.
Fraction to Decimal Converter: A online tool that converts fractions to decimals.
Mathematical Tables: A collection of mathematical tables that you can use to look up mathematical values.
Expressing Terminating Decimals as Quotients of Integers: Q&A ===========================================================

Introduction

In our previous article, we explored how to express a terminating decimal as a quotient of integers and reduce it to its lowest terms. In this article, we will answer some frequently asked questions about expressing terminating decimals as quotients of integers.

Q: What is a terminating decimal?

A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example, 0.5, 0.25, and 0.125 are all terminating decimals.

Q: How do I express a terminating decimal as a quotient of integers?

To express a terminating decimal as a quotient of integers, you need to follow these steps:

Move the decimal point: Move the decimal point of the terminating decimal to the right until you have a whole number.
Multiply by 10: Multiply the whole number by 10 for each place value that you moved the decimal point.
Write as a fraction: Write the whole number as a fraction with the denominator equal to 10 raised to the power of the number of places you moved the decimal point.

Q: What is the difference between a terminating decimal and a non-terminating decimal?

A terminating decimal is a decimal that has a finite number of digits after the decimal point, while a non-terminating decimal is a decimal that has an infinite number of digits after the decimal point. For example, 0.5 is a terminating decimal, while 0.333... is a non-terminating decimal.

Q: How do I reduce a fraction to its lowest terms?

To reduce a fraction to its lowest terms, you need to find the greatest common divisor (GCD) of the numerator and the denominator. You can use the Euclidean algorithm to find the GCD.

Q: What is the Euclidean algorithm?

The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It works by repeatedly dividing the larger number by the smaller number and replacing the larger number with the smaller number and the smaller number with the remainder.

Q: How do I use the Euclidean algorithm to find the GCD of two numbers?

To use the Euclidean algorithm to find the GCD of two numbers, you need to follow these steps:

Divide the larger number by the smaller number: Divide the larger number by the smaller number.
Replace the larger number with the smaller number and the smaller number with the remainder: Replace the larger number with the smaller number and the smaller number with the remainder.
Repeat the process: Repeat the process until the remainder is 0.

Q: What is the greatest common divisor (GCD)?

The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I use the GCD to reduce a fraction to its lowest terms?

To use the GCD to reduce a fraction to its lowest terms, you need to divide both the numerator and the denominator by the GCD.

Q: What are some examples of terminating decimals?

Some examples of terminating decimals include:

0.5
0.25
0.125
0.9375

Q: What are some examples of non-terminating decimals?

Some examples of non-terminating decimals include:

0.333...
0.666...
0.999...

Conclusion

In this article, we answered some frequently asked questions about expressing terminating decimals as quotients of integers. We hope that this article has been helpful in understanding how to express terminating decimals as quotients of integers and reduce them to their lowest terms.

References

Euclid: The Elements
Gauss: Disquisitiones Arithmeticae

Appendix

Here are some additional resources that you may find helpful:

Decimal to Fraction Converter: A online tool that converts decimals to fractions.
Fraction to Decimal Converter: A online tool that converts fractions to decimals.
Mathematical Tables: A collection of mathematical tables that you can use to look up mathematical values.