GoMim AI | What is terminating decimal

Introduction

In mathematics, understanding different types of decimals is crucial for various calculations and number representations. One such type is the terminating decimal, which often arises in basic arithmetic, algebra, and real-world applications. This article will explore the concept of terminating decimals, their significance, and how to calculate them effectively.

What is it?

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. This means that the decimal representation of the number comes to an end, or terminates, after a certain number of digits. For example, the number 0.75 is a terminating decimal because it ends after two decimal places. Mathematically, a fraction \( \frac{a}{b} \) will have a terminating decimal if the denominator \( b \) can be expressed in the form of \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

It's important to distinguish between terminating decimals and repeating decimals. A repeating decimal has one or more repeating digits or sequences of digits after the decimal point. For example, 0.333... is a repeating decimal because the digit '3' repeats indefinitely. Terminating decimals do not have this characteristic, as they have a definitive end.

Why is it important?

Terminating decimals are significant in various fields for multiple reasons:

• Mathematics Education: Understanding terminating decimals is foundational for students learning about decimal numbers and fractions. It helps them grasp the concept of converting fractions to decimals and vice versa.
• Data Analysis: In data analysis, precise representation of numbers is crucial. Terminating decimals provide exact values, reducing errors in computations.
• Financial Calculations: Terminating decimals are often used in financial calculations such as interest rates and currency conversions where precision is important.
• Engineering and Science: Accurate measurements and calculations are essential in engineering and scientific computations, where terminating decimals are preferred for their exactness.

How to Calculate it Step-by-Step

Step 1: Begin with a fraction. For example, consider the fraction \( \frac{3}{8} \).

Step 2: Simplify the fraction, if necessary, to ensure it's in its simplest form.

Step 3: Check if the denominator can be expressed as a power of 2, a power of 5, or a combination of both. Here, 8 can be written as \( 2^3 \).

Step 4: Divide the numerator by the denominator using long division to find the decimal equivalent.

Step 5: The result from the division will be the terminating decimal.

Example 1

1. Convert the fraction \( \frac{1}{2} \) to a decimal.
2. Since 2 is a power of 2, \( \frac{1}{2} \) will have a terminating decimal.
3. Perform the division: 1 divided by 2 equals 0.5.
4. The terminating decimal of \( \frac{1}{2} \) is 0.5.

Example 2

1. Convert the fraction \( \frac{7}{25} \) to a decimal.
2. Since 25 is \( 5^2 \), this fraction will have a terminating decimal.
3. Perform the division: 7 divided by 25 equals 0.28.
4. The terminating decimal of \( \frac{7}{25} \) is 0.28.

Common Mistakes

1. Not simplifying the fraction first, which may lead to incorrect conclusions about the decimal's nature.

2. Misinterpreting repeating decimals as terminating decimals, especially when the repeating part is lengthy.

Solving Tips

1. Always simplify the fraction to its lowest terms before starting any calculations.

2. Familiarize yourself with powers of 2 and 5 to quickly determine if a decimal will terminate.

3. Practice long division to improve your ability to convert fractions to decimals accurately.

Related Practice Problem

Problem: Determine if the fraction \( \frac{18}{40} \) results in a terminating decimal, and if so, find the decimal form.

Step-by-step Solution:

1. Simplify the fraction \( \frac{18}{40} \) to \( \frac{9}{20} \).

2. Check the denominator 20. It can be expressed as \( 2^2 \times 5 \), which fits the form \( 2^m \times 5^n \).

3. Since the denominator is composed of powers of 2 and 5, the decimal will terminate.

4. Divide 9 by 20 using long division to get 0.45.

5. Therefore, \( \frac{18}{40} \) as a terminating decimal is 0.45.

Related Topics

• Repeating Decimal: A decimal with one or more repeating digits after the decimal point.
• Long Division: A method used to divide numbers and find decimal representations.
• Fraction Simplification: The process of reducing a fraction to its simplest form.
• Rational Numbers: Numbers that can be expressed as a fraction of two integers.

Use GoMim Math AI Solver for terminating decimal

Step 1: Visit the GoMim website and navigate to the AI Math Solver section.

Step 2: Enter the fraction you want to convert into a terminating decimal in the input field.

Step 3: Use the 'Calculate' function to have the AI Math Solver process the fraction.

Step 4: Review the step-by-step solution provided by the AI, showing how the fraction converts to a terminating decimal.

Step 5: Utilize the insights and explanations to improve your understanding and apply them to similar problems.

Didn't get the point clearly? Try AI Math Solver Free

FAQ

Q: What is a terminating decimal?

A: A terminating decimal is a decimal number that has a finite number of digits after the decimal point, meaning it ends or terminates.

Q: How do you know if a decimal is terminating?

A: A decimal is terminating if the denominator of its fraction form can be expressed as a power of 2, a power of 5, or a combination of both.

Q: Can all fractions be written as terminating decimals?

A: No, only fractions whose denominators are composed of the prime factors 2 and 5 can be converted to terminating decimals.

Q: Is 0.25 a terminating decimal?

A: Yes, 0.25 is a terminating decimal because it ends after two digits.

Q: What is the difference between terminating and repeating decimals?

A: Terminating decimals have a finite number of digits and end, while repeating decimals have an infinite sequence of one or more digits that repeat.

Q: How can AI tools help with terminating decimals?

A: AI tools like GoMim AI Math Solver provide step-by-step solutions to convert fractions to terminating decimals, improving understanding and accuracy.

Q: Why are terminating decimals important in finance?

A: Terminating decimals provide precise values in financial calculations, reducing rounding errors and ensuring accuracy.

Conclusion

Terminating decimals play a crucial role in mathematics and real-world applications, offering precision and clarity in numerical representation. By understanding how to identify and calculate terminating decimals, learners can enhance their mathematical proficiency. Tools like GoMim AI Math Solver can further aid in mastering these concepts, making learning more efficient and effective.