Determining Terminating And Non-Terminating Decimals Without Division

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It can be tricky figuring out whether a fraction turns into a decimal that ends neatly (terminating) or one that goes on forever in a repeating pattern (non-terminating repeating). The good news is, we can tell just by looking at the fraction's denominator (the bottom number)! This article explores how to determine whether a rational number will have a terminating decimal form or a non-terminating, repeating decimal form without performing actual division. We'll break down the key concepts and apply them to several examples.

Understanding Terminating and Non-Terminating Decimals

Before diving into the method, let's solidify our understanding of terminating and non-terminating decimals.

  • Terminating decimals are decimal numbers that have a finite number of digits after the decimal point. They terminate, meaning they come to an end. For example, 0.5, 0.25, and 0.125 are terminating decimals.
  • Non-terminating decimals are decimal numbers that have an infinite number of digits after the decimal point. These can be further classified into:
    • Repeating decimals: These decimals have a block of digits that repeats indefinitely. For example, 0.333..., 0.142857142857..., and 1.666... are repeating decimals. The repeating block is indicated by a bar over the repeating digits.
    • Non-repeating decimals: These decimals have an infinite number of digits after the decimal point, but there's no repeating pattern. These decimals represent irrational numbers (numbers that cannot be expressed as a fraction of two integers), such as pi (π) and the square root of 2.

In this discussion, we are focusing on determining whether a rational number (a number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0) will have a terminating decimal form or a non-terminating, repeating decimal form. Non-repeating decimals fall outside the scope of rational numbers.

The Key Principle: Prime Factorization of the Denominator

The secret to determining whether a rational number has a terminating decimal representation lies in the prime factorization of its denominator. The core principle is this:

A rational number p/q (where p and q are co-prime, meaning they have no common factors other than 1) will have a terminating decimal representation if and only if the prime factorization of the denominator q is of the form 2n5m, where n and m are non-negative integers. In simpler terms, the denominator should only have the prime factors 2 and/or 5.

Let's break this down:

  • Prime Factorization: Expressing a number as a product of its prime factors (prime numbers that multiply together to give the original number). For example, the prime factorization of 12 is 2 x 2 x 3 (or 22 x 3).
  • Co-prime: Two numbers are co-prime if their greatest common divisor (GCD) is 1. For example, 7 and 15 are co-prime because they share no common factors other than 1. If the fraction is not in its simplest form (p and q are not co-prime), you must simplify it first by canceling out any common factors.
  • 2n5m form: This means the denominator, after prime factorization, should only contain the prime factors 2 and 5, raised to some non-negative integer powers (n and m). Either n or m can be zero, meaning only 2 or only 5 can be present in the prime factorization.

Why does this work? A terminating decimal can be written as a fraction with a denominator that is a power of 10 (e.g., 0.25 = 25/100 = 1/4). Since 10 = 2 x 5, any power of 10 will only have 2 and 5 as prime factors. Therefore, if the denominator of a simplified fraction only has 2 and 5 as prime factors, it can be converted to an equivalent fraction with a denominator that is a power of 10, resulting in a terminating decimal.

If the prime factorization of the denominator contains any prime factor other than 2 and 5, the decimal representation will be non-terminating and repeating.

Applying the Principle: Examples

Now, let's apply this principle to the examples provided.

(i) 13/3125

  • First, we need to find the prime factorization of the denominator, 3125. 3125 = 5 x 5 x 5 x 5 x 5 = 55
  • The prime factorization of the denominator is 55, which is in the form 2n5m (where n = 0 and m = 5).
  • Therefore, 13/3125 will have a terminating decimal form.

(ii) 11/12

  • Find the prime factorization of the denominator, 12. 12 = 2 x 2 x 3 = 22 x 3
  • The prime factorization of the denominator is 22 x 3. It contains the prime factor 3, which is other than 2 or 5.
  • Therefore, 11/12 will have a non-terminating, repeating decimal form.

(iii) 64/455

  • Find the prime factorization of the denominator, 455. 455 = 5 x 7 x 13
  • The prime factorization of the denominator is 5 x 7 x 13. It contains the prime factors 7 and 13, which are other than 2 or 5.
  • Therefore, 64/455 will have a non-terminating, repeating decimal form.

(iv) 15/1600

  • Find the prime factorization of the denominator, 1600. 1600 = 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 = 26 x 52
  • The prime factorization of the denominator is 26 x 52, which is in the form 2n5m (where n = 6 and m = 2).
  • Therefore, 15/1600 will have a terminating decimal form.

(v) 29/343

  • Find the prime factorization of the denominator, 343. 343 = 7 x 7 x 7 = 73
  • The prime factorization of the denominator is 73. It contains the prime factor 7, which is other than 2 or 5.
  • Therefore, 29/343 will have a non-terminating, repeating decimal form.

(vi) 23/2³5²

  • The denominator is already given in its prime factorization form: 2³5².
  • The prime factorization of the denominator is in the form 2n5m (where n = 3 and m = 2).
  • Therefore, 23/2³5² will have a terminating decimal form.

Summary of Results

Here's a summary of our findings:

  • (i) 13/3125: Terminating
  • (ii) 11/12: Non-terminating, repeating
  • (iii) 64/455: Non-terminating, repeating
  • (iv) 15/1600: Terminating
  • (v) 29/343: Non-terminating, repeating
  • (vi) 23/2³5²: Terminating

Conclusion

By understanding the relationship between the prime factorization of the denominator and the decimal representation of a rational number, we can efficiently determine whether a fraction will result in a terminating or non-terminating, repeating decimal without actually performing the division. This method provides a powerful tool for simplifying calculations and gaining a deeper understanding of number properties. Remember, the key is to express the denominator in its prime factorization form and check if it only contains the prime factors 2 and 5. If it does, the decimal representation will terminate; otherwise, it will be non-terminating and repeating.

This principle is fundamental in number theory and has applications in various areas of mathematics and computer science. Mastering this concept strengthens your understanding of rational numbers and their decimal representations, which is crucial for further mathematical studies.

Remember to always simplify the fraction to its lowest terms before applying this rule. This ensures that the numerator and denominator are co-prime, and you are working with the simplest form of the fraction. By following these steps, you can confidently predict the decimal nature of any rational number.