Have you ever encountered a number that seems to repeat endlessly? Maybe it’s a decimal that goes on and on without a clear pattern, leaving you wondering how to make sense of it. If so, you’re not alone! Repeating numbers can be tricky, but fear not, we’re here to guide you through the process of calculating a periodic number.

What is a repeating number?

A repeating number, or a periodic number, is a type of irrational number that expresses itself in a patterned sequence. Instead of terminating like rational numbers (e.g., 2.5), repeating numbers continue indefinitely with a specific pattern.

Example: Calculating a Repeating Decimal

Let’s take the number 1/3 as an example. When we divide 1 by 3, we get 0.3333… with the 3 repeating forever. This indicates that the number 1/3 is a repeating decimal. But how do we calculate it?

  • Step 1: Identify the numerator and denominator
  • Step 2: Perform the division
  • Step 3: Track the remainders
  • Step 4: Identify the repeating pattern
  • Step 5: Express the repeating pattern as a fraction

Let’s go through each step in detail.

Step 1: Identify the numerator and denominator

In our example, the numerator is 1 and the denominator is 3. These numbers will help us determine the repeating pattern.

Step 2: Perform the division

To calculate the decimal equivalent of the fraction, divide the numerator by the denominator. In this case, 1 divided by 3 equals 0.3333… Keep in mind that you might need to continue the division beyond the decimal point to identify the repeating pattern.

Step 3: Track the remainders

Each time you perform a division, you’ll obtain a remainder. In our example, the remainder is always 1. Keep track of these remainders as they will help identify when the pattern starts repeating.

Step 4: Identify the repeating pattern

At some point during the division process, you’ll notice that the remainders begin to repeat. In our example, the remainder is always 1, which indicates that the pattern repeats after every 3 digits. Thus, the repeating pattern for 1/3 is 3.

Step 5: Express the repeating pattern as a fraction

To express the repeating pattern as a fraction, set the repeating pattern over a number of nines equal to the number of digits in the pattern. In our example, the repeating pattern is 3, so we can write it as 3/9.

Putting it all together, we can express 1/3 as 0.3333… or as the fraction 3/9.

By following these steps, you can calculate the repeating decimal equivalent of any given fraction.

Calculating a repeating number may initially seem daunting, but it becomes simpler once you understand the steps involved. By identifying the numerator and denominator, performing the division, tracking the remainders, and identifying the repeating pattern, you can express any repeating number with confidence.

So, the next time you encounter a repeating number, don’t get overwhelmed. Instead, embrace the challenge and calculate that periodic number with ease!

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