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How to Put Recurring Numbers in A Calculator

Reviewed by Calculator Editorial Team

Recurring numbers, also known as repeating decimals, are numbers that have a digit or group of digits that repeat infinitely. These numbers are common in mathematics and appear in various calculations. This guide explains how to properly enter and work with recurring numbers in a calculator.

Understanding Recurring Numbers

Recurring numbers are written with a bar over the repeating digits. For example, 0.333... is written as 0.3̅, and 0.142857142857... is written as 0.142857̅.

There are two main types of recurring numbers:

  • Pure recurring decimals: All digits after the decimal point repeat (e.g., 0.3̅, 0.123̅)
  • Mixed recurring decimals: Some digits repeat while others do not (e.g., 0.166̅, 0.109̅)

Recurring numbers can be converted to fractions, which makes calculations easier. For example, 0.3̅ = 1/3 and 0.166̅ = 1/6.

Entering Recurring Numbers in a Calculator

Most scientific and graphing calculators have a specific method for entering recurring numbers. Here's how to do it:

  1. Press the "2nd" or "SHIFT" key (depending on your calculator model)
  2. Press the decimal point (.) key
  3. Enter the non-repeating digits (if any)
  4. Press the "2nd" or "SHIFT" key again
  5. Press the decimal point (.) key again to indicate the start of the repeating digits
  6. Enter the repeating digits
  7. Press the "2nd" or "SHIFT" key one more time to end the repeating sequence

Note: The exact method may vary slightly depending on your calculator model. Refer to your calculator's manual for specific instructions.

For example, to enter 0.166̅ on a TI-84 calculator:

  1. Press 2nd then . (this displays "Dec" in the display)
  2. Press 1
  3. Press 2nd then . (this displays "Frac" in the display)
  4. Press 6
  5. Press 2nd (this displays "Dec" in the display)

Converting Recurring Numbers

Converting recurring numbers to fractions can simplify calculations. Here's the general method:

  1. Let x = the repeating decimal
  2. Multiply x by 10^n where n is the number of non-repeating digits
  3. Multiply x by 10^(n+m) where m is the number of repeating digits
  4. Subtract the two equations to eliminate the repeating part
  5. Solve for x

Example: Convert 0.166̅ to a fraction

  1. Let x = 0.166̅
  2. Multiply by 10: 10x = 1.666̅
  3. Multiply by 100: 1000x = 166.666̅
  4. Subtract: 1000x - 10x = 166.666̅ - 1.666̅ → 990x = 165
  5. Solve: x = 165/990 = 1/6

Practical Examples

Here are some practical examples of working with recurring numbers:

Example 1: Adding Recurring Numbers

Calculate 0.3̅ + 0.6̅

  1. Convert 0.3̅ to fraction: 1/3
  2. Convert 0.6̅ to fraction: 2/3
  3. Add: 1/3 + 2/3 = 1

Example 2: Multiplying Recurring Numbers

Calculate 0.166̅ × 3

  1. Convert 0.166̅ to fraction: 1/6
  2. Multiply: (1/6) × 3 = 1/2 = 0.5

Example 3: Solving Equations with Recurring Numbers

Solve for x: 2x + 0.3̅ = 1.3̅

  1. Convert all terms to fractions: 2x + 1/3 = 4/3
  2. Subtract 1/3 from both sides: 2x = 1/3
  3. Divide by 2: x = 1/6 ≈ 0.166̅

FAQ

Can I enter recurring numbers directly into a standard calculator?

Standard calculators typically don't support direct entry of recurring numbers. You'll need to convert them to fractions or use a scientific calculator with the proper function.

How do I know if a number is recurring?

A number is recurring if it has a repeating pattern of digits after the decimal point. For example, 0.333..., 0.142857..., and 0.166... are all recurring numbers.

Can I use recurring numbers in financial calculations?

Yes, recurring numbers are commonly used in financial calculations, such as interest rates and loan payments. Always convert them to fractions for precise calculations.

What if my calculator doesn't support recurring numbers?

If your calculator doesn't support recurring numbers, you can manually convert them to fractions using the algebraic method described in this guide.