Periodic numbers, also known as repeating decimals, are numbers that have a repeating pattern of digits after the decimal point. They can be tricky to work with when it comes to mathematical expressions, but fear not! In this blog post, we will guide you through the steps of making expressions with periodic numbers.

What are periodic numbers?

Periodic numbers are decimals that repeat indefinitely. For example, 1/3 is a periodic number because its decimal representation is 0.3333…, where the digit 3 repeats indefinitely. Similarly, 1/7 is a periodic number because its decimal representation is 0.142857142857…, where the pattern 142857 repeats.

Periodic numbers can be represented using a bar notation. For instance, 0.3333… is written as 0.(3), and 0.142857142857… is written as 0.(142857).

Adding periodic numbers

When adding periodic numbers, the key is to align the decimal places and be aware of the repeating pattern. Let’s take the examples:

  • Add 0.3333… to 0.6666…

To add these periodic numbers, we align the decimal places:

    0.3333...
+   0.6666...
------------

Now, we can see that the repeating pattern of both numbers is the digit 3. So, we can rewrite the expression as:

    0.3333...
+   0.6666...
------------
    0.(3)
------------

Therefore, the sum of 0.3333… and 0.6666… is 0.(3).

Multiplying periodic numbers

Multiplying periodic numbers can be a bit more challenging, but with the right approach, it becomes easier. Let’s consider:

  • Multiply 0.25 by 0.6666…

The first step is to ignore the decimal point and multiply the numbers as if they were whole numbers:

         25
  × 6666...
----------

Next, count the total number of digits after the decimal point in both numbers. In this case, the first number (0.25) has 2 digits, and the second number (0.6666…) has infinite digits.

Since we are multiplying a number with infinite digits, the result will also have an infinite number of digits after the decimal point. The pattern of the repeating digits can be determined using long division. In this case:

     25.0000
  ÷  6.6666...
-------------

The quotient is 3.75, and the pattern 375 repeats indefinitely. Therefore, the product of 0.25 and 0.6666… is 0.(375).

Working with periodic numbers in expressions requires attention to detail and understanding of their repeating patterns. By aligning decimal places and following specific rules, you can confidently add and multiply these numbers. Practice is key to becoming comfortable with these calculations!

Remember to always double-check your work when dealing with periodic numbers to ensure accuracy in your calculations.

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