How do u show the group of digits that repeat ina resorting decimal
To show the group of digits that repeat in a resorting decimal, you can use a notation called the "bar notation". In this notation, a bar is placed over the group of digits that repeat. Here is an example:
1.234545454545...
In this example, the group of digits "4545" keeps repeating. To show this repetition, you can place a bar over the "45" like this:
1.234545̅
This indicates that the digits "45" repeat indefinitely in the decimal.
To show the repeating group of digits in a resorting decimal, you can follow these steps:
1. Start by converting the resorting decimal into a fraction. Let's assume the decimal is represented as x.
2. Multiply both sides of the equation by a power of 10 that is equal to the number of digits in the recurring group. For example, if the recurring group has 3 digits, multiply both sides by 1000.
3. Subtract the original equation from the new equation to eliminate the decimal point. This will give you an equation in terms of x.
4. Simplify the equation and solve for x.
5. Now that you have the value of x in the fraction form, the recurring group of digits can be identified by the numbers that repeat in the numerator of the fraction.
Here is an example to help illustrate the steps:
Let's say we have the resorting decimal 0.363636...
1. Converting it to a fraction, x = 0.363636...
2. Multiply both sides by 100, as there are two repeating digits.
100x = 36.363636...
3. Subtract the original equation from the new equation:
100x - x = 36.363636... - 0.363636...
99x = 36
4. Simplify the equation and solve for x:
x = 36/99
x = 4/11
5. In this case, the recurring group of digits is 36, which corresponds to the numerator of the fraction.
Hence, the resorting decimal 0.363636... can be represented as the fraction 4/11, and the repeating group of digits is 36.
To identify the group of digits that repeat in a resorting decimal, you can follow these steps:
Step 1: Write down the original decimal number. For example, let's consider the decimal number 0.123123123...
Step 2: Multiply the decimal number by a power of 10 to shift the repeating part to the left of the decimal point. In our example, we multiply by 1000, so we have 123.123123123...
Step 3: Subtract the original decimal number from the result obtained in Step 2. In our example, subtracting 0.123123123... from 123.123123123... gives us:
123.123123123...
- 0.123123123...
-----------------
123.000000000...
Step 4: The result obtained in Step 3 will have the repeating part removed, and you will be left with an integer. In our example, the result is 123.000000000..., but since the decimals are all zeros, we can conclude that the repeating part is 123.
Therefore, in the decimal 0.123123123..., the group of digits that repeat is 123.