How do you show the group of digits that repeat in a repeating decimal?
How do you show the group of digits that repeat in a repeating decimal?(1 point)
Responses
by underlining them
by underlining them
by putting a bar over them
by putting a bar over them
by printing them in italics
by printing them in italics
by putting them in parentheses
To show the group of digits that repeat in a repeating decimal, you can follow these steps:
Step 1: Identify the repeating pattern.
Look for a recurring sequence of digits that repeats indefinitely in the decimal representation of the number.
Step 2: Write an algebraic equation.
Assign a variable to the repeating pattern and write an equation that represents the repeating decimal.
Step 3: Solve the equation.
Solve the equation to find the value of the repeating pattern.
Step 4: Express the repeating decimal.
Express the repeating decimal by writing the non-repeating part first followed by a line over the repeating pattern of digits.
For example, let's consider the repeating decimal 0.6666...:
Step 1: Identify the repeating pattern.
In this case, the repeating pattern is "6."
Step 2: Write an algebraic equation.
If we assign "x" to the repeating pattern, we can write the equation as: x = 0.6666...
Step 3: Solve the equation.
To find the value of "x," we can manipulate the equation by multiplying both sides by 10: 10x = 6.6666...
Now, subtract the equation "x = 0.6666..." from the above equation:
10x - x = 6.6666... - 0.6666...
9x = 6
x = 6/9 = 2/3
Step 4: Express the repeating decimal.
The repeating decimal 0.6666... can be expressed as 0.6̅, where the line is placed over the "6" to indicate the repeating pattern.
To show the group of digits that repeat in a repeating decimal, you can use a bar notation or a parenthesis notation.
Bar notation: In this notation, a horizontal bar is placed over the repeated group of digits. For example, if the decimal 0.3333... repeats the digit 3, the bar notation would be written as 0.3̅, where the bar is placed over the digit 3.
Parenthesis notation: In this notation, the repeating group of digits is enclosed within parentheses. For example, if the decimal 0.363636... repeats the digits 36, the parenthesis notation would be written as 0.(36).
Both notations indicate that the group of digits immediately following the bar or within the parentheses repeats infinitely.
To determine the repeating pattern in a decimal, you can observe the pattern of digits by looking for a sequence that repeats. Another method is to calculate the remainders when dividing the decimal's numerator by its denominator repeatedly. The repeating pattern emerges when a remainder repeats. For example, when dividing 1 by 3, the decimal representation is 0.3333..., and the pattern is revealed by performing long division.
Remember to use these notations only when the repeating pattern is truly infinitely repeating, without any non-repeating digits in between.