if a repeating decimal has a repeating cycle of three digits it will convert to a rational number with what denominator?
If a repeating decimal has a repeating cycle of three digits, it can be converted to a rational number with a denominator of 999.
To see why this is the case, let's take an example of a repeating decimal with a repeating cycle of three digits, such as 0.123123123...
To convert this repeating decimal to a rational number, we can represent it as follows:
Let x = 0.123123123...
Multiply both sides of the equation by 1000 to remove the decimal point:
1000x = 123.123123...
Notice that if we subtract x from this new equation, the repeating cycle cancels out:
1000x - x = 123.123123... - 0.123123123...
This simplifies to:
999x = 123
Divide both sides by 999 to isolate x:
x = 123/999
This fraction, 123/999, is a rational number. Therefore, a repeating decimal with a repeating cycle of three digits can be converted to a rational number with a denominator of 999.