DECIMAL REPRESENTATION
Converting Recurring Decimals to p/q
We have seen that a Rational Number can produce a recurring decimal.
But can we go in the opposite direction?
Let's call the decimal x.
The repeating part has one digit, so multiply both sides by 10.
Now compare this with the original equation:
Subtract the original equation from the new one:
Therefore:
KEY IDEA
The repeating parts cancel when we subtract.
We use a power of 10 to make the repeating parts line up.
What if two digits repeat?
The repeating block is 27.
Since two digits repeat, multiply by 100:
Now subtract:
Therefore:
THE PATTERN
Multiply by a power of 10 until the repeating parts line up.
What if the decimal has a non-repeating part?
Here, 1 is the non-repeating part and 6 is the repeating digit.
First move the non-repeating digit:
Now multiply by 10 once more:
Subtract the first equation from the second:
Therefore:
The same idea works: use powers of 10 to align the repeating parts, then subtract.
NUMBER SYSTEMS
Every recurring decimal can be converted into a Rational Number.
We have now seen both directions:
This completes our journey through Number Systems.