NUMBER SYSTEMS
Why Do Some Rational Decimals Terminate or Repeat?
We have seen that Rational Numbers can have different decimal representations.
This decimal terminates.
This decimal keeps repeating.
KEY IDEA
The remainder determines what happens during division.
Consider:
After reaching 5, the remainder becomes 0.
There is nothing left to divide, so the decimal terminates.
Why does 1/2 terminate?
We can rewrite the fraction with a denominator of 10:
Since our decimal system is based on powers of 10:
So the decimal has a finite number of digits.
Now consider:
The remainder never becomes zero.
Since the same remainder keeps appearing, the same digit keeps repeating.
The remainder controls what happens next.
Why does 1/3 repeat?
During the division, the same remainder returns again and again.
The division therefore enters the same state repeatedly.
THE PATTERN
Write the Rational Number in its simplest form, then look at its denominator.
→ Terminating
→ Recurring
THE RULE
Only 2s and 5s in the denominator → Terminating decimal
Any other prime factor → Recurring decimal
Why are 2 and 5 special?
Our decimal system is based on powers of 10.
So a denominator containing only 2s and 5s can be converted into a power of 10.
That is why the decimal can terminate.
NOW LET'S GO THE OTHER WAY
We know how Rational Numbers behave in decimal form.
Can we turn a recurring decimal back into a fraction?
Let's find out.