Types of Numbers
There are many ways to represent any one fraction. How do we choose which to use? It will depend on the problem, but it's often helpful to simplify or reduce the fraction.
We know that if we start with a fraction and multiply the numerator and denominator each by the same number n, we get a fraction equivalent to . This also works in reverse: if the whole number n divides evenly into both p and q, then dividing p and q each by n will create a new fraction equivalent to . Let's call it j/b, for a change of pace.
Consider the fraction .
4 and 6 are both divisible by 2. Divide each by 2 to get a fraction equivalent to .
Easy peasy - that's .
To simplify or reduce a fraction , we carry out this process until there are no natural numbers n left (except 1) that divide both p and q. It is now in lowest terms. This fraction would win the limbo every time.
How to systematically reduce fractions:
Rather than just eyeballing a couple of numbers and using the trial and error method, we can save ourselves some time by finding the biggest number n that divides both the numerator and the denominator. Ho there - this is something we've seen before! The number n is the GCF, or the greatest common factor of the numerator and denominator! In the improv world, we call this a "callback." (We like to pretend we're a part of the improv world. Just humor us.)
To quickly figure out what the GCF of the numerator and denominator of a fraction is, write out the prime factorizations of the numerator and denominator. The GCF is what you get when you multiply all the prime factors shared by both numerator and denominator. It's all about the overlap.
We can see that 3 divides both the numerator and the denominator. When we go ahead and divide the numerator and denominator each by 3 we get . This process is also called "canceling out" the 3's from the top and bottom of the fraction. Apparently, they didn't get very good ratings. They were really just appealing to the prime number demographic.
Check the overlap - we can cancel out one 2 and one 3 from both the numerator and denominator to get
There is a huge benefit to this process of canceling out the common prime factors to arrive at a reduced fraction. Since a number is completely determined by its prime factorization, the upshot is that, even though there are many, many equivalent fractions representing the same rational number, there is only one fully reduced fraction! It's like you've been dating around and have met plenty of perfectly nice guys, but finally you've found "the one!" And all you had to do was reduce him to practically nothing! Oh happy day!