At a Glance - Rational Numbers
Integers are fine and dandy, but not everything in this world works out to a perfectly even whole number. That pair of boots you've had your eye on may not be exactly $64. That recipe may not call for exactly 1 cup of sugar. And good luck finding a book in the library using the Dewey Whole Number System.
Take a look at the problems below, and see if you can solve them by using only integers.
1. Divide 25 cookies between 5 people. How many cookies does each person get? Answer: 5, assuming Santa isn't in the building.
2. Divide 100 pennies into piles of 10. How many piles do we have? That's 10. Make cents?
3. Divide 3 cheesecakes between 4 people. How many cheesecakes does each person get?
First of all, are you not aware of the obesity problem in this country? Are there no carrots in this hypothetical refrigerator?
Okay, so we can't answer this question with a whole number, because each person gets of a cheesecake. The number is an example of a rational number: a number that can be written as the quotient of two integers. Mathematicians say a rational number can be written as a quotient p/q where q is nonzero. We call p the numerator and q the denominator. When we write a rational number in the form p/q where q is not equal to 0 or 1, we call it a fraction. The reason we can't have q equal 0 is because we'd be dividing by 0, which gives us an undefined number. Go ahead—just try looking up 0/0 in the dictionary. You won't find it. And if we divide it by one, we've just got ourselves a whole number, and not a fraction. In this case, we don't even have to worry about minding our p's and q's.
Fractions are examples of division problems where we simply don't carry out the long division. We just let the two numbers kinda hang out there, one in the top bunk and one in the bottom bunk. They could use the rest. You could, too, if you spent so much of your time getting divided up into smaller pieces like that.
Even though we don't write integers as fractions, they're still considered rational numbers. This is because we can think of all integers as a quotient with a denominator of 1. For example, we can write 5 as 5/1, 11 as 11/1, and -25,693 as -25,693/1. Note that this doesn't necessarily work in reverse, as not all rational numbers are integers—for example: 5/2. This is the only example we could think of, but since there's literally an infinite amount of rational numbers where this is the case, we probably weren't trying very hard.
Not only can we think of 5 as 5/1, but we can also write it as 25/5, or 50/10. Similarly, we can write 10 as 100/10, or as 10/1. As you can see, there are many different fractions that can be used to express any particular rational number. We'll talk later about how to tell when two different fractions are representing the same number, but don't worry your little head about that for now. Just close your eyes, go to sleep, and dream of puppies and rainbows.
All right, so switching gears...why can't the denominator q be zero?
Think about it this way. If you have 3 cheeseburgers and divide them between zero people, how many cheeseburgers does each person get? First of all, who made you the Grand High Cheeseburger Distributor? And second, how come you don't get any? That seems sort of cruel.
Third, and more to the point, if there aren't any people at all, what does it even mean to "divide the cheeseburgers among people?" This question doesn't make sense. We can put zero into 3 any number of times that we want and still have plenty of zero leftover! This number just can't be defined by any single value.
If that example didn't drive the idea home, here's another. If you have p pennies and put them in piles of zero pennies each, how many piles do you have? No matter how many piles of zero pennies you have, you'll still have zero pennies. Which is depressing, because you're never going to be able to afford the latest, greatest, and dreamiest smartphone at this rate.
Even infinitely many piles of zero pennies will not get you to the right number. So this doesn't make sense either. Hope that makes sense. That it doesn't make sense, we mean. Oy.