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You may not realize it, but you're already pretty good at recognizing a variety of different symbols that all represent the same number or concept. For example, the thing in parentheses (4) isn't actually the number four. It's just a symbol that represents the concept of the number four. To further illustrate this point, take a gander at these symbols:
(8 ÷ 2)
Each of these symbols is different, but they're all symbolic representations of the number four. Same deal with the Roman numeral IV. They're just different ways of saying the same thing.
Along the same lines, an amount can be expressed by either a fraction, decimal or percentage. For example, we're really saying the same thing whether we write or 75%. We can even draw a picture:
In each case, the symbol represents having three out of four equal parts, or 75% of the whole. If this were a football game, we'd be ready to head into the fourth quarter. If this were a dollar, we'd be 25 cents shy of being able to dry our laundry.
Rational numbers are often represented by fractions. Real numbers are usually represented as decimals. A percent is an uber-specific way to represent a fraction with a denominator of 100.
When you're dealing with a particular problem or situation, choose the representation that best goes with the situation. When working with money, $0.75 usually makes more sense than "three-fourths of a dollar." When running a poll, "three-fourths of those polled" makes more sense than saying "0.75 of those polled." If you're saying that "three-fourths of those polled feel that $0.75 a week is enough to live on," then you could also say that there's a "100% chance that the polling methods are flawed."
Writing real numbers as decimals allows us to separate the real numbers into rational and irrational numbers. This is important, 'cause they'll often start to tussle when left alone together.
That brings us to the decimal versions of rational numbers. We already know that a rational number is anything we can write as a fraction of two integers, right? Well, here's another definition: rational numbers are decimals that end or repeat. Here are some examples of decimals that are rational numbers:
Irrational numbers are decimals that go on forever without repeating. That's pretty amazing when you think about it, considering that most of our politicians can't go more than about two minutes without repeating.
We can find many, many digits of irrational numbers like π, e, or the square root of 2, but we'll never know them all.
What is a fraction?
A fraction is a number written in the form , where q is nonzero. Any rational number can be written as a fraction. Fractions are usually used to think about "parts of a whole." For example, if someone steals all but one-fifth of your ball of wax, you'll be four-fifths shy of having a whole ball of wax. Gross.
To understand fractions, it may be helpful to think about brownies. Unless you're on a diet. In which case, just mentally replace every occurrence of "brownies" in the following example with the words "veggie squares."
If we cut a pan of warm, double chocolate caramel brownies (man, those would be some hi-cal veggie squares) into q pieces of equal size and take p of those gooey brownies, the fraction of the pan of brownies we have is because we're taking the p from the q. The size of each brownie is .
So if we cut a pan of brownies into 4 pieces and take 3 brownies, we'll have of the entire pan since each piece is of the whole.
Never mind the oddly-shaped pan we used to bake these brownies. We have an old, oddly shaped oven.
Getting back to this fraction of ours, we call the number on top of the line the numerator and the number below the line the denominator.
Numerator comes from the word "numerate," meaning "to number." The numerator tells you how many pieces you have. Denominator comes from the word "denominate," meaning "to give a name to." The denominator gives a name to the pieces, according to their size (for example, "fourths" or "fifths"). Just think, if you had a big litter of nameless puppies, you could numerate and denominate them at the same time.
If the fraction we're looking at is less than 1 (the numerator is less than the denominator), the fraction is called a proper fraction. A fraction that's greater than or equal to 1 is called an improper fraction. Especially if it's using its dessert fork to eat its salad.
There are infinitely many different ways to represent the same fraction. Since we see no reason to abandon our brownie analogy, let's stick with it. Half of that pan of brownies can be represented by:
1/2 = 2/4 = 3/6 = 4/8 = ...
Different fractions that represent the same value are called equivalent. See how that word begins with "equi-"? What do you suppose that prefix means? Yep, equine. You've stumbled upon the horse-fraction connection. No, wise guy; it means "equal." Same...equal...got it?
Let's start by discussing how you can tell if two fractions are equivalent, and then we'll examine some specific cases that you may encounter. We say that two fractions and are equivalent if p × s = q × r. For example, to see if is equivalent to , you can just crunch some quick numbers: is 1 × 7 equal to 2 × 5? Since the products are not the same, these fractions represent different rational numbers. On the other hand, we know that 2/3 and 8/12 are equivalent, because 2 × 12 is equal to 3 × 8. Chew on that, and .
Here are some equivalent representations of the number 1:
In other words, if we take all the brownies (three pieces out of three, four pieces out of four, and so on), we get the whole panful all to ourselves. We're going to feel real good about that, too, until about 2 in the morning. Those brownies are some pretty vengeful concoctions.
Here are some equivalent representations of the number 0:
If we have no brownies, it doesn't matter what size the brownies are, because we still don't have any brownies. That thought's enough to make a person cry. No brownies at all? Aw, fudge.
Let's examine a little more closely why and are equivalent fractions. Start with a pan that was cut into 2 brownies, and take one of them.
Why these brownies are brick-like, we'd rather not know. Probably best not to look a gift horse in the mouth.
Anyway, this picture represents the fraction . If we cut each brownie in half, the pan will have twice as many brownies as it did before, so the denominator gets multiplied by two. Each individual brownie will also get cut in half, so we'll have twice as many brownies as we did before; therefore, the numerator also gets multiplied by 2. Anything else you want to multiply by 2 while we're at it? We're really in a groove—you sure? Nothing? All right then, let's proceed.
This picture represents the fraction , but the shaded portion is the same size as the shaded portion in the picture representing .
These pictures illustrate the fact that They also illustrate the possibility that we at Shmoop have no idea what an actual brownie looks like. We don't get out much.
Okay, so to summarize: If we start with , multiply the numerator by 2 and multiply the denominator by 2, we get a fraction equivalent to . This same idea works for any fraction and any number n; if we multiply the numerator p and the denominator q each by n, we'll get a fraction equivalent to .
Now, two final examples and one non-example of equivalent fractions:
Is equivalent to 40/48?
Yep, it totally is because .
Is equivalent to ?
That's affirmative, since 10 = 2 × 5 and 35 = 7 × 5.
Is equivalent to ?
Not even close. To get from to means multiplying the numerator by 3 but the denominator by 2.
Sometimes we want to find an equivalent fraction with a particular denominator in order to perform operations (addition/subtraction) on it. Ideally, you'd like to get as good at manipulating fractions to get them to do what you want as you are at manipulating your parents to do the same.
Just a couple more and then we'll let you off the hook.
What's a fraction with a denominator of 24 that's equivalent to ?
No prob: multiply the numerator and denominator by 4 to get .
What's a fraction equivalent to with a denominator of 121?
Multiply the top and bottom by 11 to get .
Wanna confuse your friends? Tell them you're headed to 77-121 to get a Slurpee.
We've seen a bunch of examples of fractions so far, but what happens when we mix them up with whole numbers?
As a quick reminder, a fraction whose numerator is smaller than its denominator is called a proper fraction. The point of proper fractions is that if you look at them on the number line, they all live between -1 and 1.
Let's revisit improper fractions for a sec. Because it can be hard to imagine just how much something like 354/22 is, we can represent it as a mixed number, which is a whole lot easier to wrap your head around. A mixed number is expressed as a whole number followed by a fraction. Who knows why the fraction is following the whole number? Maybe it's lost. It is pretty maze-y in here.
For example, . In other words, five halves is the same thing as two wholes plus another half. Visual aid time:
Since mathematicians like to abbreviate things, we leave out the addition symbol and just write .
To turn an improper fraction into a mixed number, let's think about the brownies again. Might as well get as much use out of them as possible before they go stale.
Write the fraction as a mixed number.
What this number is saying is that each pan of brownies is cut into 7 equal pieces, and we have 22 total pieces. That means we have 3 full trays of brownies (3 × 7 = 21), plus 1 single, lonely brownie left over.
To turn an improper fraction into a mixed number without drawing pictures (although that does take all the fun out of it), we perform long division. Remember that a fraction is a division problem in which we just haven't done the actual dividing. However, if we do that here and divide 22 by 7, using the remainder as the fractional part of our mixed number, we get the following:
Therefore, . Whaddaya know: that's the same answer we got when we did it the other way. It's like we're caught in a wormhole of perfect logic.
We're sure you're dying to learn how to do this in reverse: how to turn a mixed number back into an improper fraction. As a magician, it's one thing to cut your assistant in half, but it's always nice to be able to put her back together again as well. We'll spend some quality time converting mixed numbers into fractions when we get to the section on adding fractions. After all, the mixed number is really just , or . However, if you've got ants in your pants and absolutely can't wait, see the section below for a sneak peak. If you're a sucker for suspense, you can skip it and come back after you've learned all about adding fractions.
Okay, here we go. To turn mixed numbers back into improper fractions, remember that mixed numbers are secretly abbreviated addition problems with the plus sign surgically removed.
Since we're going to be speaking the language of halves, we want to translate everything into those terms. How many 1/2's does 6 equal? Remember last night when you overheard your mother swearing that she would only eat 1/2 of a chocolate chip cookie (hey, at least we're off brownies), and then she kept going back to the tin until she'd consumed 6 entire cookies? That was 12 halves, if you'll recall. If you don't recall, you can also get there this way:
You have 6 × 2 = 12 half-cookies plus another half-cookie, for a total of thirteen halves, or .
There are tons of ways to represent any one fraction. How do we choose which to use? It'll 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'll get a fraction that's 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 turn this alphabet soup into some actual numbers, shall we?
Can we reduce the fraction ?
Hmm...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 the GCF of the numerator and denominator, 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 into 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 3s 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's a huge benefit to this process of canceling out the common prime factors to arrive at a reduced fraction. Even though there are many, many equivalent fractions representing the same rational number, there's 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!
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