Fractions and simplification, explained
I've always found fractions to be a little bit magical; things like "doing the same multiplication to the numerator and denominator gets you a fraction of equal value" and the process of simplifying fractions have been useful, but it hasn't been very clear how they work. So in this I set out on the silly, little, but fun quest to do define the way fractions work using plain old math..
A fraction is sort of like a number made of two parts. For example, we might have a fraction made of the numbers 1 and 2, and we would write that as . We could have another fraction made of the numbers 7 and 9, and that would be written as .
The value of a fraction is simply the fraction's top number (we call this its numerator) divided by its bottom number (the denominator): , , and so on. (To evaluate a fraction just means to replace that fraction with its value.)
You can do lots of cool things with fractions, but we're interested in multiplying them.
Hopefully you already know how to multiply two normal numbers - for example, you should already know that , , and so on.
Multiplying two fractions is actually pretty simple - we just need to multiply the two numerators and the two denominators together, and we get another fraction. For example, to multiply :
We can also multiply other fractions:
..and so on.
(We've already said that fractions can be made out of two numbers. But we made the fraction above - how did that work? Well, it makes sense if you think of and as numbers. In fact you don't need to worry about what those values are (even though we know them to be 6 and 4); just that they can be used in place of numbers. If you would like to get very technical, we could say that a fraction is made of two expressions; and an expression is just a number, like 7, or a calculation, like or .)
What if we want to multiply a fraction and a normal whole number?
The trick is that you need to turn the whole number into a fraction, somehow. The fraction we are creating must have an equal value to our whole number, or else we cannot use it in place of that number.
It is true that for any number, dividing that number by 1 does not change it: , , etc. Since we already know that the value of a fraction is gotten by dividing the numerator by the denominator, we can create a fraction from any division problem: simply use the first number (5, 4, etc.) as the numerator and the second number (1) as the denominator. , , and so on.
Now we know that any number is equal to that same number , so we may put our fraction into our calculation:
And we already know how to multiply two fractions:
We're almost ready to try out something interesting, but first we need to understand one more concept - that any number multiplied by 1 is the original number: , , . (We already know this from how multiplying whole numbers always works, of course.)
We can apply the same concept to fractions using what we now know about multiplying a fraction by a whole number:
This makes sense because we know that (from our rule that for any number, we can make a fraction that is equal to that number by using it as the numerator of the fraction and 1 as the denominator).
We can proceed now to define another rule: that for any fraction where the numerator and denominator are the same, the fraction is equal to 1. For example, , , .
This makes sense because we know that any number divided by itself equals 1 (that is, that any number "fits into" itself exactly 1 time). Using our rule for the value of a fraction, we may write: , , and so on.
This means that we know how to replace any 1 in our calculations with a fraction of equal value; let's try that with our "multiplication by 1 equals itself" rule using a fraction:
Ah! How peculiar. This reveals that the fraction is actually equal to the fraction .
In fact, we can multiply any fraction and 1, or an equal value, and get a new fraction that is equivalent to the first fraction:
..And so on.
Now - we have already said that for any fraction where the top and bottom are equal, that fraction is equal to 1. What if we put two equal fractions inside of our fraction?
This is still equal to 1, because the numerator and denominator of the "big" fraction are equal. (It is true that , of course!)
Likewise, fractions such as and are also equal to 1.
Now we can use all we've learned so far to try this:
But how do we multiply those two fractions? Well, we can multiply them exactly the way we would usually multiply fractions - multiply the two numerators together and the two denominators together:
We already know how to multiply a whole number and a fraction together - just convert the whole number to a fraction by putting the whole number on top and 1 on the bottom:
We are nearly completed; we may simply insert this fraction back into our calculation:
In order to actually make this useful, we must evaluate the fractions that are in the top and bottom of our newly-created "big" fraction:
And then we may put this back into our main calculation:
As you can see, we have gone from one fraction, in this case , to a fraction of equal value but smaller numerators and denominators, . This is what is known as simplifying a fraction.
We can show these are of equal value simply by comparing their values: , and . We say that because the fractions have an equal value, they are proportional.
The greatest common divisor of two numbers is the greatest whole number you can divide both numbers by and get two resulting whole numbers. There are various methods of finding it; we write it with the notation (where and are whole number values). For example, because dividing and both get us whole numbers (4 and 3), and there is no greater number that both 20 and 15 can be divided by to get whole numbers.
The greatest common factor can be used inside of fraction simplification to get the "completely" simplified value of any fraction. For example:
(I skipped a couple of steps in multiplying the values in the "big" fraction to keep a focus on the important part, which was applying the greatest common divisor.)
This time we get the value of , which is 2, and use that as our simplified value.
All of the above can be written in elegant and general algebra-like math.
Value of a fraction:
Fractions from values using denominator 1:
Multiply two fractions:
Multiply a fraction and a value:
Simplify a fraction (completely):
(The parentheses around each step are only present to clarify the separate steps; they don't actually mean anything.)