Numerator/Denominator: The reasons beyond the names and learn how to operate and why.
If you understand something deeply, usually you can recall it for life, paradoxically people prefer to teach mathematics by applying mechanical methods, arguing that the reasons behind are too complicated. I resist believing in such a poor paradigm.
When you try to teach elemental algebra, you stumble with 3 kinds of people: those who don’t have a clue, those who learned to apply a method or formula but have no idea why, and a few who understand the concept behind it. You may think that in our modern instructed society the majority know how to operate with fractional numbers, but the truth is they remember badly a method and never asked themselves the reason. It is sad that so many people were taught mechanically, losing the possibility to appreciate the simple beauty of real mathematics. I will try here to shed some light on this issue.
We will keep things simple. Let’s understand first what represents a natural number. Suppose you have two apples. Then you add a pear. You can’t say that you have 3 apples neither 3 pears. Both cases are false. To enumerate these two apples plus a pear you have to invent a new name that covers all the possibilities. If you say 3 fruits, you will be right. Observe that this is not the only possibility, in fact, there are infinite possibilities if you realize you can negate a feature. For example, you can say that you have “three non-hammers”, or “three non-animals” or whatever.
Of course, you can forget about the origin of the numbers, and just think about abstract entities which represent anything you like, and you will be right. However, when you construct fractional numbers, this issue reapers, even with no apples or pears; then let’s call back our nice green apples to see what’s going on. Suppose you divide one apple into two and another into three equal parts, like in the picture.
Now you are curious about how much apple is 1/2 of an apple plus 1/3 of an apple. The parts you want to add are essentially different; they are still made of apples, but the portion of an apple they represent is not the same, as you can appreciate in the picture. You can always do the easy way, calculate 1/2 = 0.5 and 1/3 = 0.33333…., add it, and get 0.83333…., but doing this you can land with an inexact value, like in this case, because the decimal “tail” never ends. If you want to obtain an exact value, you need to know how to add two fractional (rational numbers is the fancy name used by the mathematicians) numbers. Surely the great majority have already got the result, 5/6, but the question is the background of the method you learned in school many years ago. As with the apples and pears, you need to find first a common name. You can say, as before, that you have “two parts of an apple” or “two non-horses”, but neither of the former common names helps in your goal to know which fraction of an apple is 1/2+1/3. You need to recognize that dividing numbers are primordial. These numbers, 2 and 3, play the same role as the apples and pears from our first example; because 1/2 is essentially different from 1/3 as you can see in the previous picture. This is the reason for the name: denominator, because these numbers define the corresponding fractions. The next question is key: through which process you can unify the names, in the same manner, you choose fruit in the first example, and simultaneously preserve the numerical value of the fractions? A picture is 1000 words worth: in the image below you will find the answer!
Now you have it, doing a new division using the other number, you finish with a new denominator which is the multiplication of the two previous: 2 x 3 = 6. As you can see in the next picture, each part is now identical to the rest, and there are 5 in total.
As I said before, 5/6 is the final result. Observe that 5 tells you the number of apple portions with the 1/6 quality; therefore, the upper number is called Numerator. As far as I know, in any language these names are preserved as a literal translation, at least it happens in Spanish, German, Hungarian, and obviously English. Please let me know in the comments, if you speak a language where these factors have other names. In the next picture, the general formulas for adding or subtracting two or more fractions are presented.
Summarizing: the secret of adding (or subtracting) two or more fractions, or rational numbers, is to find a common denominator, which is simply the product of all the denominators. Then, take the common denominator and divide by each former denominator, and multiply with the corresponding numerator. For example, in the last term of the picture: e.bdf/f = ebd. This rule can be extrapolated to any number of summing fractions. Maybe you have learned a method involving the “Least Common Multiple” or lcm. In this method you should calculate the lcm of all the denominators, in the example of the last picture would be lcm(b,d,f). Maybe you even learned that this is the only correct method to calculate the addition of fractions. The last is not true. The lcm methods just optimize to the smallest possible common denominator, but this method is time consuming and gives no clear advantages, as you can always simplify the final resulting fraction.