Of Innerstanding Fractions

Listen Siblings, I come in peace,

“Europeans and Asians corrupt our children so much that our children see them as protectors and teachers.” — Onitaset Kumat

The following article was written by me for a young child.  It shows how our mathematical textbooks should look.  But to think on this, before gaining an African Blood Siblings Community Center is counting chickens before they are hatched.  Write for more information.  Subscribe, share, love.

Of Innerstanding Fractions
By Onitaset Kumat

Written for a blessed thirteen-year-old Sister.

As an African people, fractions have a special meaning all their own.  Marcus Garvey had held that the greatest weapon used against us was disorganization; in other words, we were deliberately fragmented or “fractionized:” turned into fractions.  This reflects on our ancestry, where most of us are different fractions of Igbo, Hausa, Mandinka, Kongo, Dogon, Akan, Yoruba and much more.  If we wanted to know which part of Africa our ancestry most came from, we’d need to add the different fractions and compare the total fractions by areas.  Besides from Africa’s people, politically Africa was “fractionized,” at one point becoming fifty-four UN recognized countries, a far cry from the continent becoming one country–though with South Sudan and conflicts galore, the number is unfortunately growing.  Fractions are here useful too if we ever hear that nine out of fifty-four heads of African states love a proposal and three-twenty-sevenths like the proposal.  For by adding these two fractions, we can know how many states positively review the proposal.  Finally, for all people, fractions play an important role in adding and subtracting numbers which are not whole numbers.  This is useful when measuring distance.  For instance, if you want to take the longest route for your exercise, which route is better to take? Is it one that involves a half-mile in a forest, followed by one-third of a mile by the lake, or one that involves three-fourths of a mile by the lake?  Or suppose that you want to order tape, but no more than necessary.  If you measure one-and-a-half feet for the length and two-and-two-thirds for the width, how will you know how much tape to order without understanding how to add fractions?

To begin to understand fractions, one must understand what a fraction is.  In an informal sense, a fraction is a number of equal parts of something: for instance, three-fourths of a mile is three one-fourths of a mile.  That is, a mile is broken into four equal parts (a quarter-mile) then three of those quarter miles are added together to make three-fourths of a mile.  In a more mathematical sense, a fraction is the division of two quantities.  So three-fourths equals three divided by four.  This gives the same result as the informal sense, so it’s a notion worth internalizing before proceeding: A fraction is the division of two quantities.  Again, for any number which we will call “a” and for any number[1] which we will call “b”, a/b (read as “a over b”) equals “a” divided by “b” (in this generic fraction “a/b,” “a” is called the numerator and “b” is called the denominator.)  NOTE: For your information, in the mathematical sense, we also call fractions “ratios,” where 3/4 can be called “the ratio of three-to-four” and it can be written as “3:4.”

The addition of fractions reveals something previously unseen in the laws of addition as we knew  them.  Normally, addition is straight forward: 3 + 10 = 13.  We can count three over ten or ten over three and do that simply.  But what we do not realize in the previous example is that “3,” “10” and “13” are actually fractions.  Namely, every whole number is a fraction with a denominator of 1.  So “3” is really 3/1 (read as “3 over 1”) and “10” is really 10/1 and “13” is really 13/1.  Therefore, the reason why the addition was so easy is because the denomintor of “3” and “10” and “13” were all the same.  However, when we come to fractions with denominators different from 1, our simple addition needs to change.  For instance, 4004/4004 + 12/12 = 2, but there’s no way of seeing this without altering our rules of addition (or simplifying.)

So to add fractions, denominators need to be equal, then the numberators can be added like we did whole numbers.  It would be nice if every problem simplified, or if every problem followed a simple formula, but they do not.  Sometimes, there’s no simplification and the formula can involve heavy multiplication sometimes (which is good practice).  When doing fractions, you always want to check for simplication and lowest common denominators.  Otherwise, you may need to use the common denominator that two denominators need to have: their product.  In the example of 4004/4004 + 12/12, the common multiple that the two denominators have to have is 48004 or 4004 * 12.  Simplification makes the problem easier because 4004 divides 4004 one whole time and 12 divides 12 one whole time, so 4004/4004 =1 and 12/12 = 1 so 4004/4004 + 12/12 = 1 + 1 = 2 {Simplification involves canceling the common roots of the denominator and numerator.  Every number that is not a prime number is a product of different numbers.  Knowing the roots of numbers makes simplication easier.  For instance, the root of 8 is 2*2*2 and the root of 12 is 2*2*3.  You can rewrite the fraction 12/8 as (2*2*3)/(2*2*2).   Now you can cancel out the pair of 2’s, leaving behind 3/2; the simplified fraction for 12/8.}  Much easier than multiplying 4004 by 12.

Still, there is a general formula for solving fractions.  Suppose that there’s a fraction “a/b” and another fraction “c/d” where “a” and “c” are any numbers and “b” and “d” are any numbers[1]. Then “a/b + c/d = (ad + bc)/(bd).”    This is because (a/b) = (a/b)(d/d) = ad/bd since (d/d)=1 {And multiplication by 1 does not change the value of a quantity.}  And (c/d) = (c/d)(b/b)= bc/bd.  Therefore a/b + c/d = ad/bd + bc/bd.  Since these two have the same denominator now, a/b + c/d = ad/bd + bc/bd = (ad + bc)/(bd).  If you think that you can, show that if b = d, a/b + c/d = (a + c)/b.

This general formula will work on any fraction as long as no denominator is equal to zero, since no number can be divided by zero.  The reason for simplifying fractions is to avoid so much heavy cross-multiplications.  For instance, in the example 4004/4004 + 12/12, the “a” = 4004, the “b” = 4004, the “c” = 12 and the “d” = 12.  So “ad” = 48048, “bc” = 48048 and “bd” = 48048.  Making the formula reveal a “(ad + bc)/bd) = 96096/48048 which equals 2.  These numbers are way too big.  If we simplify 4004/4004 into 1 and 12/12 into 1, then we won’t have to multiply such large numbers to find such a small result.  Simplification, also works by finding the lowest common denominator.  For instance, a problem like 1312/8 + 3/4, can be solved in at least three ways.  Either by simplifying “1312/8” into “656/4” then adding “3/4” which, having the same denominator, just makes 659/4.  Or, we can change the denominator of “3/4” into 8, by multiplying by “1/2.”  However, if we multiply by “1/2” we should multiply by “2/1” to not change the value.  Therefore overall we are multiplying by “2/2” which is 1.  So doing this multiplication we change “3/4” into the equivalent fraction “6/8.”  With “6/8” we add it to “1312/8” and we get 1318/8, which is an equivalent fraction of 659/4 {equivalent fraction means that the two fractions are equal, differing in presentation due to a multiplication by 1}.  A next way to do it, is to use the formula.  If a/b = 1312/8 and c/d = 3/4.  Then the sum (ad + bc)/(bd) has element ad = 1312 * 4 = 5248, bc = 8 *3 = 24 and bd = 8 * 4 = 32, or (ad + bc)/(bd) = 5272/32 = 659/4.  Same result but way harder mathematically.  When working out problems, first try to simplify, then try to find the lowest common denominator, then, if all else fails, use the general formula.  These three methods make adding fractions easier.

Now for homework.

1) Rewrite 100 different numbers as a different fraction and, in addition, a product of a whole number and a fraction (do each in three different ways).

For instance, 2 = 2/1 = 2 * 1/1 = 2 * 2/2 = 2 * 1/2 * 2/1 = 14/7 = 18/9

Why?  You should understand the different ways of rewriting numbers with different denominators, so to easier simplify numbers.  For instance, in the above example, we see that 18/9 simplifies into 2/1 and thus 9/18 simplifies into 1/2.

2) Rewrite 1 in 100 different ways.

For instance, 1 = 2/2 = 3/3 = 9/9 = . . ..  Do one-hundred different ways.

Why?  You should understand that one can be written in an infinite amount of ways, and it’s very useful to re-represent one, especially since each product, for all a[1], 1 = a/a which further equals (a/1 * 1/a) showing the basis of simplification, since (1/a) represents dividing a numerator by “a” and (a/1) represents dividing a denominator by “a” {ADVANCED: Get it?–because “a/1” is really 1/(1/a).  This is because 1/(1/a) means 1 divided by the division of 1 by a.}

3) Find the roots of the first 100 numbers.

For example, 17 is a prime number so its root is 17, but 18 is not a prime number so its root is 3*3*2.

Why?  A fraction like 18/17 can not be simplified because 18 and 17 do not have any common roots.

Hints, every even number has at least one root of 2.  And every number that has digits that add to 9 has at least two roots of 3 (in other words, 9 is a root), and every number that has digits that add to a multiple of 3 has at least one root of 3.  And every number that has a last digit of either 0 or 5 has at least one root of 5.  There’s a fun reason for each.  :)

4)  What are the first 10 prime numbers greater than 100?

For instance, 101, 103 . . ..  Name 8 more.

Why? If you can do this, simplification would be easy.  Because you’ll understand which numbers can and can not be simplified.

5)  Do the problems listed on the other paper.  Show all work.

Upon completion, you should innerstand adding fractions.

[1] Besides zero

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