Partially Derivative: fractions

07 February 2011

I'm Not Making Another Fraction Pun

Fractions.

Again.

This time we're going to talk about operations on fractions. Let's start easy: multiplication. Multiplying fractions is the easiest, god damn. You multiply straight across, numerator to numerator and denominator to denominator.

$\frac{3}{4}\ast \frac{5}{2}=\frac{3*5}{4*2}=\frac{15}{8}$

THIS IS THE SIMPLEST THING YOU WILL EVER DO WITH FRACTIONS.

It's all downhill from there.

Sigh.

Look guys I don't want to keep talking about fractions. Lowest common denominators are such a pain in the ass, do you even know? Here is how we do it in calculus: "Oh, you have to add $\tfrac{7}{4}+\tfrac{29}{6}$ ? Whateverrrr just multiply the first one by $\tfrac{6}{6}$ and the second one by $\tfrac{4}{4}$ , it doesn't matter."

But for some reason, people in lower division math expect you to add them by doing more work by finding a lowest common denominator when that is really completely unnecessary.

You can tell your math teachers I said that. It means something because I am a math major on the internet.

So according to wikipedia (I stole their definition because I hate LCDs that much), "the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. It is the smallest positive integer that is a multiple of the denominators." Vulgar fractions??? WIKI STOP CONFUSING ME. Also integers are whole numbers - hey that definition was simple!

What the fuck does that actually MEAN though? It means that if your denominators are 4 and 6 as in the above example, you should be finding the lowest number that fits in both 4 and 6's multiplication tables. 4×2 is 8, you can't get that out of 6, 4×3 is 12, and - hey! 6×2 is also 12! Suddenly you have a lowest common denominator: 12. So you multiply $\tfrac{7}{4}$ by $\tfrac{3}{3}$ , and $\tfrac{29}{6}$ by $\tfrac{2}{2}$ to get $\tfrac{21}{12}+\tfrac{58}{12}= \tfrac{79}{12}$

This is really important and I cannot stress this enough: YOU NEED TO MULTIPLY THE NUMERATOR TOO. This ONLY works because any number over itself is equal to 1, and multiplying anything by 1 does not change your answer. If you forget and instead of multiplying $\tfrac{7}{4}\ast \tfrac{3}{3}= \tfrac{21}{12}$ you only multiply the denominator by 3, you will get $\tfrac{7}{12}$ AND YOUR WHOLE ANSWER WILL BE WRONG because $\tfrac{7}{12}\neq \tfrac{21}{12}$

Are we clear? We're never going to change denominators without remembering the numerators too? Okay. Good.

Unlike multiplication, which goes straight across top to top and bottom to bottom, addition is different. You don't add denominators. That's why they need to be the same number in the first place. If you add $\tfrac{1}{2}$ of a pizza and $\tfrac{1}{4}$ of a pizza, you do not have $\tfrac{2}{6}$ of a pizza.

You need to convert those fractions to the same denominator, which in this case is easy because 2×2 = 4. $\tfrac{1}{2}$ becomes $\tfrac{2}{4}$ , and when you add them you get $\tfrac{2}{4}+\tfrac{1}{4}=\tfrac{2+1}{4}=\tfrac{3}{4}$ .

I'm going to cover one more thing here and that is inverses, which I mentioned briefly in the post about subtraction and division being LIES. Every integer (whole number, remember?) can be written in the form of a fraction, as I also mentioned there. 3 can also be written as $\tfrac{3}{1}$ . Keep that in mind, because it is important for explaining division and inverses.

An inverse is something that multiplies with its original to equal one. For example:

$\frac{3}{1}\ast \frac{1}{3}=\frac{3\ast 1}{1\ast 3}=\frac{3}{3}=1$

Thus you see that $\tfrac{1}{3}$ is the inverse of 3. Basically, with fractions, you flip the numerator and denominator and TA-DA you have the inverse of the original fraction.

Remember how I said that division was really multiplying by the inverse? Guess how you divide fractions!

WHOA HOLD UP, you mean you have to DIVIDE by FRACTIONS sometimes? Yes, dear blog-reader, you do. It happens with alarming frequency in the middle of integrals, as a matter of fact! So how do you do it? You multiply by the inverse you ignorant oaf! God, have you even been paying attention?!

$\frac{5}{6}\div \frac{3}{4}=\frac{5}{6}\ast \frac{4}{3}$

And now it's a normal multiplication which, as stated in the beginning of this post, is the simplest thing you will ever do with fractions.

NEXT TIME: THE NUMBER LINE. It is way more interesting than fractions, you guys.

* Do I even really need to say this anymore? codecogs.com. In fact I'm just going to put this in the blog info and stop amending it to posts.
** If this post was insufficient (which, let's be honest, it probably was), please direct yourself here, where you will find a far more comprehensive explanation of how to find LCDs and shit.
*** Inverses are also called reciprocals when applied to fractions but I forgot to mention it because I hate fractions too much to remember that shit.

31 January 2011

A Fraction of What Lies Ahead (ha ha i'm hilarious)

Everyone hates fractions, they are so confusing, and what even does a lowest common denominator mean? Well sit yourself down and pay attention, because I am going to explain the shit out of fractions.

The first thing you need to keep in mind for fractions is that fractions are division rewritten. $\tfrac{2}{3}$ is $2\div 3$ . That's important when you need to reduce fractions, especially for mixed fractions.

(Sidenote: you scrubs in lower division math got it rough - nobody in higher division even cares about mixed fractions. I'm going to take a moment to point and laugh at you.)

What is a mixed fraction? That's something like $3\tfrac{1}{2}$ , which means three and one half. This is an "improper fraction" rewritten as a "proper fraction."

These words are in quotes because the idea is ridiculous and improper fractions are way easier to work with than proper fractions and as a math major I disagree with the classification NOBODY CARES MOVING ON

Improper fractions are fractions where the top (the numerator) is a bigger number than the bottom (the denominator). WHOA WHAT THE FUCK HOLD UP. Denominator? Numerator?

Look guys you gotta know these things, we can't do a vocab lesson every time. (Sigh, but we're going to.) The problem with these names is that they're really fucking easy to confuse with each other, so it helps to have a way to remember what the hell these words actually mean.

If the fraction bar is the surface of the ocean, the numerator is the number of dudes in a boat who are about to get dominated by the greatest great white that ever lived, the Denominator. FRACTIONS ARE JAWS.

Okay that was stupid let's move on. Improper fractions: top bigger than bottom. Proper fractions are the opposite: the denominator is bigger than the numerator. $\tfrac{7}{2}$ is an improper fraction and $\tfrac{1}{2}$ is a proper fraction. Mixed fractions are also "proper," because they are the result of rewriting an improper fraction into a proper one.

Proper no longer means anything as a word. I hate this post.

Here is a thing which is true: $\tfrac{7}{2}= 3\tfrac{1}{2}$

Why is this true? It is true because fractions are division, and to explain it I am going to have to bring up the good old standby of long division. DO YOU KNOW HOW TO LONG DIVIDE? BECAUSE SO HELP ME IF I HAVE TO EXPLAIN THAT TOO...

Fine. I can't even find a way to include a long division symbol so I'm just going to write it as a close-parenthesis I HOPE YOU'RE HAPPY.

You're dividing by the number on bottom, so $\tfrac{7}{2}$ is 2 )7 (DO YOU SEE THAT, ME BEING CREATIVE RIGHT THERE?)

2 goes into 7 three whole times - that means you can subtract 2 from the original number three times before the number being subtracted from is bigger than 2.

Okay, so what the hell do we do with that 1 that's left over? That's smaller than 2, so you can't just divide into it, but remember what division really is? ~*Fractions*~! You can write the remainder as $\tfrac{1}{2}$ , which means you can write the whole thing as $3\tfrac{1}{2}$ ! You have now turned an improper fraction into a mixed proper fraction! DON'T YOU FEEL SPECIAL.

The thing about mixed fractions, though, is that they are a pain in the ass to work with inside a problem. This is why people don't use them past Algebra II (you scrubs). So how do you turn a mixed fraction back into an improper fraction so you can actually work with it? You pretty just do what we did backwards. It's not that hard.

$3\tfrac{1}{2}$ is really $3+\tfrac{1}{2}$ , but you can't add fractions unless they have the same number in the denominator. So you need to get a 2 underneath that 3 somehow.

Hey guess what's really cool about the number 1? You can multiply it in anywhere and not change a damn thing about the problem's answer! Know what's even cooler? You can write 1 as anything over itself. And remember last post where I said that you can rewrite 3 as $\tfrac{3}{1}$ ? That's important now!

$3\tfrac{1}{2}$

$=3+\tfrac{1}{2}$

$=\frac{3}{1}+\frac{1}{2}$

$=(\frac{3}{1})(\frac{2}{2})+\frac{1}{2}$ (multiply by 1 in the form of $\tfrac{2}{2}$ to get the same denominator)

$=\frac{6}{2}+\frac{1}{2}$ (now you can add the numerators because the denominators are the same)

$=\frac{6+1}{2}$

$=\frac{7}{2}$

OH MAN WASN'T THAT AWESOME? We just worked backwards to get our original improper fraction!

Yeah we did a whole bunch of work to get back to exactly where we started. That's not important. What's important is that this applies to any mixed fraction you will ever meet, and also any case where you need to add a whole number to a fraction. This is how you write whole numbers into fractions with a specific denominator.

But I'm fucking sick of fractions, so we'll talk about lowest common denominators and shit next time.

* Again, codecogs.com is great for providing the equations EXCEPT THE LONG DIVISION ONE THAT I HAD TO MAKE MYSELF I HATE EVERYTHING