Oh, gosh. It would seem that in light of my excitement over finally hitting Spring Break, I forgot to write today's post. I said we'd do graphing, right? Man, fuck graphing, listen--I don't want to write about graphing right now because I have video games to play. And while I could segue neatly from graphing to coordinate systems in games, I'm going to talk about something you can use in your daily life instead.
Let me tell you a secret, dear reader. Math isn't about numbers.
"WHA-AAAT?" you cry, as your monocle flies off, landing in your tea with a splash.
It's true. Math isn't about numbers. In fact, most of the mathematicians I know are terrible at basic arithmetic, myself included. Your college algebra professor probably sucked with numbers, no lie.
Math is about ideas. It's about problem solving. The reason your books keep throwing word problems at you is not to trip you up and confuse you--it's because when you come across applications of math in real life, you're not going to be handed an equation. You're going to be thrown a complex situation that you'll need to build your own equation from.
This isn't a post on word problems (though I am working on one). This is a post on an application of math as a problem solving tool in real life. You'll find the techniques are similar.
So what am I talking about when I say using math in real life? You know the common lament of the high school math student: "I'm never going to use any of this in real life!" High school math student, you have no imagination. There are so many ways for you to use math in real life.
The most common? Groceries. If you're shopping on a budget (and if you're a college student, you probably are), knowledge of math is vital for maximizing your dollar when you're grocery shopping.
Your total cost is the sum of all your costs. Figuring those out is pretty easy. Say you're picking up top ramen at ten cents a bag. If one bag is $.10, two bags are $.20, so on--you write that as $.10x, where x is how many bags of top ramen you're buying. The price per item times how many items. So 10 bags of top ramen would be $1.00.
If you're buying other stuff--and you should be, because a person cannot live on top ramen alone--you use the same method to figure out the cost. Price times quantity. Then you add all those products together to get the total cost of your items. If you build the equation first, you can see how many of each you can afford before throwing them in your cart and getting surprised by the cost at the end.
We all know to look for the cheapest version of a product when we're trying to save money, but here's a little-known secret--the lowest price on the shelf is not always the cheapest product. It is usually more cost effective to buy the bigger box than the smaller box.
For example, say you're picking up some frozen waffles because that's all you have time for before rushing off to that one morning class you hate. You go to your frozen foods isle and find that a box of 12 waffles costs $2, a box of 24 costs $3.50, and a big box of 36 is $5. Hopefully you are thinking "whoa wait up here," because it should be obvious that 12*3=36 but $2*3≠$5.
You can easily figure out the cost per waffle. Remember how I said the price of all your top ramen is the price times the quantity? $.10x=y, where y is your total cost. If c is your cost per item, that means your general equation is cx=y. Here, we're starting with the total cost, y=$2, and for a box of 12, our number of items x is 12, so our equation is c=$2/12, and c=~$.17. So, 17 cents per waffle for a box of 12.
A box of 36 is $5, so your total cost is y=$5 and your number of items is x=36. c=$5/36, c=~$.14. For a box of 36 waffles, the cost is 14 cents per waffle, 3 cents cheaper than for a box of 12. Thus, getting the box of 36 might be more expensive, but it is a better value. Your dollar goes farther.
You can do this for everything. Check the net weight of an item and figure out its cost per gram, or FL, or lb. Oftentimes you will find that you're throwing away money by buying the lowest-priced item on the shelf because it's the worst value per measurement.
Okay, so you know how to figure out the best value, and how to calculate your total cost. What about the sales tax, Guindo? How do I figure out that?!
First you need to know what your sales tax is. It's different in different states, obviously, but 8%~9% seems to be the average. If you want to play it safe, use 10% to calculate, and you should always end up with an end price slightly lower than you expected.
If you don't know how to calculate a percentage, it's actually pretty easy. Elementary school arithmetic tricked you into thinking it was hard. I'll do a post on percentages later I guess, but for now know that to get 10% of something, you multiply it by .10. 10% of 100 is 10, 100 times .10 is 10. (.10 is also the same as 1/10, if you find it easier to work with fractions or don't have a calculator handy.)
But finding your sum, finding 10% of it, and then adding those two together is a pain. Don't do that. There is an easier way. Think of it like this: if y is your total cost, and the sales tax is 10% of your total cost, then your final total is y + .10y. The total cost plus the sales tax. This is the same as 1y + .10y. Remember like terms from the variable post? You can add those together, and get 1.10y. So multiplying your total cost by 1.10 will give you the final total of all your items plus sales tax. Pretty neat!
So, in summary, the cost of your groceries is going to be somewhere around 1.10(price*number of items + price*number of items + price*number of items....) = final total.
Math! You do use it in real life!
Graphing next time. Maybe.
Showing posts with label arithmetic. Show all posts
Showing posts with label arithmetic. Show all posts
18 April 2011
04 April 2011
The Root of All Math Puns
Fucking exponents, how do they work???
Actually this post is about exponents AND roots, because - get ready to have your mind blown - they are the same thing.
WHOA. BACK THE FUCK UP, GUINDO. What do you mean ROOTS and EXPONENTS are the same thing?
I proved addition and subtraction were the same fucking thing and you STILL doubt me? Sheeeeesh you people are ridic.
Okay, before we go any farther let me explain what I mean by roots and exponents. Exponents are those neat little superscripts like the 2 in x2. Roots are square roots, cube roots, and so on, that weird looking long division symbol over a number:
.
What an exponent means is "multiply this by itself this many times." 22 means 2*2 and 23 means 2*2*2, and so on. This is also called "power of [number]." Squares are powers of two, cubes are powers of three, etc.
A root means the opposite. What times itself this many times gets you this number?
means "what times itself three times gives you 8?" (Hint: the answer is 2.)
Roots and exponents are inverse functions. That means if you take the square root of something squared, both the root and the exponent cancel out. BUT WHY DOES THAT HAPPEN? It really isn't "because I said so," I promise.
The first thing that needs to be explained is this: a root is really a fractional exponent. That's right,
. Likewise, 
, and so on for whatever number you might put on top of that little spike (what the fuck is that thing even called anyway).
The second thing is this: when you take something with an exponent to an exponent, you multiply both exponents. When you take the square root of both sides of an equation, what you're really doing is taking both sides to a power of 1/2. So what does that mean in practice? Let's check it out.
As you can see, the exponents end up multiplying out to become 1, which means both your root and your exponent can drop out.
You can also use this knowledge to re-write stuff into a doable form. You can't really take something to the 3/2's power without a calculator, but you can take the square root of the cube of something.
First you split up the exponent into ones you know how to do (a cube and a square root), and then you calculate.
Pretty cool! I think. And my opinion is the only one that matters.
>=|
BUT WAIT! THERE'S MORE!
("There's MORE? Auuuugh.")
Silence, hapless reader. There are RULES about exponents. These rules are pretty dang strict and if you break them YOUR ANSWER WILL BE WRONG FOREVER. So let's talk about exponent rules.
You CAN NOT split up an added quantity with an exponent. This is not true and don't do it ever. If you have multiple terms inside a quantity taken to an exponent, you HAVE to either get rid of the exponent algebraically or expand the quantity by multiplying it out using the definition of an exponent (which is a thing you can do and I will cover it eventually).
On the other hand, you CAN do this with multiplication. What this rule is really handy for is reducing roots. If you end up with, say,
, you can reduce it using this rule. Let me show you how:
Expanding on the multiplication rule, it applies to division as well. Which shouldn't surprise you because division is the same as multiplication.
If your bases are the same (and ONLY if your bases are the same, DO NOT TRY THIS IF YOUR BASES ARE DIFFERENT), then you add the exponents together. Shove the following into your calculator to confirm if you're curious:
And the one we've already discussed, if you have exponents to an exponent, then you multiply them.
But gosh, Guindo, you may ask. What about adding stuff with exponents?
Here's an unfortunate truth. You can only add stuff with exponents if that stuff has the same exponent. You cannot, CAN NOT, add x3+x2. Can't do it. On the other hand, x2+x2 CAN be added, and it is 2x2. The exponent doesn't change. You can plug in some values for x and test it out if you don't believe me.
Next time, oh, I don't know, let's talk a little bit about using letters and shit to stand in for numbers. NEXT TIME: VARIABLES!
Actually this post is about exponents AND roots, because - get ready to have your mind blown - they are the same thing.
WHOA. BACK THE FUCK UP, GUINDO. What do you mean ROOTS and EXPONENTS are the same thing?
I proved addition and subtraction were the same fucking thing and you STILL doubt me? Sheeeeesh you people are ridic.
Okay, before we go any farther let me explain what I mean by roots and exponents. Exponents are those neat little superscripts like the 2 in x2. Roots are square roots, cube roots, and so on, that weird looking long division symbol over a number:
.What an exponent means is "multiply this by itself this many times." 22 means 2*2 and 23 means 2*2*2, and so on. This is also called "power of [number]." Squares are powers of two, cubes are powers of three, etc.
A root means the opposite. What times itself this many times gets you this number?
means "what times itself three times gives you 8?" (Hint: the answer is 2.)Roots and exponents are inverse functions. That means if you take the square root of something squared, both the root and the exponent cancel out. BUT WHY DOES THAT HAPPEN? It really isn't "because I said so," I promise.
The first thing that needs to be explained is this: a root is really a fractional exponent. That's right,
. Likewise, , and so on for whatever number you might put on top of that little spike (what the fuck is that thing even called anyway).The second thing is this: when you take something with an exponent to an exponent, you multiply both exponents. When you take the square root of both sides of an equation, what you're really doing is taking both sides to a power of 1/2. So what does that mean in practice? Let's check it out.
As you can see, the exponents end up multiplying out to become 1, which means both your root and your exponent can drop out.
You can also use this knowledge to re-write stuff into a doable form. You can't really take something to the 3/2's power without a calculator, but you can take the square root of the cube of something.
First you split up the exponent into ones you know how to do (a cube and a square root), and then you calculate.
Pretty cool! I think. And my opinion is the only one that matters.
>=|
BUT WAIT! THERE'S MORE!
("There's MORE? Auuuugh.")
Silence, hapless reader. There are RULES about exponents. These rules are pretty dang strict and if you break them YOUR ANSWER WILL BE WRONG FOREVER. So let's talk about exponent rules.
You CAN NOT split up an added quantity with an exponent. This is not true and don't do it ever. If you have multiple terms inside a quantity taken to an exponent, you HAVE to either get rid of the exponent algebraically or expand the quantity by multiplying it out using the definition of an exponent (which is a thing you can do and I will cover it eventually).
On the other hand, you CAN do this with multiplication. What this rule is really handy for is reducing roots. If you end up with, say,
, you can reduce it using this rule. Let me show you how:Expanding on the multiplication rule, it applies to division as well. Which shouldn't surprise you because division is the same as multiplication.
If your bases are the same (and ONLY if your bases are the same, DO NOT TRY THIS IF YOUR BASES ARE DIFFERENT), then you add the exponents together. Shove the following into your calculator to confirm if you're curious:
And the one we've already discussed, if you have exponents to an exponent, then you multiply them.
But gosh, Guindo, you may ask. What about adding stuff with exponents?
Here's an unfortunate truth. You can only add stuff with exponents if that stuff has the same exponent. You cannot, CAN NOT, add x3+x2. Can't do it. On the other hand, x2+x2 CAN be added, and it is 2x2. The exponent doesn't change. You can plug in some values for x and test it out if you don't believe me.
Next time, oh, I don't know, let's talk a little bit about using letters and shit to stand in for numbers. NEXT TIME: VARIABLES!
28 March 2011
Operations to Order
You've probably heard the acronym PEMDAS at some point in your academic career. If you haven't taken a math class since the bare minimum to graduate your last level of schooling, you've probably forgotten what it means because who even fucking uses MATH in REAL LIFE, JEEZ. (Everyone btw, you philistine.)
PEMDAS is supposed to be a mnemonic to help you remember order of operations. (If you don't know what a mnemonic is,get out bookmark dictionary.com or something.) When you see an equation, you're supposed to evaluate it in a certain order. PEMDAS is an acronym for that order:
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Why is there a rule for this? Because if we didn't agree on the rules before we started talking about something, nobody would know what the fuck they were doing. Imagine trying to play monopoly with five different sets of house rules and everyone constantly arguing over shit the other players were doing. Eventually somebody robs the bank, the board gets flipped, and the Waterworks are on fire.
To avoid that, you lay out your agreed-upon house rules BEFORE you start playing. Math's the same way. We lay out our rules, and then we work based on them. So, PEMDAS is our rule for turn-order. Here's how it works.
PARENTHESES - first, evaluate whatever is in parentheses, starting with the innermost ones and moving to the outermost if you have multiple sets of parentheses. An operation inside parentheses is commonly referred to as a quantity. (x+2)2 is read as x plus 2 quantity squared.
EXPONENTS - next, whatever exponents are involved.
MULTIPLICATION/DIVISION - these are basically the same operation, remember? But if you're not going to the trouble of re-writing all your division as fractions (and why aren't you? It's so much easier that way), multiply first and then divide.
ADDITION/SUBTRACTION - also basically the same operation and it doesn't even require much rewriting. This comes last, after everything else has been figured out. See above examples for further examples.
Let's do an example that brings everything together.
DON'T TRIP, YOU GUYS. THIS IS GONNA LOOK REALLY FUCKING COMPLICATED.
What.
What did I tell you last time? Start at the top and just keep working until you get something you know how to solve. Math is all about breaking problems down and working in steps.
So let's start with the parentheses and go down through PEMDAS from there.
Click the image for a step-by-step walkthrough
[NOTE: I wrote (7-3)3 here but calculated (7-3)4. It's been changed in the equations, but the .jpg still says 3 until I get around to changing it.]
Knowing how to apply PEMDAS is vital as a basis for fucking everything else you will ever do in math.
YOU'RE WELCOME.
PEMDAS is supposed to be a mnemonic to help you remember order of operations. (If you don't know what a mnemonic is,
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Why is there a rule for this? Because if we didn't agree on the rules before we started talking about something, nobody would know what the fuck they were doing. Imagine trying to play monopoly with five different sets of house rules and everyone constantly arguing over shit the other players were doing. Eventually somebody robs the bank, the board gets flipped, and the Waterworks are on fire.
To avoid that, you lay out your agreed-upon house rules BEFORE you start playing. Math's the same way. We lay out our rules, and then we work based on them. So, PEMDAS is our rule for turn-order. Here's how it works.
PARENTHESES - first, evaluate whatever is in parentheses, starting with the innermost ones and moving to the outermost if you have multiple sets of parentheses. An operation inside parentheses is commonly referred to as a quantity. (x+2)2 is read as x plus 2 quantity squared.
Example:
EXPONENTS - next, whatever exponents are involved.
Example:
or, if there are no parentheses to be done first:
MULTIPLICATION/DIVISION - these are basically the same operation, remember? But if you're not going to the trouble of re-writing all your division as fractions (and why aren't you? It's so much easier that way), multiply first and then divide.
Example:
ADDITION/SUBTRACTION - also basically the same operation and it doesn't even require much rewriting. This comes last, after everything else has been figured out. See above examples for further examples.
Let's do an example that brings everything together.
DON'T TRIP, YOU GUYS. THIS IS GONNA LOOK REALLY FUCKING COMPLICATED.
What.
What did I tell you last time? Start at the top and just keep working until you get something you know how to solve. Math is all about breaking problems down and working in steps.
So let's start with the parentheses and go down through PEMDAS from there.
Click the image for a step-by-step walkthrough
[NOTE: I wrote (7-3)3 here but calculated (7-3)4. It's been changed in the equations, but the .jpg still says 3 until I get around to changing it.]
Knowing how to apply PEMDAS is vital as a basis for fucking everything else you will ever do in math.
YOU'RE WELCOME.
21 February 2011
I'm Absolutely Positive About This
OH MAN YOU GUYS TODAY WE ARE TALKING ABOUT THE NUMBER LINE!!! I AM SO EXCITED.
What is the number line? It is a visual represesntation of all real numbers arranged in the form of a horizontal line. What is a real number? ONE THAT ISN'T IMAGINARY, OF COURSE.
That wasn't a joke. Imaginary numbers are a thing.
Anyway the number line looks something like this. It's basically a graph with only one axis, or a "one-dimensional graph" if you want to sound like a mathematically knowledgeable douchebag about it. (I will of course be calling it a one-dimensional graph.)
The right side goes all the way to positive infinity, and the left side goes all the way to negative infinity, as indicated by the arrows on either end, and zero sits there in the middle being way too fucking smug about the whole thing.
The cool thing about the number line is that it's a really handy visual for explaining concepts! Remember that post about how subtraction is a lie? The number line serves as a way to show that. I am probably way more excited than one man should ever be about a one-dimensional graph. I just do not understand why teachers glaze over this and show it once and then never bring it up again - IT IS SO USEFUL FOR ILLUSTRATING SO MANY IDEAS.
HOW TO GRAPH POINTS ON A NUMBER LINE: Start at 0, first off. You're starting out with nothing.
Then look at the point you're trying to graph. If it's positive, you move to the right (positive direction), and if it's negative, you move to the left (negative direction.)
This is 5 and -3:
Five steps in the positive direction
Three steps in the negative direction
This totally doesn't sound as cool as advertised, does it? That's just putting points on a line, what is there even to get excited about?
God, quit being such a killjoy. You jerk.
Look here's how addition works on a number line. Take 5+3. Let's plot that on the line. We start at 5 and then move 3 spaces in the positive direction:
Daaaang that was an unnecessarily convoluted way to show something you already know how to do! LET'S DO IT AGAIN! This time we'll subtract, 5 - 3 = 2. Start at 5 and then move 3 spaces in the negative direction:
Okay whatever, what is the SIGNIFICANCE of that? Well, when you add you move in the positive direction, and when you subtract you move in the negative direction. What happens when you add a positive number (right of zero) and a negative number (left of zero)? Let's check it out:
5 + (-3)
(-3) + 5
Oh shit, did I just prove that subtraction is addition of negatives and that it is TOTALLY commutative? I THINK I DID.
(But Guindo nobody was even questioning you about that--SHUT UP)
Now let's talk about something that most people think they understand but they actually don't because it was explained to them in a very simplistic, watered down way: absolute value. "Wait, I know how absolute value works!" you are thinking to yourself. "That makes things positive!"
And you would be WRONG.
If 3 - 5 is really 3 + (-5), and absolute value makes things positive, then wouldn't |3 + (-5)| become |3 + 5|, making |3 - 5| = 8 ? No. No, that is not correct at all. (This is not a wholly made-up example by the way, I have had college math students come to me thinking that this is how it worked because of the shitty "makes things positive" explanation their teachers gave them.)
What absolute value actually means is "distance from zero," which brings us back to the number line! First you perform whatever operation is inside the absolute value bars, in this case we're using 3 - 5:
3 - 5 = -2
Then, to figure out the absolute value of that, count how many spaces away from 0 your answer is:
3 - 5 = -2, which is 2 spaces away from 0, which means |3 - 5| = 2.
And now you understand what absolute values are and what your math teacher is actually asking you for when you see those goofy straight-bars in a problem! Absolute value isn't telling negative signs to take a hike, it's giving a distance.
This post was maybe still not as exciting as advertised so here is the absolute value of my cat's awesomeosity:
NEXT TIME: The behaviour of this blog as time approaches infinity.
* "Why all the purple in this post?" BECAUSE PURPLE IS AWESOME FUCK YOU
What is the number line? It is a visual represesntation of all real numbers arranged in the form of a horizontal line. What is a real number? ONE THAT ISN'T IMAGINARY, OF COURSE.
That wasn't a joke. Imaginary numbers are a thing.
Anyway the number line looks something like this. It's basically a graph with only one axis, or a "one-dimensional graph" if you want to sound like a mathematically knowledgeable douchebag about it. (I will of course be calling it a one-dimensional graph.)
The right side goes all the way to positive infinity, and the left side goes all the way to negative infinity, as indicated by the arrows on either end, and zero sits there in the middle being way too fucking smug about the whole thing.
The cool thing about the number line is that it's a really handy visual for explaining concepts! Remember that post about how subtraction is a lie? The number line serves as a way to show that. I am probably way more excited than one man should ever be about a one-dimensional graph. I just do not understand why teachers glaze over this and show it once and then never bring it up again - IT IS SO USEFUL FOR ILLUSTRATING SO MANY IDEAS.
HOW TO GRAPH POINTS ON A NUMBER LINE: Start at 0, first off. You're starting out with nothing.
Then look at the point you're trying to graph. If it's positive, you move to the right (positive direction), and if it's negative, you move to the left (negative direction.)
This is 5 and -3:
Five steps in the positive direction
Three steps in the negative direction
This totally doesn't sound as cool as advertised, does it? That's just putting points on a line, what is there even to get excited about?
God, quit being such a killjoy. You jerk.
Look here's how addition works on a number line. Take 5+3. Let's plot that on the line. We start at 5 and then move 3 spaces in the positive direction:
Daaaang that was an unnecessarily convoluted way to show something you already know how to do! LET'S DO IT AGAIN! This time we'll subtract, 5 - 3 = 2. Start at 5 and then move 3 spaces in the negative direction:
Okay whatever, what is the SIGNIFICANCE of that? Well, when you add you move in the positive direction, and when you subtract you move in the negative direction. What happens when you add a positive number (right of zero) and a negative number (left of zero)? Let's check it out:
5 + (-3)
(-3) + 5
Oh shit, did I just prove that subtraction is addition of negatives and that it is TOTALLY commutative? I THINK I DID.
(But Guindo nobody was even questioning you about that--SHUT UP)
Now let's talk about something that most people think they understand but they actually don't because it was explained to them in a very simplistic, watered down way: absolute value. "Wait, I know how absolute value works!" you are thinking to yourself. "That makes things positive!"
And you would be WRONG.
If 3 - 5 is really 3 + (-5), and absolute value makes things positive, then wouldn't |3 + (-5)| become |3 + 5|, making |3 - 5| = 8 ? No. No, that is not correct at all. (This is not a wholly made-up example by the way, I have had college math students come to me thinking that this is how it worked because of the shitty "makes things positive" explanation their teachers gave them.)
What absolute value actually means is "distance from zero," which brings us back to the number line! First you perform whatever operation is inside the absolute value bars, in this case we're using 3 - 5:
3 - 5 = -2
Then, to figure out the absolute value of that, count how many spaces away from 0 your answer is:
3 - 5 = -2, which is 2 spaces away from 0, which means |3 - 5| = 2.
And now you understand what absolute values are and what your math teacher is actually asking you for when you see those goofy straight-bars in a problem! Absolute value isn't telling negative signs to take a hike, it's giving a distance.
This post was maybe still not as exciting as advertised so here is the absolute value of my cat's awesomeosity:
NEXT TIME: The behaviour of this blog as time approaches infinity.
* "Why all the purple in this post?" BECAUSE PURPLE IS AWESOME FUCK YOU
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.
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
? Whateverrrr just multiply the first one by 
and the second one by 
, 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
by 
, and 
by 
to get 
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
you only multiply the denominator by 3, you will get 
AND YOUR WHOLE ANSWER WILL BE WRONG because 
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
of a pizza and 
of a pizza, you do not have 
of a pizza.
You need to convert those fractions to the same denominator, which in this case is easy because 2×2 = 4. becomes 
, and when you add them you get 
.
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
. 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:
Thus you see that
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?!
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.
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.
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
? Whateverrrr just multiply the first one by and the second one by , 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
by , and by to get 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
you only multiply the denominator by 3, you will get AND YOUR WHOLE ANSWER WILL BE WRONG because 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
of a pizza and of a pizza, you do not have of a pizza.You need to convert those fractions to the same denominator, which in this case is easy because 2×2 = 4.
becomes , and when you add them you get .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
. 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:
Thus you see that
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?!
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.
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