Last entry we had determined that: 1. PEMDAS as a saying doesn't make a whole lot of sense, 2. The MDAS is ok, but ignores the "I'll write my name with <3 on the back of my hand" relationship between Multiply and Divide (also Add and Subtract, but not Multiply and Subtract), and 3. Sheesh Mathman, you're pretty loquacious.
What we didn't deal with was: What about the PE a.k.a. Parenthesis and Exponents part of the Order of Operations?
First, the Parenthesis are always first. DO IT NOW! is their mantra. Don't ask questions. Just Do It. (Actually just saying Parenthesis are the only bossy symbols is not a correct statement. Brackets also mean the same thing.)
I don't know about you, but I'm not a fan of people constantly bossing me around. The grouping symbols, ( ), [ ], and { }, kind of have their own thing going on. They live in their own little dreamworld and don't really have anyone else to hang out with since they're so full of themselves and don't really care about others- unlike Multiply <3 Divide or Subtract <3 Add. Ahh, special relationships...
So the pressing question is now, what about E? I mean, are Exponents to be left forever by themselves? In PEMDAS they're right after the bossy ones and before the googly eyed pairs. Don't they have any prospects for a long term relationship?
First up, Exponents deserve to be right after the bossy ones. Exponents are a shortcut for repeated multiplication which was at the forefront of the MDAS (but may be second to divide). An Exponent- like 2^5 is a shorter way of writing 2*2*2*2*2, just like 2*5 is a shorter way of writing 2+2+2+2+2 (multiplication is a shortcut for repeated addition). I guess that makes exponents kind of the big brother to multiplication and the even bigger brother to addition. Does that mean exponents are repeated repeated addition?
But what about their prospects for BFFs or eternal love? They're at the top (not counting the brats) of the Order of Operations, but should they have to be alone?
In short, no.
Exponents do have a special relationship going on, but they're so secure in it that they don't really advertise it on the back of their hand or with a sketchy tattoo. They've got roots. No really, they've got roots.
Tune in next week to find out more.
Until then: Stay Radical. (Ooh, That's the perfect tagline in this case.)
Tuesday, September 22, 2009
Monday, September 14, 2009
Would You Like Fries With That Order?
Dear Mathman,
I've been learning about the "Order of Operations" in Math class. I don't like following orders. Do I really need to? Why?
Thanks,
Ivanna Havitall
Dear Ivanna,
Order is a fabulous concept. I mean without orders, how would you get what you want at a restaurant?
Oh wait, I just reread your question. I guess that wasn't the proper answer. Let's try again:
First a bit of history. Actually, a bit of your history. What was the first thing that you learned in math class after you learned how to count. (Please don't say "flush" or "wash hands"). I hope you said "Add". Those were the days... It wasn't too long before the next thing to do showed up on the scene: Subtract. Now you probably didn't like subtract as much as you liked add, because we're all a bit selfish and don't want things taken away. Wasn't it better to say, "You have 3 m and m's (not the rapper) and I give you two more - How many do you have?" rather than "You have 5 m and m's and I take away two." Simple so far, eh?
The next amazing thing you learned to do was multiply. I don't know about you, but when I was back in school I found multiplication to be a fantastic shortcut when I had to repeatedly add a collection of numbers. Example: 3+3+3+3+3+3+3+3+3+3+3+3 = 3 x 12 (the 3+3 was hard to even type -try it!). Just after getting pretty good at multiplication, division came along and ruined the whole party. I mean seriously, subtraction was hard enough - now equal groups?! Come on!
Let's recap: Divide, Multiply, Subtract, Add- those are the operations in reverse order we learned them. If you learned a saying for the "Order of Operations" it probably was PEMDAS (which isn't even a word) or "Please Excuse My Dear Aunt Sally." Personally, I'm not a fan of either. PEMDAS sounds like a contagious disease (Did you hear about Joe? He didn't wash his hands and came down with a nasty case of PEMDAS.) And "Please Excuse My Dear Aunt Sally?" doesn't make any sense to me at all. I don't have an Aunt Sally, and if I did, why would she need continually excused? Is she uber-rude? Does she have a gatrointestinal disorder? I might as well say, "Prickly Elvises Maim Deer And Studebakers" or even "Portly Elephants Make Doilies After September." All nonsense.
Not really, but the saying is incomplete. It's missing something. We don't always do things in the order of Multiply, Divide, Add, and Subtract. If we take the "big 4" and slightly switch the order we get the MDAS of the end of that saying. I personally think (in regards to the saying) we think Multiply before Divide because we prefer to Multiply first. Likewise with Add and Subtract. (It does make sense that the Multiply comes before Add because Multiplication is the "shortcut" for repeated addition.)
In reality, sometimes Divide comes before Multiply and Subtract before Add. You see, Multiply and Divide are in a "special relationship". They are equal and opposite. It's like they're "going out"- but permanently (unlike typical junior high "special relationships" which may not even outlast a sock change). When you encounter Multiply and Divide in a problem remember they <3 each other. Do the operation that comes first when you read the problem normally (left to right).
Add and Subtract have the same thing going on between them. This is getting too mushy so I'll stop there in explaining their situation.
So (hopefully) we've got the MDAS covered. We haven't dealt with the PE part of the saying (Note: PE doesn't stand for gym class). Since this post is getting a bit long, we'll save PE for another time.
Ivanna, hopefully this starts to answer your question a bit.
Stay radical.
I've been learning about the "Order of Operations" in Math class. I don't like following orders. Do I really need to? Why?
Thanks,
Ivanna Havitall
Dear Ivanna,
Order is a fabulous concept. I mean without orders, how would you get what you want at a restaurant?
Oh wait, I just reread your question. I guess that wasn't the proper answer. Let's try again:
First a bit of history. Actually, a bit of your history. What was the first thing that you learned in math class after you learned how to count. (Please don't say "flush" or "wash hands"). I hope you said "Add". Those were the days... It wasn't too long before the next thing to do showed up on the scene: Subtract. Now you probably didn't like subtract as much as you liked add, because we're all a bit selfish and don't want things taken away. Wasn't it better to say, "You have 3 m and m's (not the rapper) and I give you two more - How many do you have?" rather than "You have 5 m and m's and I take away two." Simple so far, eh?
The next amazing thing you learned to do was multiply. I don't know about you, but when I was back in school I found multiplication to be a fantastic shortcut when I had to repeatedly add a collection of numbers. Example: 3+3+3+3+3+3+3+3+3+3+3+3 = 3 x 12 (the 3+3 was hard to even type -try it!). Just after getting pretty good at multiplication, division came along and ruined the whole party. I mean seriously, subtraction was hard enough - now equal groups?! Come on!
Let's recap: Divide, Multiply, Subtract, Add- those are the operations in reverse order we learned them. If you learned a saying for the "Order of Operations" it probably was PEMDAS (which isn't even a word) or "Please Excuse My Dear Aunt Sally." Personally, I'm not a fan of either. PEMDAS sounds like a contagious disease (Did you hear about Joe? He didn't wash his hands and came down with a nasty case of PEMDAS.) And "Please Excuse My Dear Aunt Sally?" doesn't make any sense to me at all. I don't have an Aunt Sally, and if I did, why would she need continually excused? Is she uber-rude? Does she have a gatrointestinal disorder? I might as well say, "Prickly Elvises Maim Deer And Studebakers" or even "Portly Elephants Make Doilies After September." All nonsense.
Not really, but the saying is incomplete. It's missing something. We don't always do things in the order of Multiply, Divide, Add, and Subtract. If we take the "big 4" and slightly switch the order we get the MDAS of the end of that saying. I personally think (in regards to the saying) we think Multiply before Divide because we prefer to Multiply first. Likewise with Add and Subtract. (It does make sense that the Multiply comes before Add because Multiplication is the "shortcut" for repeated addition.)
In reality, sometimes Divide comes before Multiply and Subtract before Add. You see, Multiply and Divide are in a "special relationship". They are equal and opposite. It's like they're "going out"- but permanently (unlike typical junior high "special relationships" which may not even outlast a sock change). When you encounter Multiply and Divide in a problem remember they <3 each other. Do the operation that comes first when you read the problem normally (left to right).
Add and Subtract have the same thing going on between them. This is getting too mushy so I'll stop there in explaining their situation.
So (hopefully) we've got the MDAS covered. We haven't dealt with the PE part of the saying (Note: PE doesn't stand for gym class). Since this post is getting a bit long, we'll save PE for another time.
Ivanna, hopefully this starts to answer your question a bit.
Stay radical.
Friday, September 4, 2009
Have it Your Way?
Sheesh, I take the summer off and the Inbox is full. Time to get a bit caught up. Here's a good one:
Dear Mathman,
I have an iPod shuffle. I really like the thing as it allows me to ignore everyone around me and get exactly what I want when I want it- except for when I use the shuffle mode. Half the time I do it just ruins my day as I get stuck listening to a mix that just Dysons. Seriously, why doesn't this thing just give me what I want?
Sincerely,
Cody H.
Dear Cody,
Bigger isn't necessarily better. Especially when it comes down to doing the iPod shuffle. The more songs you have on that iPod, the greater the chances that the song you want to hear won't be coming up next. It makes sense if you think about it, but did you ever stop to think what the chances are you'll get a mix you want?
Let's think back to a time before the mp3 format. Let's go back to when "Don't Stop Believin'" first came out on 8 track, cassette, or LP. At that time you would have to buy the entire album of 10 or so songs in the order that the band intended it to be. You couldn't change the order that the songs were in unless you did some type of physical work. The "shuffle" option (or "random" in the CD world) didn't exist. You were forced to listen to the album or you had to physically change the media- you couldn't just push a button to access another band's work (unless you bought the K-Tel compilations).
The point of this? There was only one easy choice to hear the album (2 if you count side A and side B).
When CDs popped into the mix, things began to change. Now you could hit the "random" function and you could hear the songs in a bunch of different orders. This was good and bad since it's like reading a book by skipping around from chapter to chapter not necessarily staying in order.
How many orders could you hear all of the songs? Get in the math zone.
If you had a single, it would typically have 3 or 4 songs on it. (So how's that a single?) Let's figure out how many orders you could listen to the songs in with various numbers of songs: (Let's just say the first song starts with "A", the second with "B", the third with "C", ...)
One song : One way
Two songs : Two ways (A then B or B then A)
Three songs : Six ways-
ABC
ACB
BAC
BCA
CAB
CBA
Four songs : 24 ways. Think of it as three songs with a different starting song each time. Ex: DABC DACB DBAC DBCA DCAB DCBA. Note that the D comes first, but there are 6x4 ways since A,B, or C could have been as first as well.
Five songs: 5 x 24 = 120
Six songs: 6 x 120 = 720
Seven songs: 7 x 720 = 5,040
Eight songs: 8 x 5,040 = 40,320
Nine songs: 9 x 40,320 = 362,880
Ten songs: 10 x 362,880 = 3,628,800 ways
Whoa.
That's just for a 10 song CD placed on "random". Imagine an iPod with 500 songs on it. Yikes!
Let's look at the pattern again to figure out what is happening:
1 song : 1 way
2 songs : 2 ways you could pick the first, the second is decided; 2x1=2
3 songs : 3 ways to choose the first, 2 for the second; 3x2x1=6
4 songs : 4 ways for the first then hit me baby one more time with the results from the 3 song calculations; 4x3x2x1
5 songs : 5x4x3x2x1
6 songs : 6x5x4x3x2x1
and on and on.
There is a term for this, it's called the factorial. It looks like this: ! It means, HOLY COW THAT'S A WHOLE LOT!
so 10 songs is 10! = 10x9x8x7x6x5x4x3x2x1.
With over 3 million ways to listen to just 10 songs, no wonder why you've got a good chance for a bad mix!
stay radical.
Dear Mathman,
I have an iPod shuffle. I really like the thing as it allows me to ignore everyone around me and get exactly what I want when I want it- except for when I use the shuffle mode. Half the time I do it just ruins my day as I get stuck listening to a mix that just Dysons. Seriously, why doesn't this thing just give me what I want?
Sincerely,
Cody H.
Dear Cody,
Bigger isn't necessarily better. Especially when it comes down to doing the iPod shuffle. The more songs you have on that iPod, the greater the chances that the song you want to hear won't be coming up next. It makes sense if you think about it, but did you ever stop to think what the chances are you'll get a mix you want?
Let's think back to a time before the mp3 format. Let's go back to when "Don't Stop Believin'" first came out on 8 track, cassette, or LP. At that time you would have to buy the entire album of 10 or so songs in the order that the band intended it to be. You couldn't change the order that the songs were in unless you did some type of physical work. The "shuffle" option (or "random" in the CD world) didn't exist. You were forced to listen to the album or you had to physically change the media- you couldn't just push a button to access another band's work (unless you bought the K-Tel compilations).
The point of this? There was only one easy choice to hear the album (2 if you count side A and side B).
When CDs popped into the mix, things began to change. Now you could hit the "random" function and you could hear the songs in a bunch of different orders. This was good and bad since it's like reading a book by skipping around from chapter to chapter not necessarily staying in order.
How many orders could you hear all of the songs? Get in the math zone.
If you had a single, it would typically have 3 or 4 songs on it. (So how's that a single?) Let's figure out how many orders you could listen to the songs in with various numbers of songs: (Let's just say the first song starts with "A", the second with "B", the third with "C", ...)
One song : One way
Two songs : Two ways (A then B or B then A)
Three songs : Six ways-
ABC
ACB
BAC
BCA
CAB
CBA
Four songs : 24 ways. Think of it as three songs with a different starting song each time. Ex: DABC DACB DBAC DBCA DCAB DCBA. Note that the D comes first, but there are 6x4 ways since A,B, or C could have been as first as well.
Five songs: 5 x 24 = 120
Six songs: 6 x 120 = 720
Seven songs: 7 x 720 = 5,040
Eight songs: 8 x 5,040 = 40,320
Nine songs: 9 x 40,320 = 362,880
Ten songs: 10 x 362,880 = 3,628,800 ways
Whoa.
That's just for a 10 song CD placed on "random". Imagine an iPod with 500 songs on it. Yikes!
Let's look at the pattern again to figure out what is happening:
1 song : 1 way
2 songs : 2 ways you could pick the first, the second is decided; 2x1=2
3 songs : 3 ways to choose the first, 2 for the second; 3x2x1=6
4 songs : 4 ways for the first then hit me baby one more time with the results from the 3 song calculations; 4x3x2x1
5 songs : 5x4x3x2x1
6 songs : 6x5x4x3x2x1
and on and on.
There is a term for this, it's called the factorial. It looks like this: ! It means, HOLY COW THAT'S A WHOLE LOT!
so 10 songs is 10! = 10x9x8x7x6x5x4x3x2x1.
With over 3 million ways to listen to just 10 songs, no wonder why you've got a good chance for a bad mix!
stay radical.
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