Confession time: I suck at math. I'm not proud of that, but there it is.
I recognize math's importance. You might even say I find it interesting and even fun. I just have no math instinct. The math gene failed to blossom in my bosom. This may have had something to do with the curse a gypsy put on mom and dad on the night of my conception. However, it probably had more something to do with watching kids who liked math getting regularly beat-up in grades K–12 (thank you, crumbling American public school system).
Yet, if you were to suss out LabKitty's secret identity and go look up the articles I've published (of which there are literally several), you would find pages and pages of equations in them. How can that be?
It's a mystery, it is. An enigma wrapped inside a riddle.
A philosophical turducken, so to speak.
Well, I didn't say I was stupid, I just said I suck at math. Perhaps a more equitable description is that I have no natural talent for math. But instead of giving up, I took to overcoming my handicap through sheer effort. It's like that nice policeman said to me that one time: you just don't know when to quit, do you?
How does one improve one's math skills? Ask any nerd, and s/he will tell you the way to get good is to work lots of problems. So it went. There is on my bookshelf an oppressive collection of Schaum's outlines and REA Problem Solvers which I have methodically worked though over many an evening, weekend, and holiday. My Math Fu indeed gained potency. However, my experience taught me there is another dimension to problem solving beyond just grinding through exercises.
Buried among the hundreds of problems I worked were recurring tricks. A stratagem. A contrivance. An idea that, while not necessarily complicated, often seemed to come out of nowhere. And it was the key to solving the problem. The sort of trick that once revealed -- especially if you had given up and flipped to the solution in frustration -- might make you exclaim: Oh, come on! How would anyone think to do THAT?
In math circles, these sort of tricks are known as crux moves (apparently it's rock climbing jargon). Remembering a crux move may be all that separates you from a weekend spent beating your head against the wall. And remembering a collection of crux moves will make you a fearsome problem-solver indeed. True mathozoids adsorb crux moves by osmosis in the course of problem solving, subconsciously adding them to their problem-solving arsenal. However, you may not have taken note of these tricks, especially while neck-deep in hundreds of practice problems (a phenomenon I call "Schaum's Disease"). What to do?
Hey, you know what would be swell? If someone put together a list of crux moves and posted it on their boss web-site blog.
It is in this spirit of nerd giving that I give you: LabKitty's Big List of Crux Moves. Whatever you call them -- crux moves, tricks, illusions, whatever -- I've collected a bunch that pop up again and again which I will be posting in the coming weeks and months. I'll provide a clever (or not) mnemonic, a brief description, and an example application or two. Once we a get a few up and running, I'll start one of those Blogger listy things on the front page to keep a running tally for easy review.
Some of them you've no doubt seen before. Some of them you may find obvious or trite. Some you may not even realize are a trick. No worries. Remember: the only trick that matters is the one that solves your problem.
TL;DR: Here's a bunch of tricks that come up over and over when doing math. If you worked lots of problems, you'd probably eventually notice them. However, I took the time to notice them for you and made a list. If you commit this list to memory, it may save you hours of frustration in your future mathematical endeavors. Better grades, higher-paying job, quack quack quack.
We begin, not surprisingly, with Crux Move #1.
We may describe CM #1 as: Add and subtract the same quantity (aka "add zero"). Or: divide and multiply by the same quantity (aka "multiply by one").
This trick is probably 99% of mathematics (the other 1% is whiskey). If you ask a mathematician what they do for a living, or what they write on their 1040, they will say "I add and subtract the same quantity." In fact, you probably do this so often you don't even think of it as a trick. But it is.
E X A M P L E 1
Let's dig way back, and derive the Quadratic Formula.
We're given: ax2 + bx + c = 0
4a2x2 + 4abx + 4ac = 0
4a2x2 + 4abx + 4ac + b2 – b2 = 0
4a2x2 + 4abx + b2 = b2 – 4ac
(2ax + b)2 = b2 – 4ac
2ax + b = ± √ b2 – 4ac
Hence x = [ –b ± √ b2 – 4ac ] / 2a
E X A M P L E 2
Integrate: ∫ 1 / (ex + 1) dx
Add and subtract ex to/from the numerator. We have:
∫ [ (1 + ex – ex) / (ex + 1) ] dx
∫ [ (1 + ex) / (ex + 1) – ex / (ex + 1) ] dx
∫ [ 1 – ex / (ex + 1) ] dx
x − ln(ex + 1) + C
E X A M P L E 3
Of course, you typically do complex division by multiplying/dividing the expression by its conjugate:
( 2 + i ) / ( 3 – 4i )
= ( 2 + i ) / ( 3 – 4i ) ] ⋅ [ (3 + 4i ) / ( 3 + 4i ) ]
= 2/25 + 11/25 i
Next Crux Move: Substitute a Taylor Series
I recognize math's importance. You might even say I find it interesting and even fun. I just have no math instinct. The math gene failed to blossom in my bosom. This may have had something to do with the curse a gypsy put on mom and dad on the night of my conception. However, it probably had more something to do with watching kids who liked math getting regularly beat-up in grades K–12 (thank you, crumbling American public school system).
Yet, if you were to suss out LabKitty's secret identity and go look up the articles I've published (of which there are literally several), you would find pages and pages of equations in them. How can that be?
It's a mystery, it is. An enigma wrapped inside a riddle.
A philosophical turducken, so to speak.
Well, I didn't say I was stupid, I just said I suck at math. Perhaps a more equitable description is that I have no natural talent for math. But instead of giving up, I took to overcoming my handicap through sheer effort. It's like that nice policeman said to me that one time: you just don't know when to quit, do you?
How does one improve one's math skills? Ask any nerd, and s/he will tell you the way to get good is to work lots of problems. So it went. There is on my bookshelf an oppressive collection of Schaum's outlines and REA Problem Solvers which I have methodically worked though over many an evening, weekend, and holiday. My Math Fu indeed gained potency. However, my experience taught me there is another dimension to problem solving beyond just grinding through exercises.
Buried among the hundreds of problems I worked were recurring tricks. A stratagem. A contrivance. An idea that, while not necessarily complicated, often seemed to come out of nowhere. And it was the key to solving the problem. The sort of trick that once revealed -- especially if you had given up and flipped to the solution in frustration -- might make you exclaim: Oh, come on! How would anyone think to do THAT?
In math circles, these sort of tricks are known as crux moves (apparently it's rock climbing jargon). Remembering a crux move may be all that separates you from a weekend spent beating your head against the wall. And remembering a collection of crux moves will make you a fearsome problem-solver indeed. True mathozoids adsorb crux moves by osmosis in the course of problem solving, subconsciously adding them to their problem-solving arsenal. However, you may not have taken note of these tricks, especially while neck-deep in hundreds of practice problems (a phenomenon I call "Schaum's Disease"). What to do?
Hey, you know what would be swell? If someone put together a list of crux moves and posted it on their boss web-site blog.
It is in this spirit of nerd giving that I give you: LabKitty's Big List of Crux Moves. Whatever you call them -- crux moves, tricks, illusions, whatever -- I've collected a bunch that pop up again and again which I will be posting in the coming weeks and months. I'll provide a clever (or not) mnemonic, a brief description, and an example application or two. Once we a get a few up and running, I'll start one of those Blogger listy things on the front page to keep a running tally for easy review.
Some of them you've no doubt seen before. Some of them you may find obvious or trite. Some you may not even realize are a trick. No worries. Remember: the only trick that matters is the one that solves your problem.
TL;DR: Here's a bunch of tricks that come up over and over when doing math. If you worked lots of problems, you'd probably eventually notice them. However, I took the time to notice them for you and made a list. If you commit this list to memory, it may save you hours of frustration in your future mathematical endeavors. Better grades, higher-paying job, quack quack quack.
We begin, not surprisingly, with Crux Move #1.
CRUX MOVE #1
The Change of No Change
The Change of No Change
We may describe CM #1 as: Add and subtract the same quantity (aka "add zero"). Or: divide and multiply by the same quantity (aka "multiply by one").
This trick is probably 99% of mathematics (the other 1% is whiskey). If you ask a mathematician what they do for a living, or what they write on their 1040, they will say "I add and subtract the same quantity." In fact, you probably do this so often you don't even think of it as a trick. But it is.
E X A M P L E 1
Let's dig way back, and derive the Quadratic Formula.
We're given: ax2 + bx + c = 0
4a2x2 + 4abx + 4ac = 0
4a2x2 + 4abx + 4ac + b2 – b2 = 0
4a2x2 + 4abx + b2 = b2 – 4ac
(2ax + b)2 = b2 – 4ac
2ax + b = ± √ b2 – 4ac
Hence x = [ –b ± √ b2 – 4ac ] / 2a
E X A M P L E 2
Integrate: ∫ 1 / (ex + 1) dx
Add and subtract ex to/from the numerator. We have:
∫ [ (1 + ex – ex) / (ex + 1) ] dx
∫ [ (1 + ex) / (ex + 1) – ex / (ex + 1) ] dx
∫ [ 1 – ex / (ex + 1) ] dx
x − ln(ex + 1) + C
E X A M P L E 3
Of course, you typically do complex division by multiplying/dividing the expression by its conjugate:
( 2 + i ) / ( 3 – 4i )
= ( 2 + i ) / ( 3 – 4i ) ] ⋅ [ (3 + 4i ) / ( 3 + 4i ) ]
= 2/25 + 11/25 i
Next Crux Move: Substitute a Taylor Series
No comments:
Post a Comment