Oh Ben ….I LOVE your blog …it makes me laugh out loud …just got up and saw this ….it tally’s with my own last blog post ….if you’d care to take a look maybe you would like to comment as I could do with some clarification to ensure I’m on the right track ….they don’t come any ‘simple”r than me :D:D:D
I really hated those kinds of questions. I usually managed to find an answer the teacher would find acceptable, by using who-knows-what basic human intuition, but it is still maddeningly vague.
I know how very human and ambiguous these kinds of questions are because I once wrote a (small, elementary) computer algebra system in which I had a function called “simplify.” I tried my best and I still don’t think it was bad, but it wasn’t anywhere good enough for actual use by someone trying to actually do algebra.
This design is spectacular! You definitely know how to keep a reader amused. Between your wit and your videos, I was almost moved to start my own blog (well, almost…HaHa!) Fantastic job. I really loved what you had to say, and more than that, how you presented it. Too cool!
I think the first two examples especially illustrate this. 😉 (Although most careful sources I’ve seen explicitly state “factorize” and “distribute” rather than saying “simplify”.)
In 37 years, in all my all my time as a grade school student, high school student, undergraduate student, graduate student, and college professor, I have never seen a single textbook or instructor use the word “simplify” to mean “factor”. I think you’ve missed this one, Ben.
Agreed. Simplify means to perform all indicated operations that can be performed. If simplify meant “make it look better”, the instruction would be entirely subjective.
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I think that, at its core, “simplify” means “write in the clearest way possible”, which CAN mean factoring in many cases. For that matter, 13/52 might be more “simplified” than 1/4 if the point is to look at probabilities and poker decks. We too often insist that “simplify” means to carry out a specific directionality of steps, even if that means that the result is less clearly useful for whatever practical purpose we’re doing the math for.
But that is not how it is traditionally used. 13/52 is not simplified, but 1/4 is. Nobody has ever claimed that to simplify means to make it more useful. Depending on the class, I don’t make students reduce fractions or rationalize denominators or distribute factors, but I never claim that these other answers are “simplified”. Math terms do not always agree with lay connotations.
Perhaps the moral is that we should avoud “simplify” if there is a clearer word. E.g., “reduce” is clear when it comes to fractions, so use it instead of “simplify”.
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Removing redundancy, as in 5x*(3x+1)/(3x+1) = 5x, should always count as simplification. Factoring and distributing are more mechanical changes and may not always have the same utility. Is n*(n-1) better than n^2-n? Depends on context. But changing 5 to 5+2-1-1 is silly unless you’re setting up to do something tricky in the next step.
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reply to nuezjr on January 15, 2016 at 11:27 am said:
“Removing redundancy, as in 5x*(3x+1)/(3x+1) = 5x …”
careful, x cannot be -1/3 on the left side. do not lose this information by simplifying.
e.g. 5x*(3x+1)/(3x+1) = 5x and x unequal -1/3
I always used to use “factor,” but “factorize” is more common in the UK (well, really “factorise”) where I work now, so these days I slip between them.
I like that. Sometimes normalizing makes things bigger, for example conjunctive normal form in logic (or electronics); there’s definitely a distinction to be made between simplifying and optimizing but I’m only sure I understand what one of those words means.
1/(a + bi) is generally considered less simple than (a – bi)/(a^2 + b^2), because of the common “best practices” rule that denominators ought to be real.
If you’re counting characters as symbols: 1/8 uses three symbols; 0.125 uses five. But I think there are many contexts where 0.125 is clearer than 1/8.
Those are nice examples – clearly it’s simpler to have a complex number in a form where you can identify real and imaginary parts than it is as a reciprocal (even at the price of concision).
In Boolean algebra, where the distributive laws run both ways, there are two normal forms, conjunctive normal form (express everything as products of sums) and disjunctive normal form (express everything as sums of products). Each has its advantages.
I see you have adopted the UK method of writing “x”, from the little gaps between your arcs in some of the above examples. Do you find it quicker to use?
There’s a weird causal chain of notation here: because Brits often write the decimal point in the middle of the line, they’re less inclined to use the dot for multiplication; thus, they use an X; thus, they need to distinguish their variable x by writing it swooshy.
(And if you give a mouse a cookie…)
Anyway, I’ve had to relearn to write my x’s, and now it’s habit.
For awhile now, I’ve thought that we should just use the circled x (tensor product symbol) for multiplication of numbers. It’s consistent, so it wouldn’t hurt tensor products to adopt this convention.
That sounds plausible but is actually false: Brits learn to write the letter “x” as two arcs long before they have to worry about distinguishing it from the × symbol. They are just repurposing a distinction that already exists.
Similarly, it looks plausible that editor was formed by adding the suffix -or to the pre-existing word edit. But actually editor came first by about 150 years, and edit was created by an analogy like actor:act::editor:X. (Linguists call this back formation.)
Do you know what? If I keep looking at your blog, I may even learn to like Math. And the “bad” drawings make it look so human, in these days when Clip art hides one’s personality, it’s refreshing to see stick figures. Long may you thrive.
J
Reblogged this on Mean Green Math and commented:
A very thoughtful post on the notion of simplification. There are several excellent discussions in the comments section as well.
“Simplify” is a convention that encompasses lots of little rules. There is no simple definition. It means to write polynomials, fractions, radicals, complex numbers, etc in certain prescribed forms. There really isn’t a unifying principle, no matter how much some people think there ought to be. And even those of us who accept this loose conventional definition of “simplify” do not always prefer it or enforce it or find it “simpler”.
Notation and terminology are frequently inconsistent, so I would not bother hoping for consistency in “simplify”. Just like 3^(-1)=1/3 but sin^(-1) x is not csc x. Just like sin^2 x = (sin x)^2 but for a general function f(x), f^2 instead means f(f(x)). I wish I could fix it all, but we’re stuck with a mess. And that includes “simplify”.
Interesting – I think we have similar ways of thinking about the underlying mathematics, but different ways of using this word.
To me, “simplify” is an eminently avoidable word (you can always say “distribute,” “rationalize,” “express in standard form,” etc). So why bother using this sort of ambiguous, preachy term? Why not avoid it entirely? Because, I think, it expresses mathematicians’ desire for clarity and concision in their use of symbols.
As you say, the symbolic manoeuvres it refers to are diverse and local, rather than uniform and global. But I still think there’s a unifying theme.
(Also, FWIW, most textbooks I’ve seen avoid using “simplify” to mean “distribute” OR “factor,” since it’s so ambiguous which is really simpler. But I’ve heard lots of humans use it in the course of speech to mean “distribute,” “factor,” and other things besides.)
(But – I should add – it’s interesting that you’ve always heard it used to mean distribute and never factor. I think the usage I’m thinking of, where it might mean factor, is after, say, computing a derivative, when you need to clean up the mess. This cleaning is a process I’ve always heard called “simplifying,” but which may involve factoring rather than distributing.)
Every algebra and precalculus textbook I’ve seen has used “simplify” to mean “fully distribute” when giving instructions regarding polynomials. It’s fairly ubiquitous in print.
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I rarely see “simplify” in written instructions for courses in Calculus and above, but when I do, it’s for polynomial distribution and computational introductions to exponentials and logarithms.
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By the time you get to a Calculus course you’re expected to know how to clean up after yourself. ; )
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Mmm, that makes sense.
I suspect that more specific technical words would be better in these settings, too (I like “condense” and “expand” for logarithms), since what’s “simpler” is a matter of context/taste/judgment, but reflecting on what I’ve seen in textbooks, you’re right that this is where the word “simplify” comes up.
This is a good point – Paul Hartzer brings up the same thing below.
This is probably a little blasphemous, but I think there’s often a tradeoff between clarity and precision in the way we present mathematics, and often I’m willing to sacrifice a bit of precision.
For example, it used to bug me that trig textbooks present 1/tanx = cotx without discussing that the left-hand-side is undefined at odd multiples of pi/2, while the right-hand-side equals 0 at those values.
But as long as we’re careful about ironing out those technicalities when they become relevant, I’m not too worried about ignoring them when they’re not.
I’ll double down your heresy: Since 0/0 is technically speaking indeterminate not undefined, it can equal any number just as much as it can equal any other number. So in x^2/x, 0/0 can equal 0; in (x + 1)/(x + 1), it can equal 1.
Singularities that create discontinuities are one thing, but I don’t see why it’s that important (particularly at the high school level) to dwell on singularities that don’t create discontinuities.
(Despite what I said below. I like arguing with myself.)
True, but then it’s no longer true that simplification doesn’t result in a different equation, since (as I said) the domain is different. Unless, of course, the domain of the original function was already stated to exclude the singularity.
The EQUIVALENT equation would be “f(x) is indeterminate when x = -1/3 and 5x elsewhere”, which is more complicated than “f(x) = 5x*(3x+1)/(3x+1)”.
All I can say is wow. Your demonstration of passion on this topic is unsurpassed. This is some of the best informative content I have ever seen online. Thank you.
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How about ‘what the = sign means’? The two sides are not identical, they don’t appear the same. In what sense can they be the same?
That’s a good question! They’re “equal” in the same sense that 3 + 4 and 5 + 2 are equal. The symbols don’t look the same. But the symbols are just stand-ins for quantities, and those quantities are equal.
Oh Ben ….I LOVE your blog …it makes me laugh out loud …just got up and saw this ….it tally’s with my own last blog post ….if you’d care to take a look maybe you would like to comment as I could do with some clarification to ensure I’m on the right track ….they don’t come any ‘simple”r than me :D:D:D
So…for polynomials, which is the simplest form? Expanded, or factorised? (but, didn’t we just..) 😛
Depends! Do you want a sum, or a product?
If I’m differentiating, I want a sum.
If I’m looking for roots, I want a product!
I see!
Though, some question just asks “Simplify *put an expression here*” XD
Most of the times it seems such questions are asking for a sum form…
I really hated those kinds of questions. I usually managed to find an answer the teacher would find acceptable, by using who-knows-what basic human intuition, but it is still maddeningly vague.
I know how very human and ambiguous these kinds of questions are because I once wrote a (small, elementary) computer algebra system in which I had a function called “simplify.” I tried my best and I still don’t think it was bad, but it wasn’t anywhere good enough for actual use by someone trying to actually do algebra.
Great work . I hope many kids learn maths.
Very nice.
This design is spectacular! You definitely know how to keep a reader amused. Between your wit and your videos, I was almost moved to start my own blog (well, almost…HaHa!) Fantastic job. I really loved what you had to say, and more than that, how you presented it. Too cool!
maths is supposed to make sense, and you always demonstrate that. thank you.
Lovely! The idea that algebra is “the language of math” is one of the dumbest ideas around.
Beautiful.
Too often, “simplify” just means “Do what we’ve been doing this week,” but I DO like the paraphrase notion.
I think the first two examples especially illustrate this. 😉 (Although most careful sources I’ve seen explicitly state “factorize” and “distribute” rather than saying “simplify”.)
“Simplify” means carry out the operations. “Factoring” is the OPPOSITE of simplifying.
Otherwise, good stuff!
I dunno, I think factoring often simplifies, e.g.,:
(3x^2 + 4)(5x + z) + (3x^2 + 4)(6x^3 + y)
Factoring out the (3x^2 + 4) seems like the right way to simplify! A lot better than distributing everything.
In 37 years, in all my all my time as a grade school student, high school student, undergraduate student, graduate student, and college professor, I have never seen a single textbook or instructor use the word “simplify” to mean “factor”. I think you’ve missed this one, Ben.
Agreed. Simplify means to perform all indicated operations that can be performed. If simplify meant “make it look better”, the instruction would be entirely subjective.
I think that, at its core, “simplify” means “write in the clearest way possible”, which CAN mean factoring in many cases. For that matter, 13/52 might be more “simplified” than 1/4 if the point is to look at probabilities and poker decks. We too often insist that “simplify” means to carry out a specific directionality of steps, even if that means that the result is less clearly useful for whatever practical purpose we’re doing the math for.
But that is not how it is traditionally used. 13/52 is not simplified, but 1/4 is. Nobody has ever claimed that to simplify means to make it more useful. Depending on the class, I don’t make students reduce fractions or rationalize denominators or distribute factors, but I never claim that these other answers are “simplified”. Math terms do not always agree with lay connotations.
Perhaps the moral is that we should avoud “simplify” if there is a clearer word. E.g., “reduce” is clear when it comes to fractions, so use it instead of “simplify”.
Removing redundancy, as in 5x*(3x+1)/(3x+1) = 5x, should always count as simplification. Factoring and distributing are more mechanical changes and may not always have the same utility. Is n*(n-1) better than n^2-n? Depends on context. But changing 5 to 5+2-1-1 is silly unless you’re setting up to do something tricky in the next step.
reply to nuezjr on January 15, 2016 at 11:27 am said:
“Removing redundancy, as in 5x*(3x+1)/(3x+1) = 5x …”
careful, x cannot be -1/3 on the left side. do not lose this information by simplifying.
e.g. 5x*(3x+1)/(3x+1) = 5x and x unequal -1/3
“to Factorize” or “to Factor”?
I prefer factor.
I always used to use “factor,” but “factorize” is more common in the UK (well, really “factorise”) where I work now, so these days I slip between them.
‘Simplify’ usually means ‘express the same value using fewer symbols’. Does anyone have an example of improving an expression by making it bigger?
That’s a really good question.
Hmm, does expand partial fraction count? i.e. expressing 9/7 into 1 and 2/7, but in algebraic terms. That helps in differentiating.
or is it integrating ←_←
I like that. Sometimes normalizing makes things bigger, for example conjunctive normal form in logic (or electronics); there’s definitely a distinction to be made between simplifying and optimizing but I’m only sure I understand what one of those words means.
1/(a + bi) is generally considered less simple than (a – bi)/(a^2 + b^2), because of the common “best practices” rule that denominators ought to be real.
If you’re counting characters as symbols: 1/8 uses three symbols; 0.125 uses five. But I think there are many contexts where 0.125 is clearer than 1/8.
Those are nice examples – clearly it’s simpler to have a complex number in a form where you can identify real and imaginary parts than it is as a reciprocal (even at the price of concision).
You remind me: Technically, the “simplest” form of 1/(a + bi) is a/(a^2 + b^2) – b/(a^2 + b^2) i, which is many more symbols.
Unless you are going to be multiplying by a+bi at some later stage of the process! Simplification is so very context sensitive.
Also, “simplifying” radicals. E.g., sqrt(8) = 2 sqrt(2).
In Boolean algebra, where the distributive laws run both ways, there are two normal forms, conjunctive normal form (express everything as products of sums) and disjunctive normal form (express everything as sums of products). Each has its advantages.
I see you have adopted the UK method of writing “x”, from the little gaps between your arcs in some of the above examples. Do you find it quicker to use?
Now I’m going to have to check Xs all the time to see whether they’re drawn as crossed lines or opposed parentheses.
You’ve spotted the infection!
There’s a weird causal chain of notation here: because Brits often write the decimal point in the middle of the line, they’re less inclined to use the dot for multiplication; thus, they use an X; thus, they need to distinguish their variable x by writing it swooshy.
(And if you give a mouse a cookie…)
Anyway, I’ve had to relearn to write my x’s, and now it’s habit.
For awhile now, I’ve thought that we should just use the circled x (tensor product symbol) for multiplication of numbers. It’s consistent, so it wouldn’t hurt tensor products to adopt this convention.
a while, not awhile. sorry
That sounds plausible but is actually false: Brits learn to write the letter “x” as two arcs long before they have to worry about distinguishing it from the × symbol. They are just repurposing a distinction that already exists.
Similarly, it looks plausible that editor was formed by adding the suffix -or to the pre-existing word edit. But actually editor came first by about 150 years, and edit was created by an analogy like actor:act::editor:X. (Linguists call this back formation.)
Ah, makes sense! I’ve heard that linguistic process described before but had forgotten the name – which gives me a new favorite Wikipedia page…
https://en.wikipedia.org/wiki/List_of_English_back-formations
Of course, back formation applied to “back formation” gives us the verb “to back form” or even “to back formate”. Got it? Now you’re ready for The Final, Complete, Authoritative List of Self-Describing Linguistic Expressions.
Do you know what? If I keep looking at your blog, I may even learn to like Math. And the “bad” drawings make it look so human, in these days when Clip art hides one’s personality, it’s refreshing to see stick figures. Long may you thrive.
J
Thanks for reading! 🙂
Reblogged this on Mean Green Math and commented:
A very thoughtful post on the notion of simplification. There are several excellent discussions in the comments section as well.
This is the most profound description of algebra I have ever read. I will sharing it with my students.
Thank you for sharing.
Thanks for reading!
“Simplify” is a convention that encompasses lots of little rules. There is no simple definition. It means to write polynomials, fractions, radicals, complex numbers, etc in certain prescribed forms. There really isn’t a unifying principle, no matter how much some people think there ought to be. And even those of us who accept this loose conventional definition of “simplify” do not always prefer it or enforce it or find it “simpler”.
Notation and terminology are frequently inconsistent, so I would not bother hoping for consistency in “simplify”. Just like 3^(-1)=1/3 but sin^(-1) x is not csc x. Just like sin^2 x = (sin x)^2 but for a general function f(x), f^2 instead means f(f(x)). I wish I could fix it all, but we’re stuck with a mess. And that includes “simplify”.
Interesting – I think we have similar ways of thinking about the underlying mathematics, but different ways of using this word.
To me, “simplify” is an eminently avoidable word (you can always say “distribute,” “rationalize,” “express in standard form,” etc). So why bother using this sort of ambiguous, preachy term? Why not avoid it entirely? Because, I think, it expresses mathematicians’ desire for clarity and concision in their use of symbols.
As you say, the symbolic manoeuvres it refers to are diverse and local, rather than uniform and global. But I still think there’s a unifying theme.
(Also, FWIW, most textbooks I’ve seen avoid using “simplify” to mean “distribute” OR “factor,” since it’s so ambiguous which is really simpler. But I’ve heard lots of humans use it in the course of speech to mean “distribute,” “factor,” and other things besides.)
(But – I should add – it’s interesting that you’ve always heard it used to mean distribute and never factor. I think the usage I’m thinking of, where it might mean factor, is after, say, computing a derivative, when you need to clean up the mess. This cleaning is a process I’ve always heard called “simplifying,” but which may involve factoring rather than distributing.)
Every algebra and precalculus textbook I’ve seen has used “simplify” to mean “fully distribute” when giving instructions regarding polynomials. It’s fairly ubiquitous in print.
I rarely see “simplify” in written instructions for courses in Calculus and above, but when I do, it’s for polynomial distribution and computational introductions to exponentials and logarithms.
By the time you get to a Calculus course you’re expected to know how to clean up after yourself. ; )
Mmm, that makes sense.
I suspect that more specific technical words would be better in these settings, too (I like “condense” and “expand” for logarithms), since what’s “simpler” is a matter of context/taste/judgment, but reflecting on what I’ve seen in textbooks, you’re right that this is where the word “simplify” comes up.
Simple!
some of your examples to simplify lose information!
e.g. (x²/x) is the same as (x with the condition that x0) (this condition was implied in writing the fraction)
“x0” was supposed to mean “x unequal to 0”
This is a good point – Paul Hartzer brings up the same thing below.
This is probably a little blasphemous, but I think there’s often a tradeoff between clarity and precision in the way we present mathematics, and often I’m willing to sacrifice a bit of precision.
For example, it used to bug me that trig textbooks present 1/tanx = cotx without discussing that the left-hand-side is undefined at odd multiples of pi/2, while the right-hand-side equals 0 at those values.
But as long as we’re careful about ironing out those technicalities when they become relevant, I’m not too worried about ignoring them when they’re not.
I’ll double down your heresy: Since 0/0 is technically speaking indeterminate not undefined, it can equal any number just as much as it can equal any other number. So in x^2/x, 0/0 can equal 0; in (x + 1)/(x + 1), it can equal 1.
Singularities that create discontinuities are one thing, but I don’t see why it’s that important (particularly at the high school level) to dwell on singularities that don’t create discontinuities.
(Despite what I said below. I like arguing with myself.)
… except your example of simplification changes the potential domain of the function, which isn’t really simplifying things.
That was meant as a reply to neuzjr’s comment beginning “Removing redundancy…”.
“Real numbers” is simpler than “Real numbers but not zero”. ; )
True, but then it’s no longer true that simplification doesn’t result in a different equation, since (as I said) the domain is different. Unless, of course, the domain of the original function was already stated to exclude the singularity.
The EQUIVALENT equation would be “f(x) is indeterminate when x = -1/3 and 5x elsewhere”, which is more complicated than “f(x) = 5x*(3x+1)/(3x+1)”.
All I can say is wow. Your demonstration of passion on this topic is unsurpassed. This is some of the best informative content I have ever seen online. Thank you.
How about ‘what the = sign means’? The two sides are not identical, they don’t appear the same. In what sense can they be the same?
That’s a good question! They’re “equal” in the same sense that 3 + 4 and 5 + 2 are equal. The symbols don’t look the same. But the symbols are just stand-ins for quantities, and those quantities are equal.
Reblogged this on Ancien Hippie.
Reblogged this on 183 Questions by Benjamin Freitag.