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Thabang from Lesotho writes, "how do we rationalize a denominator consisting of a cube root with another constant added to it or subtracted from it?"
Good morning, Thabang, and thank you for your question. This is actually something I don't remember ever seeing before, so I had to give it some thought before answering.
What you're looking for is, how do we rationalize the denominator, if the denominator is something like "The cube root of three, plus two" or "the cube root of three, minus two"?
In order to solve this, it's important to remember two factoring rules you may have learned in an Algebra class:
x3 + y3 = (x + y)(x2 - xy + y2)
x3 - y3 = (x - y)(x2 + xy + y2)
Let's say your denominator is the cube root of three, plus two. Then I'm going to do the following substitutions:
Let x = the cube root of three, let y = 2.
Now your denominator is x + y, and if you multiply the numerator and denominator of the fraction by (x2 - xy + y2), you will have turned the denominator into x3 + y3 = 3 + 8 = 11, which is rational.
That was using the first factoring rule shown above. If the denominator had a subtraction (the cube root of three, minus two), we'd just use the second factoring rule, and multiply by (x2 + xy + y2).
Thanks again for asking, Thabang.
Navya asks: "Why do we have names for numbers?"
There are two answers to this question. The first answer is: because it's impossible to talk about numbers verbally unless you have names for them. If we didn't have the names "one", "two", "three" and so forth, how would we ever say "I have five apples"?
That explanation is sufficient for why we have names for the numbers from zero to nine, but it's not sufficient for numbers like eleven, twelve, and so on. After all, if we didn't have the name "eleven", we could still say the number by saying "one one."
Thus, we would count like this (starting at ten): "one zero, one one, one two, one three..." and so forth.
And in some cases, that would be quicker. The name "eleven" has three syllables, while "one one" just has two. Even worse would be a number like "three thousand, four hundred, sixty three" which takes nine syllables instead of the four syllables required for "three four six three".
So, since the number names aren't always quicker to say than just reciting off the digits, why do we bother? The answer is that using number names allows us to get an immediate order-of-magnitude sense for how big the number is. Look at it this way - if I say "seven million, two hundred twenty three thousand, four hundred twelve," the moment I said "seven million" you had a very good sense for how big that number is. But if instead I had said "seven two two three four one two" you would not have any way of determining the number's magnitude until I was all done reciting the digits, and you knew how many digits there were. And if you lost track of how many digits there were, you still wouldn't have a good sense for how big the number is!
So number names are very helpful for order-of-magnitude sense of the size of a number.
Sarah asks, "On your site you asked the following question and i believe your answer is incorrect. "I have a drawer with 10 socks. 6 are blue and 4 are red. I draw a blue sock randomly and then I draw a red sock randomly. Are these independent or dependent events". You answered even though it is without replacement these are independent. I believe the answer should be dependent, since there is one less sock in the draw when you pick the red one. Am I correct?"
Hi Sarah, thanks for asking this question. I went back and looked at the page your question refers to (Independent and Dependent Events) and realized that the question you were asking about may be a bit ambiguous in how it's worded. Does it mean:
- I drew a sock specifically from the set of blue socks (in other words, I looked in the drawer to find the blue socks, and then randomly selected from that subset) or...
- I reached into the drawer without looking, and randomly pulled out a sock from the entire set, and that randomly selected sock happened to be blue.
I write competition math problems for various math leagues, and I always hate writing probability problems, because they can be so easily written in an ambiguous way (my proofreader hates proofreading them for the same reason). In this case, let's take a look at these two possible interpretations of the problem.
In the first case, the two events are clearly independent; it doesn't matter which of the blue socks was chosen; there are still 4 red socks, and the probability of choosing any particular red sock is 1/4. Thus, the second event is not affected by the first event.
In the second case, I randomly pulled a sock from the drawer, but now we're given the additional information that this sock happened to be blue. So this means that when I reach back into the drawer, there are now nine socks to choose from (not four, as in the previous case, because we assume I'm picking from the entire contents of the drawer.) Since we know that the sock I first chose is blue, there are still 4 red socks, so the probability of choosing a red is 4/9. We get a different answer if we read it this way, but we still have two events that don't affect each other. Since we know the first sock was blue, it doesn't matter which blue sock it was. The specifics of the draw don't affect the outcome of the second draw.
Here's how to make these two events dependent: don't specify that the first draw was blue. Now the result of the second draw is very much dependent on whether or not the first draw was blue. That's probably the situation you were thinking of.
Thank you for asking the question - as a result of your question, I'm going to do some tweaking in the wording of that problem. I don't want it to be ambiguous - especially since the second reading of the problem delves into conditional probabilities, which I don't address on that page!
The greatest irony of the meme shown below...
...is that the person who created it clearly has not YET started paying attention...in logic class, or in science class.
Maybe I should create a web series of these adventures:
"If I'm not supposed to eat the entire contents of a saltshaker at once, why does my wife keep putting salt in my food?"
"If water is so good for you, how come people keep drowning in the ocean?"
"If arsenic is a deadly poison, why aren't all the people who eat rice dead?"
"Why am I not supposed to eat the whole jar of Metamucil if Metamucil keeps me regular?"
"If carbon dioxide can cause convulsions and loss of consciousness, why do we allow so many trees on our planet?"
"If One A Day vitamins are so good for you, why are you limited to only one a day?"
Once Guy-Who-Just-Started-Paying-Attention figures out the answers to these questions, he should probably start looking into the dangers of dihydrogen monoxide...
Seventh grader Brooke from Pennsylvania writes: "Professor Puzzler, I finished your article about stressed and unstressed syllables, and it has helped me a lot! I’m in the process of writing my first sonnet and trying to juggle rhyme, iambic pentameter, and the structure of the sonnet is super difficult—especially since I still struggle with finding the stressed syllable. My main questions are: how do you know if single syllable words are stressed or unstressed? Is there anyway to check? Thank you!"
Hi Brooke, that's a great question. I'm so pleased to hear of students working to develop sonnets; the hard work you put in on such a daunting task will serve you well in your future poetry endeavors. What seems like a struggle and a juggle now will eventually become more and more natural to you. Eventually the meter will flow with much less conscious thought on your part.
But, in the meantime, I'll attempt to answer your question. There are certain words in the English language which are deemed "less important." I don't mean that we can get by without them; many sentences would be incomprehensible without them. But they are words that we often don't even consciously notice*. They are words like articles (a, an, the) prepositions (on, of, in, etc.), and conjunctions (and, but, or, that, which, etc.). Linking verbs (is, are, etc.) can be added to this list. These "lesser" words, if they have only one syllable, will typically will be unstressed in the context of a sentence. In contrast, one-syllable words which play a significant role in the sentence (such as nouns and non-linking verbs) will most likely be stressed in a poem.
One of my favorite example poems to look at is "The Night Before Christmas" -- consider the first line with unstressed syllables capitalized:
'twas the NIGHT before CHRISTmas and ALL through the HOUSE
Notice those single-syllable words that are not stressed -- there are so many prepositions, articles, and conjunctions!
Now, I said that this is typically true, but that's not a hard-and-fast rule. Context plays a very key part in identifying the stressed syllables. Consider the following two sentences:
The CAR is ON the gaRAGE.
The CAR IS on the gaRAGE.
In the second sentence, the verb "is" has the stress, but in the first sentence, the preposition "on" is stressed. Why is that? Presumably in the first sentence, someone is startled because they didn't expect the car to be ON the garage (rather than IN it). But in the second sentence, it sounds as though someone has disagreed with them, so they are emphatically declaring the truth of the statement by emphasising the verb.
There's a bit of flexibility with these "lesser" words; the meaning you intend to convey can control whether they are stressed, but also, if you have two or three of them in a row, the natural rhythm of the sentence may dictate where the stress goes. Consider my rewrite of the first line of "The Night Before Christmas":
it WAS the NIGHT beFORE the YULEtide, AND all THROUGH the HOUSE
What have I done to this line? I've rewritten it so it has an iambic meter. Read that out loud, putting the emphasis on my upper-case syllables. Now compare my line to the actual anapestic line. I've included many of the same words, but the rhythm dictated that they be emphasized differently. "Before" gets one of its syllables stressed in my line, but not the other. In my line, "and" and "through" get the stress, while "all" does not, and this is exactly the opposite of the original line where "all" gets the stress, while "and" and "through" do not. Why do I get away with doing this? Because I'm not messing with the stress of "important" words like "night," "yuletide," and "house."
How do you tell what you can get away with? You read your line aloud, and listen to hear whether you are naturally emphasizing certain syllables or not. Then read it with the stress you'd like the syllables to have, and see if it sounds awkward.
Eventually you'll stop analyzing stresses syllable-by-syllable and go straight to listening to how it sounds.
* Book Scrounger notes that many of these "lesser" words are words which are not capitalized in headlines and titles. If you look at the title of this blog post, you'll notice that there is one uncapitalized word in the title. It's the conjunction "and." The exception to this is linking verbs, which are always capitalized in headlines and titles.