Wednesday, July 9, 2014

Too Far from the Average

Too Far from the Average

By Bobby Neal Winters
Buster Williams was sitting in his Elementary Statistics class struggling not to go to sleep.  Sometimes he bit his lip, sometimes he slapped himself on the neck--a trick he’d learned while driving all night to get to Panama Beach--but nothing he did seemed to work.  The problem was at the front of the classroom: the Standard Deviation, or Dr. Bottlebutt as he was known in polite society.
Professor Bertram Bottlebutt, covered from head to foot with chalk dust, wrote one last equation on the board.  He turned for the first or second time during class and asked, “Are there any questions?
Twenty pairs of glazed eyes, Buster’s included, didn’t even blink in response. This was a trick considering he’d been drifting off to sleep before.
“In that case,” he said, “class dismissed.”
Buster felt a rush of adrenaline. His heart soared.  Existence was worth continuing again.
For the first time in an hour, the students in the room showed life. Notebooks were shut, slid into backpacks, and motions were made toward the door.
“Wait, wait, wait,” he called out.  It was something students had heard before.  Indeed several of them did passible imitations of Dr. Bottlebutt, and competed against each other with their “Wait, wait, waits”.
“I need to talk to ... uh... Williams and ... uh ... Young,” he said.
Buster, hearing this, uttered an expletive to himself.
Since being in Dr. Bottlebutt’s class, he’d gotten used to being called “Uh...Williams.” Everyone who took a class from Bottlebutt had the same first name. Students also worked uhs into their imitations of Dr. Bottlebutt.  This was a man who could give hour upon hour of lecture of mathematics composed entirely of abstruse equations on the blackboard without any notes at all but could remember Bob, Emily, or Buster.
The room emptied of students with the expressions of those who, having eaten their morning live frog, knew everything would be better for the rest of the day.  All but Buster and Young who remained.
“Uh...Young” was actually named Betty.  Buster had been sitting in a desk one behind and to the left all semester just because he wanted to be able to see her.  She was...gorgeous. Everything that nature equipped young women with to get the attention of young men was present in Betty not in abundance but in perfect proportion.  And everything was presented in such a way that Buster could find no way to ignore.  As he stood, working on an inventory of Betty starting at her ankles, working up her calves to her thighs...
“Mr. Williams,” came Dr. Bottlebutt’s voice. “Could I please have your attention?’
“Uh...” oh God it must be catching he thought, “Sorry.”
“I was saying my assistant lost your and Miss Young’s grades,” said the professor, seeming quite chagrinned. “I always put the papers in alphabetical order, and apparently he lost the two on the bottom.”
Professor Bottlebutt had his own way of dealing with exams.  He would grade them himself and work up the statistics for the class.  He would then hand back the papers to the class for a few minutes to allow the students to look them over.  Then he would put them into alphabetical order and give them to his assistant to record.
“If you tell me your correct scores, I will have my assistant record them,” he said.
Buster was astonished.  Dr. Bottlebutt was willing to take his word.  This was an opportunity for him to improve his grade.  He was straining through his brain for a number when he heard Betty’s voice from next to him.
“Ninety-eight,” she said.
Mesmerized by the sweet sound of her voice and not being the sharpest tool in the said, Buster followed suit.
“Yeah, me too, ninety-eight,” he said.,
The professor, who was sitting at his desk, glanced upward at both of them with a look on his face that was not indicative of credulity.
“Ninety-eight,” he echoed.
“Uh...,” Buster began but he didn’t get far.
“You know,” Bottlebutt said, “I don’t think that is exactly believable.  Let’s see.  The class average, which I calculated before your papers were lost, was about 70 and the standard deviation was about 6.  This means 98 is just over 4 standard deviations from the mean. That is unlikely for any population, but we have only 20 students in class.  By Chebyshev’s Rule, only 1 over 4 squared of the data can be that far from the average, that only 1 in sixteen. Now 20/16 is between 1 and 2.  So there is some chance that one of you can have that grade, mathematically speaking, but by Chebyshev’s it is mathematically impossible for both of you to have a grade that far from the average.”
Buster stood with his mouth open.  He noticed that Betty didn’t move a muscle. Not even her eyes moved.  Her chest was still moving in and out with a beautiful rhythm...
Butster looked down at the professor and noted that he was still staring up at them.
“Does either of you have anything more to say?”
Buster glanced Betty’s way and her expression was as innocent as a dove.  It made him ashamed of lying, but before he could own up, the professor continued.
“It might be well for you to know that since I’d figured up the statistics for the class  I was able to figure out some information from the rest of the papers,” he said. “The sum of your scores is 140.”
Buster thought of the score he’d really made, 70, and then, in spite of not being the sharpest tool in the shed, figured what Betty’s score had to be. She’d made 70 too.  Two thoughts raced to share his mind. One: They’d made the same score and had something in common.  Two: She was lying her gorgeously perfect backside off.
Buster looked at Betty.  The expression was so slight that he didn’t know whether it was there or whether he was imagining it.  It was in the eyes; it was around the curve of her beautiful lips. Please don’t give me away.
“Uh...Dr. Bottlebutt, did I say 98? I must have misspoke,” Buster said as he frantically tried to do mental arithmetic. “My real score is...”
“Forty-two?” the professor interrupted him.
Buster did a quick 98+42=140 and answered, “Uh...yeah.”
“There is a problem with that,” Dr. Bottlebutt said. “Forty-two is also more that four standard deviations from the average.  There can only be one that far away, high or low.  Do you want to stick with that?”
Buster looked at Betty, who was the most beautiful example of the human female that he’d stood in the presence of.  She was still holding that expression. It was so subtle, but so compelling.
“Yes,” he said, “sure.”
Bottlebutt looked up at both of them.
“You know,” he said, “I could take ten or twenty minutes and work this out. It’s not hard. I’ve got all the information I need.  But I think there is a learning opportunity to be had here, so I am going to take both of your words as true.  Good day.”
With that, they were dismissed.
Buster walked into the hall with Betty.
“Betty,” he began.  His next words were going to be about maybe going to get a latte or something, but no one was listen.
Betty walked over to the tallest, most broad shoulder man Buster had ever seen.  She put her arm around him and he put his hand in her back pocket. And the strolled away.
Buster, more than a bit crestfallen, got kind of a bitter taste in his throat. He turned and saw Bottlebutt who had clearly taken in the whole scene.
Buster stood there wordless and didn’t expect the professor to say anything, but he did.

“There are some lessons harder in the learning that math,” he said.

Monday, June 16, 2014

Converting Fahrenheit to Celsius

By Bobby Neal Winters
I am still in the process of learning Brazilian Portuguese. When you are learning a new language, numbers are always a problem because they are entities that require the same sort of manipulation that words do and everything gets clogged up.  This happens especially when the numbers you are dealing with are temperatures and you are going to a country where they measure temperature in Celsius instead of Fahrenheit.  
There are simple formulas for converting from one to another:
F=(9/5)*C+32
C=(5/9)(F-32)
where F is degrees Fahrenheit and C is degrees Celsius.  I say simple. To a mathematician, these are simple because they are linear. Linear is just one level of complication above constant.  The problem is that if you are trying to work with a new language, there is probably too much going on in your head to do this.  And to be frank, there is probably too much even if you are working in your own language because there are fractions.
The simple way to deal with the Celsius scale is to learn it on its own terms.  We will be get with some facts.  In Celsius, 0 is freezing and 38 is body temperature.  Also, 50 degrees Fahrenheit is 10 degrees Celsius; 68 Fahrenheit is 20 degrees Celsius;86 Fahrenheit is 30 Celsius.  Finally, both scales are the same at -40.  
So think about it like this.  At 40 degrees Celsius, you are above body temperature, so it’s pretty hot.  It is a typical summer day in Oklahoma or Kansas or Rio.  Folks from the north or from the coasts, wonder how people live in such weather.  They come to the conclusion they can’t, so whatever lives in it can’t be human.
At 30 degrees, an Okie will say it’s pleasantly warm, and a Californian will complain and, perhaps, pray for death.  At 20 degrees, the girls in Rio will put on light sweaters over their summer dresses; Kansas girls will think they are in paradise; Californians will complain because it’s cold.
At 10 degrees, it’s cool. The girls in Paraguay are wearing every stitch of clothing that they own.  The Californians are filing lawsuits. A Kansan has put on a windbreaker. The folks in Wisconsin are wearing shorts.

Zero degrees is cold.  The Brazilians are experiencing hypothermia, but are hoping in their hearts that they might see snow before they die. The Californians are curled up in little balls, scratching their wills into the frost. The Kansans have on long sleeves under their windbreakers. The guys from Wisconsin are still wearing shorts.

Monday, May 19, 2014

Front Porch Math


Front Porch Math 


By Bobby Neal Winters
The following is a real problem that a carpenter with a great respect for trigonometry came to me with.  Say you are going to add a porch of width L to the side of a house and want to over it by adding to the roof as depicted in the picture below:
The question is how long do you want your rafter to  be, i.e. what is the length of the segment D.  As it is there is not enough information in the problem.  You have to know that the slope of the current roof is 7/12, (i.e. over a run of 12 feet the roof will rise 7) and they desire the porch roof to have a slope of 3/12 (i.e. over a run of 12 feet it will rise 3.)  Yes, we could say that is a slope of 1/4, but these are how the numbers came to me, and it will all come out in the wash anyway.

I must confess that this problem took longer for me to solve than it might have because I was guided initially by the carpenters love of trigonometry.  It can be solved that way, but this is a nice case study of how imposing a coordinate system can simplify the problem.  Consider the following:
We've put on coordinates so that the outside point of the porch roof is a (0,0), the point where the old roof touches the house is (L,0), and the point where the porch roof meets the old roof is (x*, y*).

The equation for the line representing the porch roof is y=mx and for the old roof is y=nx-nL.  It is an exercise to work out that x*= nL/(n-m) and y*=mnL/(n-m).  We may use the Pythagorean Theorem (or the distance formula) to calculate the value of D2=(x*)2+(y*)2 which after substitutions simplifies to:
D2=(1+m2)n2L2/(n-m)2   or D=(1+m2)1/2 nL/(n-m).

For the given values of n and m, this works out to D=2.02598L.

Having done it this way, I also worked it by trigonometry, but I despair of reproducing those calculations without specialized mathematical software.

Monday, March 31, 2014

A Squirrel’s Tale

A Squirrel’s Tale

By Bobby Neal Winters
I received the following report from the Bureau of Unlikely Seeming Happenings, Bulsh for short.
Cletus T. Gnasher was out hunting squirrels one day in order to put a little meat on the table when he came upon a strange sight. He saw a squirrel go into one end of a ten foot long hollow limb.  He took out his trusty 22 rifle and aimed it toward the squirrel.  One minute later the squirrel came out the other end.  
Cletus was about to squeeze off a round when the squirrel ducked back into the hollow limb.  Not about to give up this easily, Cletus kept his eyes on the tree.  At the end of 30 seconds, the squirrels head was out of the other end.  Again, before Cletus could squeeze off a shot, the squirrel had ducked back inside the hollow limb.
The squirrel went again to the other end of the hollow limb and this time in only took 15 second before the little head poked out the other side.
It is at the point some began to doubt the veracity of the report because Cletus said that something funny began to happen. The squirrel continued to go back and forth  between the ends of the hollow limb until at one point it appeared he was looking out of both ends at once.  How long did this take?
This was presented to the Bulsh board and reports were requested.  The first report came from a mathematician.  It was very succinct.  This is an infinite series problem.  To determine the time expended, sum one minute plus one-half minute plus one-fourth minute and so on, i.e. 1+ 1/2 + 1/4+ ... .  This is a geometric series whose sum is 2.
The second response was from another mathematician who didn’t offer a solution, but said that this one couldn’t be correct because it assumed the squirrel would be transversing the tree limb an infinite number of times.  As the length of the limb was fixed, being 10 feet, transversing it at infinite number of times would require the squirrel to go an infinite distance.  That is to say that while the infinite series for time-passed converged, the series for distance traveled diverged.
At this point, a physicist chimed in, saying that the problem was reached long before infinity.  The first time the squirrel went through the limb, he was going at a speed of 0.1136 miles per hour.  However, by halving the length of time through the limb each time, he was doubling his speed. Taking the liberty of converting this to 0.05784 meters per second to make sure no one forgot he was a physicist, he said that after 33 more times through the limb, the squirrel would be going 4.36 times 10 to the 8 meters per second which is greater than the 3.0 times 10 to the 8 meters per second that light travels.  At this point, one of the mathematicians corrected him to 2.998 times 10 to the 8 meters per second.  The physicist wrote back something that best not be repeated here and a flame war ensued.  
After the verbal abuse subsided, the physicist calculated that, because of relativity, a one pound squirrel would weigh 1.45 pounds after 32 trips through the limb.  One of the mathematicians asked him whether he meant a one kilogram squirrel would have the mass of 1.45 kilograms, but fortunately the irony was lost on him, and then he was distracted by the following question from an engineer.
The engineer noted that Cletus only said that it looked as if the squirrel were looking out of both ends at the same time.  An image persists on the retina for one-sixteenth of a second.  According to his calculations, it would only take 10 trips through the limb to be making it in that amount of time.  This is a speed of 58 meters per second.
The first mathematician then calculated that this would take 1.998 minutes which was only 2 one thousands off his original estimate. The second mathematician said, “Yeah, your original WRONG estimate.”
This was beginning to degenerate into something nasty then the physicist said, “You know this would require an acceleration of almost 12 times the force of gravity to accomplish.  The squirrel would be  mush.”
At that point, one of the animal-rights folks on the committee said this was something you couldn’t even talk about doing not even to a theoretical squirrel. As this was a supposedly real squirrel, an investigation would have to be launched.
One of the mathematicians said, “By academic freedom..."
And the reply was that he could be free to seek another academic job if he said one more word.

Monday, January 20, 2014

On One Hand But Then on the Other

On One Hand But Then on the Other

By Bobby Neal Winters
I was approached yesterday by a friend of mine at church in Opolis in the following way.
“You’re a math professor, so I got a question for you,” he said.  “It’s from my grandson.”
I remained calm outside, but on the inside I had the reaction a gunfighter in the Old West had whenever someone said, “They say you’re pretty fast.”  You never know when the person asking the question might be faster.
I decided to take my chances and listen.  He held out his right hand and began to count extending his fingers one by one:
“One, two, three, four, five,” he said, extending his pinky last. “And five is ten,” he said extending all of the fingers of his left hand at once.
I nodded because I knew we weren’t at the hard part.
He then held out the fingers of his left hand and started extending them thumb first:
“Ten, nine, eight, seven, six,” he said, giving emphasis to the six. “And five is eleven, so you got eleven fingers.  How can I explain to my grandson why this is wrong.”
And my friend knows it’s wrong.  But knowing it’s wrong and being able to explain why are two different things.  And I could go on to a digression about politics here, but my point is math, or at least the explaining of math.
My friend knows this is wrong because we have ten fingers and ten does not equal eleven, no matter how fast you talk.
Let’s first analyze how this is presented because that is very important to how the confusion comes in.  We start off with something that is true: one, two, three, four, five, and five is ten.   What has happened there?  We’ve listed off the names of the first five numbers that we learned when we learned how to count.  We’ve set up a correspondence between those names and our fingers.  
The next part is where something subtle is done which sets us up for the confusion. When he says, “And five makes ten,” he’s made a very subtle shift.  He’s using the name five to refer to the quantity of fingers on his left hand.”  The five he said when he held out his pinkie was referring to a place in order; the five he said when he held out all his fingers at once was referring to the quantity of fingers.  Mathematics call the first one ordinality and the second cardinality; think of these as order and quantity.
In listing the numbers in standard order, “one, two, three, etc,” the name of the last number listed is also the name of the quantity of the items in the list.
This does not work when you count backwards, and that is one of the things that the example my friend brought me illustrates. In counting backwards from ten, there is no point at which the number counted with also be the quantity of things counted. (Ironically, if he’d started counting at eleven it could be made to work, but you can’t do that because you have ten fingers. It is the center of this trick that you start counting backwards at ten because you know there are ten fingers.)   
When you hold out the pinkie on your left hand as you count backwards from ten until you get to six, you are giving its order as if all the fingers were counted beginning with the other hand.  It can refer to a quantity, but that quantity would be the pinkie itself and the fingers on the left  hand.
Now, you must understand that I didn’t tell my friend all this.  I said, “It’s a confusion between cardinality and ordinality.  Don’t let your grandson play poker with Bill.”
Bill is a man who’s given me poker lessons. After $15 worth, I learned: don’t play cards will Bill. But that is a different story.

Monday, December 30, 2013

Casting Out Mathematical Demons

Casting Out Mathematical Demons

By Bobby Neal Winters
The other day on his Word of the Day blog, Anthony Esolen made reference to the divisibility test for 9 and 3.  The basic idea is this: take the base 10 expression of a number and add up its digits.  That is called the digit sum/ mathematicians don’t waste their imagination on names for things.  If the digit sum is divisible by 9 (or by 3) then the original number was.  
For example, 5643 is divisible by 9 because 5+6+4+3 is 18 and 18 is divisible by 9.  If you aren’t good enough at arithmetic to know that 18 is divisible by 9, note that 1+8=9.  If you don’t know that 9 is divisible by 9, then I can’t help you.  
There is a name for this beyond the Division Test for Divisibility by 9.  It’s called Casting Out Nines.  The reason for this name might become apparent after you’ve done a lot of testing.  Say that you have a number 9954968 and you want to determine whether 9 divides it.  You only need to calculate 5+4+6+8 because those 9s will not add any information and will only add complication to the addition,  so you cast them out.  Indeed, as that 5+4 is a pat 9, you can leave them out too and only worry about the 6+8=17 which is not divisible by 9.
This test is so good we sometimes there are other division tests.  
  • The number 5 divides a number only if the last digit is 0 or 5;
  • the number 2 divides a number only if the last digit is even;
  • the number 4 divides a number only if the the number that is last two digits are divisible by 4;
  • the number 8 divides a number only if the number that is the last three digits are divisible by 8;
  • the number 6 divides a number that number is divisible by 3 and 2; see the tests above
This takes care of all of the numbers from 2 to 9 with the exception of 7; this demon 7 became a mystery to me until I become math major and was given the keys to the priesthood.  I am now going to share those with you because I don’t think you will believe me, and if you do believe me, you will just forget about it anyway.  
I dare you to believe me. I double dog dare you.
Okay, take the number you want to divide by and call it d and take the number you want to test and call it T. I’ve lost a bunch of people right there because I’ve used variables. Tough.  When you divide T by d you get a remainder r.  The number T is divisible by d exactly when r is 0. I’ve not said anything now that ought to surprize you.  This process is called reducing modulo d. Forget that bit I said about not wasting imagination on names.
The surprize comes next.  You can reduce modulo d in an amazingly simple way. It is called modular arithmetic.
Let’s go back to our first example.  Is 5543 divisible by 9?  Well, 5643= 5x1000+6x100+4x10+3.  Take every term and factor of this an replace it with its remainder when divided by 9.  It is easy in this case because the 5, the 6, the 4, and the 3 don’t change and the 1000, the 100, the 10, and the 1 are all 1.  So 5x1000+6x100+4x10+3=5x1+6x1+4x1+3x1 which is exactly what we had done before.
Now consider the test for divisibility by 4 which only uses the last two digits of a number.  Why?  Well consider that the third digit will be multiplied by 100, the fourth by 1000, the fifth by 10000, and so on.  Each of these number is evenly divisible by 4 and so leaves a remainder of 0. So 5643= 5x1000+6x100+4x10+3 reduces to 5x0+6x0+4x10+3=43. Here, I am not using the full power of the technique for the sake of demonstrating the classical test.  The 4x10+ 3 would reduce to 4x2+3=11 which would reduce to 3.
You can see how the test for 5 would work out neatly as well because all powers of 10 leave a remainder of 0 when divided by 5.
Let me now go though a more complicated example to show you the full power.  Is 5643 divisible by 7?  I don’t know of any classical test for this. If you do, let me know.  Note that 10 divided by 7 leaves a remainder of 3;  then 100=10x10 will leave the same remainder as 3x3=9 which is 2; so 1000=100x10 will leave the same remainder as 2x3=6, i.e. 6. So 5643=5x1000+6x100+4x10+3 reduces to 5x6+6x2+4x3+3=30+12+12+3 which reduces to 2+5+5+3 which reduces to 5+3 which reduces to 1.  
Therefore 5643 is not divisible by 7, and, indeed, leaves a remainder of 1.   A very short calculation on a calculator will confirm this.
If you have a number like 75867 you can forget about the 7s and reduce it to 5860 which will reduce to 5160 which will reduce to 5x1000+ 100+6x10 which reduces to 5x6+2+ 6x3 which reduces to 2+2+4 which again reduces to 1.  
While I find this to be a very entertaining activity, there does come a point where simply dividing by 7 by hand becomes easier and I suspect that is why this is not taught to mere mortals.

Thursday, November 21, 2013

Long Division

Long Division
By Bobby Neal Winters
Long division is hard.  I want to state that up front. I remember when I first encountered it in grade school that it hurt my head.  There were a lot of rules and you had to be able to know your multiplication tables and guess and try and guess again. I learned how to act pitiful to get my mother to do it for me.
More than forty years later, I will admit that it’s a useful technique.  Let’s talk about long division long enough to get into trouble. Consider the easiest case: dividing a number by a smaller number.  For example, 7 divided by 2.
Two time three is six is the largest multiple of two that is less than seven, so we put down a 3 on the top line and subtract 6 from this leaves 1.   When you are a little kid who doesn’t know about decimals, you stop there and say there is a remainder of 1, but when you know about decimals, you put a decimal point after the 3 on the top line and you put a 0 down after the one to make it ten.  Two times five is ten exactly, so you put down a 5 after the decimal and stop.  This yields that 7/2=3.5, which, of course, is the correct answer.  You can always check your answer--but our students never do--by multiplying 3.5 by 2 to get 7.  
This is the easy case not because we are dividing a bigger number by a smaller number, but because the process terminates after a finite number of steps.  Consider a case that doesn’t like two divided by seven.  This is illustrated below:

Note that since seven is larger than two, we have to put down a 0 followed by a decimal on the top line. We then put zero times seven on the line below 2 getting--surprise, surprise--0.  We then take zero from two and put down 2 on the next line.  We put a 0 by it to make it twenty.  We then note that seven times two is fourteen so we put 14 below the 20.  Taking fourteen from twenty leave six, so we put down 6 on the next line and follow it by a 0 to make sixty.  Now seven times eight is fifty-six, so we put 56 below the 60.
You know the drill.  This will literally go on forever.  But before too long this will begin to repeat so that we get 2/7 is equal to 0.285714285714285714285714... .
This works nicely because we have a place-value system to represent numbers.
Mathematicians are never satisfied with just numbers, however.  We like to use letters too.  We work with entities (I almost wrote things there.  Things is sloppy writing, so I used entities there instead.  It means things, by the way.) called polynomials.  If you’ve had an algebra class you’ve seen something like 2+x or 2-x-x2.  You can divide on of these by the other two.  For example, let’s divide 2-x-x2 by 2+x:

I was tempted to explain this division. Indeed I wrote a paragraph of it, but I lost consciousness in the process.  If you’ve had the course, you can do it, but if not, I am not going to abuse your good nature by teaching it here.  Let it be sufficient to say that we go through the same motions as we do when we do the long division of numbers.
Just as in the case of long division of numbers, we can have non terminating cases here too.  Consider 2+x divided 2-x-x2:

Here the final quotient is the infinite series 1+x+x2+x3+x4+... .  This might ring a bell for some of you because not only is it 2+x divided by 2-x-x2 it is also 1/(1-x).  As PeeWee Herman used to say, “I meant that to happen.”
The equation 1/(1-x)=1+x+x2+x3+x4+... is quite famous among math geeks.  It is called the Geometric Series.  I can literally talk for hours about this.  My students will testify to this, as will the piles of legs in the classroom which my students have gnawed off like coyotes in futile attempts at escape.
You can have a lot of fun with this equation.  Let x=0.5.  The left hand side, i.e. 1/(1-x), becomes 2 while the right hand side becomes 1+0.5+ (0.5)2+ (0.5)3+... .  If you add enough terms of that right hand side, you get as close to the sum of 2 as you desire.  
If you let x=1, then the left hand side becomes 1/0 by zero while the right hand side becomes 1+1+1+.... which is infinity.  If you make yourself believe that 1/0 is equal to infinity, then this is not a problem.  Let’s push it further, though.  Let x=-1.  Then the left hand side is equal to 1/2.  The right hand side, however, becomes 1-1+1-1+1-1+.... . Let’s refer to this right hand side as S.  It is possible to argue that S is equal to zero; it is possible to argue that S is equal to one.  All of the arguments make the assumption that you can treat arithmetic operations done an infinite number of times the same way you can those done a finite number of times.
What happened to resolve this is a good example of how mathematicians operate.  We have a formula here that gives good answers in certain circumstances and nonsense in others; we determine exactly what those conditions are.  In this case, we get good answers exactly when the absolute value of x is less than one and nonsense at all other times.  There is a whole language created to talk about it using words like convergent and divergent and the Greek letters epsilon and delta.
So we began with a method for dividing one number by another.  We then extend that method to polynomials.  This led to an infinite series where work had to be done to delineate what could be said.  It would appear to be the end of the matter.  Yet.
Given the theory developed to talk about convergence,  the idea that the series S=1-1+1-1+... might be equal to anything appears to be nonsense, but consider the calculation below.  It is like a twisted coda at the end of a horror movie, Jason lives:

What was it that Scotty said, “If one man calls you an ass, pay him no mind. If two men do, buy a saddle.”
One can talk about Cesaro sums. These are like running averages of series.  The Cesaro sum of S is 1/2.
Let’s go back to the equation 1/(1-x)=1+x+x2+x3+x4+... .  Those who know a little calculus will recognize that taking the derivative of each side will yield 1/(1-x)2=1+2x+3x2+4x3+... .
Putting in x=0.5 makes the left hand side equal to 4 and the right hand side 1+2(0.5)+3(0.5)2+4(0.5)3+... can be made as close to 4 as we desire. This is all good.  It becomes more interested when we let x=-1.  Then the left hand side becomes 1/4 and the right hand side becomes 1-2+3-4+5-... .
But this can be made meaningful as well.  There is a theory of divergent series that has been worked out.  It is not my point here to expound on it--though if I learn any more I might--but to bring it up as an example of what mathematics sometimes do.
We take a technique in one area where it is clearly defined and makes sense and then apply it--sometimes playfully--in another area, where sometimes we get a mixture of good results and nonsense.  We then perform investigations to separate the good results from the nonsense.

It’s not a bad way of doing business.