Saturday, March 21, 2009
The Trick
First we can see that all such numbers are multiples of 9:
Having seen that, note that the multiples of 9 are all always marked with the same symbol. They are tricky because the symbol changes every time and there are numbers marked with that symbol that aren't multiples of 9, but every multiple of 9 is so marked.
Having seen that, note that the multiples of 9 are all always marked with the same symbol. They are tricky because the symbol changes every time and there are numbers marked with that symbol that aren't multiples of 9, but every multiple of 9 is so marked.
Sunday, March 15, 2009
Saturday, March 14, 2009
Flipping the sign
A narrow view of numbers
Negative numbers are a source of confusion for many who are getting their feet wet in mathematics. The problem comes with having too narrow an idea of number, i.e. restricting the notion to embody only quantity.
We are introduced to numbers, even negative numbers, through quantity. If I have 5 apples and eat three of them, then I have only 2 left. If I loan 3 of them to you and you eat them, then you have negative 3 apples and I will send Bubba after you with a tire iron to get them back if I have to. It doesn’t take much imagination to figure out that this can be tied to money and once things get tied to money they’ve got a way of capturing our attention. We like to keep track of our money. One device to keep track of money is the number line.
I’ve indicated it above with dots draw on it to indicate the integers, a zero to give us a starting point, and an arrow to indicate the positive direction. We can use it to keep track of our money. If I have $5, then I can indicate that by marking it as below.
However, if I buy something that costs $7, then I go $2 in the hole and move the marker backwards, in a negative direction.
Thus, the line is a device to keep track of personal wealth, with numbers to the right of 0 denoting what I have and those to the left what I owe. Of course, as I’ve set this up, it is particularly egocentric. The person I borrowed from still counts himself as having that $2 and would remind himself of that by noting it on his number line.
Introducing Geometry
By bringing in the number line, I’ve introduced geometry to the situation. Zero is my starting point and the (+) and (-) signs indicate the direction which I am going away from zero. I can think of the line as being First Street, which runs in front of my house and zero as being the corner of First Street and Catalpa, which I can see from the window of my home-office. By setting East as the positive direction, I can then tell people I am going 5 blocks and they would be able to figure out that my destination is the corner of First and Walnut, while if I told them I was going negative 2 blocks they would be able to discern I was headed to First and Georgia.
This works fine for addition. As for multiplication, if I am multiplying a positive number by a positive number, I can think of that as repeated addition; if I am multiplying negative by positive, I can think of it as repeated subtraction; and while I am convinced I could do something similar for negative times a negative, thinking about it makes my head swim. Fortunately, I don’t have to because we’ve introduced geometry to the problem.
Let us go back to borrowing versus lending. When Bubba borrows from Billy Bob, Bubba owes, but Billy Bob is owed. The same transaction is positive from one point of view, but negative from the other. Given Bubba’s predilection for non-repayment, the negative end is often hard to discern, but I digress. The point is the positivity or negativity is simply one of point of view. We could view multiplication has having two effects: one on size and one on direction. Multiplication by a positive number might change size but it preserves direction while multiplication by a negative might preserve size, but it changes direction. The negative may be viewed as flipping a number over to the other side. Hence, a negative times a negative is a positive.
Numbers as two pieces
When we multiply numbers, we can think of them as having two pieces. The first of these is their absolute value, i.e. their length, and the other is their sign. Effectively, this is what we do: (5)(-3)=(+5)(-3)=(+)(-)(5)(3)=(-)(15)=-15 or (-2)(-4)=(-)(-)(2)(4)=+8=8.
More Geometry
In the movie Paint Your Wagon, the innocent farm boy utters the line, “But I’ve never lied to my parents before.” To this Ben Rumson answers, “Well when you do it opens up a whole new world!”
When geometry is introduced to arithmetic, a whole new world is opened up. For example, I can expand my explorations beyond First Street by introducing complex numbers. I can take the complex number x+yi and take it to mean go x units in the east-west direction and go y units in the north-south direction. Recall that i is the square root of -1. I can add x+yi to u+vi by taking the sum of the two to be (x+u)+(y+v)i. This works out well geometrically because if I first go x units east-west and y units north-south and follow this by u units east-west and v units north-south this gets me to the same place as going x+u units east-west and y+v units north-south. Go for a walk and see for yourself.
Thus again we are called to expand our notion of number as being simply quantity. On the plane, as we are, we can think of the complex number x+yi as being the directed line segment from the origin of our axes to the number x+yi, as below.
Complex numbers mimic real numbers in that they are best thought of as multiplying in two pieces as well. They have their length, the distance from the origin to the point (x,y), and their direction, which is the angle from the positive real axis to the directed line segment which is associated to x+yi. This is labeled by the Greek letter alpha below.
When two complex numbers are multiplied together, the length of the product is the product of the lengths of the two numbers involved and the angle of the product is the sum of the angles of the two numbers involved. So, for example, the imaginary number i has a length 1 and angle of 90 degrees. So i times i has a length of 1 and an angle of 180. Notice that the number -1 fits this description exactly. Therefore i times i is equal to -1 just like it is supposed to.
This is how the formal multiplication works out. To prove this is a beautiful exercise in trigonometry, but it would put some folks off. I am willing to show you, but only if you beg me.
Friday, March 6, 2009
The Two Towers
We are siloed. Those of us who live in the nations breadbasket might understand it best. Silos are towers in which grain is stored. Silos have high walls, walls so high that none may hope to peer over them into the outside world.
It is an evocative phrase and an apt metaphor for the state of the academic disciplines at the universities. Each of us is within our own world, perhaps a master of it, or desiring to give that appearance, at least, but most are blissfully unaware of the vast storehouse of knowledge that is available in the silos of the surrounding landscape.
This was brought most recently to my attention through a reading group in which I am involved. We’ve been reading a book entitled Proust was a Neuroscientist by Jonah Lehrer. The basic premise of the book is that certain artists have anticipated discoveries by neuroscientists. He does this by proposing several putative examples of such, and whether he succeeds in any particular case I leave his readers to decide for themselves.
In my opinion, he has pointed to something much more important than what he believes he has.
Artists and scientists live in the same world and each of these groups attempts to describe that world in its own language. An illuminating example of this occurs in Lehrer’s chapter on Whitman, who had experienced the Civil War first hand. As a part of this, he took part in amputations and discovered that amputees often still feel the amputated limbs. This information, apparently, didn’t appear in scientific literature until much later.
I find it hard to believe this was unknown until Whitman’s day. Indeed, I can imagine a caveman having had an arm chewed off by a saber tooth tiger still feeling the arm and believing that its spirit still lingered.
It is instructive because it allows us to see what Whitman did. He raised the phenomenon into a higher level of conversation. He raised its talk among small groups of army surgeons, amputees, etc into the broader world. When scientists wrote on the subject, this is exactly what they did as well, but when they did it was in their own language.
The language of the arts is different from that of the sciences. The one thrives on ambiguity and multiplicity of interpretation, and the other on precision, but each is observing the same world and, as a result of these, both will stumble upon the same truths from time to time.
The book group to which I belong consists of two mathematicians, a psychiatrist, a historian, a political scientist, a physicist, a retire high school social science teacher, and a retired professor of Spanish. Proust was a Neuroscientist resulted in many fruitful conversations because of the topics it juxtaposed, not because of the information it offered.
Lehrer is a journalist, not a scholar, and members of the group are more comfortable with scholarly books which cite sources in a more scholarly way. He is also relatively young, so there are areas in which his reach far exceeds his grasp. Even at that, he is a writer who has the talent to convince the reader he knows what he is talking about even when this is, in fact, not true.
This is a talent much valued among journalists.
But even as I criticize the author, I have to ask whether such journalistic pidgin might be necessary when facilitating exchange between two groups with such different languages. To truly understand art as an artist does, does one need to be an artist? To grasp science in a full way does one need to be a scientist?
I believe the answer is no, but I also believe to deal well with both simultaneously would require something I call mileage. This is a quality I don’t see Lehrer possessing in spite of his talent as a writer.
It is an evocative phrase and an apt metaphor for the state of the academic disciplines at the universities. Each of us is within our own world, perhaps a master of it, or desiring to give that appearance, at least, but most are blissfully unaware of the vast storehouse of knowledge that is available in the silos of the surrounding landscape.
This was brought most recently to my attention through a reading group in which I am involved. We’ve been reading a book entitled Proust was a Neuroscientist by Jonah Lehrer. The basic premise of the book is that certain artists have anticipated discoveries by neuroscientists. He does this by proposing several putative examples of such, and whether he succeeds in any particular case I leave his readers to decide for themselves.
In my opinion, he has pointed to something much more important than what he believes he has.
Artists and scientists live in the same world and each of these groups attempts to describe that world in its own language. An illuminating example of this occurs in Lehrer’s chapter on Whitman, who had experienced the Civil War first hand. As a part of this, he took part in amputations and discovered that amputees often still feel the amputated limbs. This information, apparently, didn’t appear in scientific literature until much later.
I find it hard to believe this was unknown until Whitman’s day. Indeed, I can imagine a caveman having had an arm chewed off by a saber tooth tiger still feeling the arm and believing that its spirit still lingered.
It is instructive because it allows us to see what Whitman did. He raised the phenomenon into a higher level of conversation. He raised its talk among small groups of army surgeons, amputees, etc into the broader world. When scientists wrote on the subject, this is exactly what they did as well, but when they did it was in their own language.
The language of the arts is different from that of the sciences. The one thrives on ambiguity and multiplicity of interpretation, and the other on precision, but each is observing the same world and, as a result of these, both will stumble upon the same truths from time to time.
The book group to which I belong consists of two mathematicians, a psychiatrist, a historian, a political scientist, a physicist, a retire high school social science teacher, and a retired professor of Spanish. Proust was a Neuroscientist resulted in many fruitful conversations because of the topics it juxtaposed, not because of the information it offered.
Lehrer is a journalist, not a scholar, and members of the group are more comfortable with scholarly books which cite sources in a more scholarly way. He is also relatively young, so there are areas in which his reach far exceeds his grasp. Even at that, he is a writer who has the talent to convince the reader he knows what he is talking about even when this is, in fact, not true.
This is a talent much valued among journalists.
But even as I criticize the author, I have to ask whether such journalistic pidgin might be necessary when facilitating exchange between two groups with such different languages. To truly understand art as an artist does, does one need to be an artist? To grasp science in a full way does one need to be a scientist?
I believe the answer is no, but I also believe to deal well with both simultaneously would require something I call mileage. This is a quality I don’t see Lehrer possessing in spite of his talent as a writer.
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