MT00702/SL26601
Ideas in Mathematics: The Grammar of Numbers
Brief notes

1. 1/12/98
2. 1/14/98
3. 1/16/98
4. 1/21/98
5. 1/23/98
6. 1/26/98
7. 1/28/98
8. 1/30/98

1/12/98

• Review of syllabus.
• Summing powers of 2 to illustrate the concept of mathematical proof.
• Definition of language and grammar.

1/14/98

• Common occurrences of the number 9:
  • 9 lives of a cat
  • Rotate 9 by 180 degrees to get 6
  • J.S. Bach: "J+S"=9+18=27
  • String intervals: 8:9=major second, 9:10=minor second
  • 9 innings in a baseball game; 9 players on a side
  • Beijing: city center has 8 streets leading from it
  • Chinese word for 9 is a homophone for the word for "long"
  • 9 fathers of mankind in Chinese tradition (3 in Biblical tradition)
  • Other Chinese uses of 9: 9 areas of the sky, 9 countries of the earth, each country has 9 mountain ranges, with 9 passes between them. 9 islands in the ocean, 9 rivers (which are also 9 dragon heads)
  • Plato died at age 81
  • 9 planets. Mnemonic: "My very educated mother just showed us nine planets."
  • Ptolemaic cosmology: 9 spheres in heaven
  • Troy was besieged for 9 years
  • Ulysses wandered for 9 years
  • Pregnancy lasts approximately 9 months
  • Luke 15 (preceding the parable of the prodigal son): 99 sheep are mentioned explicitly, 9 found coins are inferentially noted
  • 9 vowel sounds (eight plus schwa)
• Testing divisibility by 9 by adding up the digits of a number
• Introduction to casting out nines

1/16/98

• More about divisibility by 9: Write n%9 for the remainder when n is divided by 9. For example, 23%9 = 5, and 45%9 = 0.

  Then let abc be a 3-digit number. We know that abc really means 100a + 10b + c. Rewrite that as 99a + a + 9b + b + c, and re-order that to become (99a + 9b) + a + b + c, and then 99a + 9b is a multiple of 9.

  Therefore, if a+b+c is a multiple of 9, so is abc, and vice versa. Moreover (100a+10b+c)%9 = (a+b+c)%9 by the same reasoning.

  That clarifies the idea behind casting out nines:

  523	-->	5+2+3=10	-->	1+0=1
  +131	-->	1+3+1= 5		5
  654	-->	6+5+4=15	-->	1+5=6

  What is really happening is that we are checking that 523%9 + 131%9 = 654%9.
• Other facts about 9: "the full 9 yards," "dressed to the nines."
• A novena is nine days of devotion.
• "Nine" and "new" seem to have similar roots in Latin and in many Indo-European languages. One possible explanation is that if one counts on fingers, ignoring thumbs, then 8 is as high as one can get, and so 9 is the "new" number after the eight fingers.
• Magic squares go back to ancient Chinese history. The oldest one seems to be

  1	2	3
  4	5	6
  7	8	9

  In this magic square, both diagonals add to the same number (15), and so do the middle row and middle column (4+5+6=15, 2+5+8=15). We call this an "imperfect" magic square, because the other rows and columns do not add up to 15.

  In a perfect magic square, the rows, columns, and diagonals all add up to the same number. Our first job is to figure out what that number will be.

  If the magic square contains the numbers 1-9, then the sum of all three of the rows will be 1+2+3+4+5+6+7+8+9. How do we add up that sum?

  Write

  S =	1	+ 2	+ 3	+ ... +	7 +	8 +	9
  S =	9	+ 8	+ 7	+ ... +	3 +	2 +	1
  2S =	10	+ 10	+ 10	+ ... +	10 +	10 +	10

  That means that 2S = 90, so S=45. Therefore, each row adds up to S/3 = 15.

  Now some trial and error shows that the number 5 must be in the center of the magic square. If instead, we tried putting 6 in the center, we would get

  	6

  and now there is no place to put the number 9. Similar reasoning rules out the numbers 7, 8, and 9 in the center.

  On the other hand, putting 4 in the center gives
  

  	4

  
  and now there is no way to put 1 in the magic square. Similar reasoning rules out putting 1, 2, and 3 in the center.

1/21/98

• An interesting numerical fact about 9:
  

  0*9	+ 1 =	1
  1*9	+ 2 =	11
  12*9	+ 3 =	111
  123*9	+ 4 =	1111

  
  and so on.
• Another interesting numerical fact about 9:
  

  12345679 *	9 = 111...
  12345679 *	18 = 222...
  12345679 *	27 = 333...

  
  and so on.
• Take a 3-digit number abc, where a is not the same as c. Reverse the digits to get cba, and subtract the smaller number from the larger. In your answer, the middle digit will always be 9, and the ones and hundreds digit will add to 9.
• There is a staircase method which generates magic squares of any odd size. Here is a brief sketch of the method.

  Start by placing 1 in the top row in the middle. Then proceed by placing numbers sequentially heading in the north-east direction. Whenever you "fall off" the magic square, just continue on the other side. Whenever you find that you can't go north-east, head south instead.

  For example, start with
  

  	1

  
  and head north-east (and wrap around from top to bottom) to get
  

  	1
  		2

  
  Again, head north-east (and wrap around from right to left) to get
  

  	1
  3
  		2

  
  At this point, we can no longer head north-east, and so we head south instead to get
  

  	1
  3
  4		2

  
  Now we can go north-east twice to get
  

  	1	6
  3	5
  4		2

  
  We again have to head south, to get
  

  	1	6
  3	5	7
  4		2

  
  Now going north-east and wrapping around from right to left gives
  

  8	1	6
  3	5	7
  4		2

  
  and finally we head north-east and wrap around from top to bottom to get
  

  8	1	6
  3	5	7
  4	9	2

  
  

• There are also multiplication squares, in which the product of rows, columns, and diagonals are constant. The smallest such constant is 216, and the square is
  

  12	1	18
  9	6	4
  2	36	3

  
  There are division squares, in which the product of the outer 2 numbers divided by the central one is a constant. For example,
  

  3	1	2
  9	6	4
  18	36	12

  
  has constant 6.
• Another interesting cultural fact about 9 concerns Beethoven's ninth symphony. Haydn and Mozart had each written far more than 9 symphonies, but Beethoven's nine symphonies, and particularly his ninth, exerted a power over many nineteenth century composers. Mendelssohn, Schumann, and Brahms all wrote far fewer than nine symphonies.
  Anton Bruckner apparently experienced writer's block, and spent more than five years unsuccessfully trying to finish his ninth symphony, working on it up to the day that he died. Trying to avoid this same fate, Gustav Mahler called the work after his eighth symphony "Das Lied von der Erde" (The Song of the Earth) rather than numbering it as a symphony. Mahler then wrote another symphony, and died before either The Song of the Earth or his ninth symphony were performed.
• The phrase "nine tailors make the man" refers to a medieval English custom, in which 9 bell tolls announced the death of a man, and 6 bell tolls announced the death of a woman. Eighteen bell tolls is obviously ambiguous, since it can announce either the death of 2 men or 3 women. On the other hand, 20 bell tolls must show a mistake on the part of the listener, since 20 is not a multiple of 3.
  If we change the system to 9 and 7 tolls respectively, we see that there are far fewer ambiguous numbers.
• Let's construct a magic square more slowly. Start with 5 in the center:
  

  	5

  
  and try to see where we can put in 1 and 9.

  If we try
  

  1
  	5
  		9

  
  see that there is no number that can be placed in the upper-right corner (or the lower-left, for that matter) that can make the square work. This shows that 1 and 9 cannot go in the corner of any square.

  Try instead
  

  2
  	5
  		8

  
  and there is no obvious problem yet. Try to place the number 1 in the square, and we can't have either
  

  2	1
  	5
  		8

  
  or
  

  2
  1	5
  		8

  
  But we can try instead
  

  2
  	5	1
  		8

  
  and now there is no choice how to proceed. We must have
  

  2		6
  9	5	1
  		8

  
  and that forces
  

  2	7	6
  9	5	1
  4		8

  
  and in turn that means that we must conclude with
  

  2	7	6
  9	5	1
  4	3	8

  
  Now the only thing to do is check and see if this works.

• We also notice that rotation of a magic square by ninety degrees gives a magic square. Call that r. Also, taking the mirror image of a magic square gives a magic square. Call that m, for mirror image. Combinations of r and m give a total of 8 magic squares in all.

1/23/98

Indian mathematicians learned many tricks to simplify multiplication of numbers. They are named after the first word of a sutra. Today's class will talk about a method called nikhilam. navata´s' caramam. da´satah, which translates as "continually from 9, the last from 10." The name refers to the method one uses to subtract a number from a power of 10.

For example, to subtract 657 from 1000, one subtracts 6 from 9, 5 from 9, and 7 from 10, to get 343.

Nikhilam works when subtracting two numbers with the same number of digits which are both close to some base, which we will take as a power of 10. In the following, the base will be indicated in parentheses, and will be denoted symbolically by b. The two numbers to be multiplied are called the two givens, and will be written g1 and g2. The difference between the base and the givens is called the deficit, and will be denoted d1 and d2.

Suppose that the two givens are 9 and 7. Here's how to compute their product using nikhilam, using 10 as the base.


(10)
g1 =	9	---	1	= d1
g2 =	7	---	3	= d1
g1 + g2 - b or b - (d1 + d2) or g1 - d2 or g2 - d1	= 6	|	3	= d1 * d2

Similarly,


(10)
7	---	3
6	---	4
3	|	12


giving a result of 30 + 12 = 42.

Another example with larger numbers, and a base of 100:


(100)
93	---	7
94	---	6
87	|	42


giving a result of 8,742.

Another example with a base of 100:


(100)
25	---	75
98	---	02
23	|	150


giving a result of 2300 + 150 = 2,450.

One large example:


(10,000)
9997	---	0003
9997	---	0003
9994	|	0009


giving a result of 99940009.

1/26/98

• Review of 9 tailors homework.
• How many branches are there on a menorah?
• More methods for construction of a magic square. Let the order of a magic square be the number of rows (or columns).
• The knight's move method is to begin with a 1 in the middle of the top row, and then make knight's moves (2 up, 1 to the right) to fill in consecutive numbers. If the square is occupied, drop down one rather than make the knight's move.
  Conceivably, the "fetch" (knight's move) has to be adjusted depending on the order of the square, so that it might be more or less than 2 squares vertical depending on the size of the square.
• Another method is the pyramid method. Take the magic square, and extend it outwards in a pyramid, so that
  

  *	*	*
  *	*	*
  *	*	*

  
  becomes
  

  		*
  	*	*	*
  *	*	*	*	*
  	*	*	*
  		*

  
  Then fill in consecutive numbers in diagonals from lower left to upper right:
  

  		3
  	2	*	6
  1	*	5	*	9
  	4	*	8
  		7

  
  and then place the numbers in the 3x3 square as usual to get
  

  		3
  	2	7	6
  1	9	5	1	9
  	4	3	8
  		7

  
  
• The synthetic method involves creating 2 squares A and B, and then combining them.

  For square A, take the number (order-1), (Note: See correction below.) and place it in a square from the upper left to the lower right:
  

  2
  	2
  		2

  
  Then complete each row, by taking the series 1,2,3 (in general, 1,2, ..., order), and repeating it cyclically (1, 2, 3, 1, 2, 3, 1, 2, 3, ...):
  

  2	3	1
  1	2	3
  3	1	2

  
  That gives square A.

  For square B, take the number (order-2), (Note: See correction below.) and place it in a square from lower left to upper right:
  

  		1
  	1
  1

  
  and now repeat the series 0, 1, 2 cyclically (in general, the series will be 0, 1, ..., order-2):
  

  2	0	1
  0	1	2
  1	2	0

  
  Now take A + (B*order) to get the magic square:
  

  8	3	4
  1	5	9
  6	7	2

  
  

  This method was invented by de la Hire in 1705. It can be sped up by initially putting into the second square the series 0, 3, 6 (in general, 0, order, 2*order, ...) to simplify the arithmetic.

• Indian mathematics gives a quick way of squaring numbers, called "yavadunam tavadunikrtya varganca yojayet" (as much the deficit, so much less doing, square, and let one add).

  Call the number to be squared the given g, and the deficit d. To compute the square of 9, we have:
  

  (10)
  g =	9	---	1	= d
  g - d =	8	|	1	= d * d

  
  giving a result of 81.

  Similarly, to square 8, we have
  

  (10)
  8	---	2
  6	|	4

  
  giving 64, and squaring 7 means computing
  

  (10)
  7	---	3
  4	|	9

  
  or 49.

  We can also do this method with numbers larger than the base, in which case the number to the left of the virgule (|) is produced by adding the given and the surplus:
  

  (10)
  11	---	1
  12	|	1

  
  or 121, and
  

  (10)
  19	---	9
  28	|	(8)1

  
  giving 280+81=361 as an answer.

  That last problem can also be done using a base of 20:
  

  (20)
  19	---	1
  18	|	1

  
  where the answer must be interpreted as 18*20 + 1, or 361.

• Another note about number names. Consider the following chart:
  

  Sanskrit	nava	da'sa
  Indo-European	newm	de'km
  Russian	devjat'	desjat
  	9	10

  
  The Russian word for 9 should begin with the letter "n", but in fact it begins with the letter "d". What happened? The "d" that begins the word for 10 has migrated to the word for 9.

1/28/98

• A bit of terminology about magic squares. We have been using the term "magic square" to refer to what is often called a perfect magic square, in which the rows, columns, and both diagonals all add up to the constant. An associate square is a perfect magic square in which peripheral numbers (that is, numbers which are symmetrically placed around the central square) add up to a fixed number. A pan-diagonal magic square is one in which broken diagonals add to the constant. In the diagram below, a diagonal is marked with the letter "d", and a broken diagonal with the letter "b".
  

  d	b
  	d	b
  		d	b
  			d	b
  b				d

  
  
• A correction to the synthetic method discussed in the last class. In the A square, the number to be placed on the diagonal from upper left to lower right is (order+1)/2, and then, as before fill in each row with the numbers from 1 to order cyclically. In the B square the number to be placed on the diagonal from lower left to upper right is (order-1)/2, and then, as before, fill in each row with the numbers from 0 to order-1 cyclically.
• Why does the synthetic method work? Because it is so algebraic, we can see a lot of the reason for the success of the method. (In contrast, the knight's move method involves a lot more geometry, and checking it is more elaborate.)

  Before beginning, remember the formula 1+2+...+n = n(n+1)/2, which can be derived using the methods of an earlier class. This formula allows us to figure out the constant for an order n square. We know that the square contains the numbers 1, 2, ..., n^2, and so the sum of all the rows has to be 1+2+...+n^2 = n^2(n^2+1)/2. Since there are n rows, the sum of each row has to be n(n^2+1)/2. Therefore, the constant is n(n^2+1)/2.

  Consider the A square first. Each row and each column contains the numbers from 1 to n. Therefore, each row and each column add up to 1+2+...+n = n(n+1)/2.

  In the B square, each row and each column contains the numbers from 0 to n-1, and so each row and each column add up to 0+1+...+(n-1)=(n-1)n/2.

  Remember that we are supposed to get the magic square by computing A+n*B. Therefore, in the resulting magic square, each row and each column add up to
  

  n(n+1)/2 + n[(n-1)n]/2 = (n^3+n)/2.
  A bit of checking shows that this is the constant that we need.

  Finally, what about the diagonals? Start with the diagonal from upper left to lower right. In the A square, each number on that diagonal is (n+1)/2, and so the sum of that diagonal is n(n+1)/2. In the B square, every number from 0 to n-1 occurs on that diagonal, and so the sum of the diagonal is 0+1+...+n-1=(n-1)n/2. Do the algebra, and you see that the diagonal of A+n*B again gives the right constant.

  Take the diagonal from upper right to lower left. On that diagonal, the A square contains every number from 1 to n, and so the sum is 1+2+...+n=n(n+1)/2. The B square contains the constant (n-1)/2 in each spot on that diagonal, and so the sum of the B diagonal is n(n-1)/2. Again, the sum of the diagonal on A+n*B gives the right constant.

• Why does nikhilam work? First, show that all 4 formulas for the number in the left-hand column are equivalent. The formulas are above. The one that we will use is b-d1-d2.

  We are multiplying g1*g2. Write that as (b-d1)*(b-d2), and then use the algebraic formula (x-a)*(x-b)=x*(x-a-b)+ab. We get
  

  (b-d1)*(b-d2) = b*(b-d1-d2) + d1*d2.
  The d1*d2 is the number on the right side of the vertical line, and the other term is the number on the left side of the vertical line.
• A few other nine phrases to wrap things up. A nine-day wonder is another term for fleeting celebrity. Nine times out of ten refers to most of the time. A nine-killer is another term for a shrike. And 999 is the British system for phoning the police, similar to the American 911 (but introduced in 1937, much earlier than the American number).

1/30/98

• Quiz.
• Here's an interesting party trick which requires memorizing one particular order 3 magic square. Suppose that you have memorized
  

  8	1	6
  3	5	7
  4	9	2

  
  At a party, you can then draw an empty square, and challenge someone to fill in any number that they choose, any place that they choose. Suppose that they fill in 68 like this:
  

  		68

  
  You can then quickly figure out that 68 is 62 more than 6, which is the corresponding number in the magic square that you have memorized, and add 62 to each number in your original magic square to get:
  

  70	63	68
  65	67	69
  66	71	64

  
  which is a magic square.
• A bit more on the synthetic method. For an order 5 magic square, the method discussed in class was to create an A square containing the number 3 on one diagonal, with the numbers 1 through 5 filled in cyclically to get:
  

  3	4	5	1	2
  2	3	4	5	1
  1	2	3	4	5
  5	1	2	3	4
  4	5	1	2	3

  
  The B square was supposed to contain the number 2 on the other diagonal, and the numbers 0 through 4 filled in cyclically, yielding
  

  3	4	0	1	2
  4	0	1	2	3
  0	1	2	3	4
  1	2	3	4	0
  2	3	4	0	1

  
  and then compute A + 5*B to get the magic square.

  But in fact, there is far more latitude in creating the A and B squares. You can fill in any first row you like for the A square, provided that it begins with the number 3, and then shift it one square to the right each time to get the next row. For example, the A square could be
  

  3	5	1	4	2
  2	3	5	1	4
  4	2	3	5	1
  1	4	2	3	5
  5	1	4	2	3

  
  

  Similarly, the B square can contain any first row, provided that it ends with the number 2, and each row underneath is gotten by shifting the previous row one to the left. For example, the B square could be:
  

  3	0	1	4	2
  0	1	4	2	3
  1	4	2	3	0
  4	2	3	0	1
  2	3	0	1	4

  
  And then A + 5*B will still give a magic square.
• There are also several methods of getting a new magic square from an old one. When we talked about order 3 magic squares, we discussed that rotation and mirror image can give new magic squares. But there are other things that can be done with order 5 (in fact, with any odd order and sometimes with any order) magic squares to create new ones.

  Notice first that switching any two rows will not affect the row sums or the column sums of a magic square. However, usually the diagonal sums will be completely altered. Imagine, though, first switching the first and last rows of a magic square, and then the first and last columns. This will not affect the diagonal sums.

  Even more complicated, you can switch the first and second rows, and the fourth and fifth rows. This completely alters the diagonal sums. However, if you now switch the first and second columns, and the fourth and fifth columns, the diagonal sums will be restored to their constant value.

• One can also swap quadrants of a magic square to get a new one, though this is a bit trickier than what was described in class. Label the quadrants like this:
  

  I		1	II
  		7
  4	6	13	20	22
  III		19	IV
  		25

  
  Then swapping quadrants I and IV, and quadrants II and III, will keep the diagonal sums correct. You then need to swap the two halves of the center row, and the two halves of the center column, to make a new magic square.
• Given an order 5 magic square, it is easy to create an order 25 magic square by taking the basic order 5 magic square, and repeatedly entering it into the order 25 magic square, adjusting the numbers accordingly as you go along.
• A bit more about nikhilam. Suppose that we want to compute
  

  29

  + 48

  
  We can instead think of 29 as 30-1, and write it as 31, where we will use color to substitute for a line over the 1. The line over the 1 is called a vinculum, which is Latin for link. We would then have
  

  31

  + 48

  77

  
  

  Remember how we would multiply 103*104:
  

  (100)
  103	+3
  104	+4
  107	| 12

  
  giving an answer of 10712.

  What about 106*97? We have
  

  (100)
  106	+6
  97	-3
  103	| 18

  
  and we have to interpret 10318 as 10300-18=10282.

February notes.
March notes.
April notes.