1.0 Fun with Numbers:
Before
we study numbers and their properties, let us familiarise few interesting
things about numbers
Telling
a number:
Can
you tell a number a friend has in his mind without him revealing that number to
you?
Here
are the steps
|
Step No |
Action |
Example |
|
1 |
Ask your friend to keep
a number in his mind |
Say 27 |
|
2 |
Ask him to multiply
that number by 2 |
54 |
|
3 |
Ask him to add 4 to that |
58 |
|
4 |
Ask him to divide the
number by 2 |
29 |
|
5 |
Ask him to reveal the
number to you |
29 |
Now
you subtract 2 from that number and
tell him that his number is 27!
What
is the mathematics behind that?
Let
x be the number.
|
Step No |
Action |
Example |
|
1 |
Ask your friend to keep
a number in his mind |
Say x |
|
2 |
Ask him to multiply that
number by 2 |
2x |
|
3 |
Ask him to add 4 to that |
2x+4 |
|
4 |
Ask him to divide the
number by 2 |
x+2 |
|
5 |
Ask him to reveal the
number to you |
x+2 |
When
you subtract 2 from that you get x
which is the number he had in his mind
Can
you construct a different method by using different multiplication factor in
stead of 2 and also different
additional factor instead of 4?
There
are innumerable numbers of ways, we can formulate above type of tricks/puzzles.
Let
us now look at some interesting facts about some numbers.
Speciality
of numbers from 1 to 9:
1.
Consider
the number 123456789. Sum of individual digits of this number is
45(=1+2+3+4+5+6+7+8+9)
2.
Now
multiply 123456789 by 2 and we get 246913578. Again sum of individual digits of
this number is also 45(=2+4+6+9+1+3+5+7+8) Also note that no digit repeats in
the result and no digit is missing between 1 and 9. What do you observe when
123456789 is multiplied by 4,5,7,8?
3.
Let
us look at the multiples of 9: They are 9, 18, 27, 36, 45, 54, 63, 72, 81,
90,99,108,117..( If we add the individual digits of these multiples we get the
sum as 9(1+8=9,2+7=9.. )
Properties
of squares:
What did we notice?
The result of squaring a number can also be arrived at by
progressively adding consecutive odd numbers as shown above
Is
this surprising? There is mathematics behind it if we know the formula for
formula for (a+b)2 which we learn later in algebra
Let
us look at the expansion for (n+1)2 for any number.
We
learn later that (n+1)2= n2+2n+12= n2+(2n+1).
Note
here that here 2n+1 is a consecutive odd number after the odd number in n2
Meru Prastara: Observe the following arrangement of numbers:
In
this triangle type of figure, at the tip is a small square containing the
number 1, which makes up the 0th row. In each row, value at the
extreme ends of the row on both sides are said to have value 0. The first row
(1 & 1) contains two 1's, both formed by adding the two numbers above them
to the left and the right, in this case 1 and 0. Do the same to create the 2nd
row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1;
1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle
go on forever. A number in the triangle can also be found by nCr
(n Choose r) where n is the number of the row and r is the order of element in
the square corresponding to that row. For example, in row 3, 1 is the 0th
element, 3 is first element, the next 3 is the 2nd element, and the
last 1 is the 3rd element. The formula for nCr
is: 
(We
learn the concept in the chapter on Permutations and Combinations later in a
higher class)
First reference to this type of arrangement was formed by
Pingala(Period : 2nd BC). He used this concept in Chandas Shastra(
Study of syllables). It was called Meru Prastara( Steps of
Mountain Meru ). Halayudha a 10th Century
Mathematician provided commentary on this arrangement. As is the case with
crediting all the discoveries of Indian mathamaticians after westerners, this triangle
is now called Pascal triangle named after Pascal(17th Century) mach
after its discovery and relevance 1900 years ago!
Now lets us look at the some of the properties of this
Pingalas Meru Prastara.
Properties:
Let n be the row number starting from 0
1.
Sum
of the numbers across a horizontal row is always 2n .( Ex 1 = 20
1+1=2=21, 1+2+1=4= 22 1+3+3+1=8=23,,
Denoted by line marked as blue)
2.
The
numbers when arranged across a horizontal row represents 11n ( Ex:
1= 110,11 = 111 ,121= 112, 1331= 113Denoted
by line marked as red )
3.
The
numbers across a diagonal represent natural numbers (1,2,3. Denoted by squares
in red colour)
4.
The
numbers across a diagonal when added with previous numbers is a perfect square
(1= 12, 1+3=4=22, 3+6=9=32, 6+10=16= 42,.
Denoted by squares in green colour)
(Perfect square is a number made by squaring a whole
number)
5.
If
the 2nd element in a row is prime number other
than 2, then all next
numbers in that row are multiples of it( 3, 3 in Row 3, 5,10,10,5 in Row5,
7,21,35,35,21,7 in Row 7)
Let
us look at the above arrangement from a different perspective
The
numbers indicated in red are the sum of
numbers in diagonals. They are
1
1 2 3 5 8 13 21 34 55 89
Except
for first number 1, each of the subsequent number is sum of previous 2 numbers
(13=5+8, 21=8+13, 34=13+21..)
Let
us now look at another interesting mathematics concept called Magic square
What do we observe in the figure? The figure has 3 rows
(R1, R2 and R3). It also has 3 columns (C1, C2 and C3) and in all has 9 squares
(3*3). Squares are filled with numbers from 1to 9 without any number repeating
and without any number missing from 1 to 9. We also observe
Sum of numbers in every Row (R1, R2 and R3) is 15
Sum of numbers in every column (C1, C2 and C3) is 15
Sum of numbers across 2 diagonals (joined by red line) is
also 15.
15 is called magic sum. Magic sum is the sum of all
numbers in any row/column/diagonal in a magic square.
Observe the below mentioned figure
Above is another magic square whose magic sum is 30 and
squares have all even numbers without any repetition and without any missing
number starting from 2 to 18.(In all 9 even numbers)
Construction of magic square is very simple if you can remember
the right angled triangle
For easy remembrance rule can be simply stated as
Diagonal, Down, Left
Rules:
Start with placing the number in the top row and middle
column in an odd numbered square, and next number in the diagonal square in the
top
|
Rule No |
Action |
|
1 |
If
there is no row and no column for a number then place the number immediately
below it else place
it in the bottom most square in the same column |
|
2 |
If the
square is already occupied, then place it immediately below in the same
column |
|
3 |
If
there is no column then place the number in extreme left square in the same
row |
Let us see now how to use the above rules to construct the
Magic square given below
|
Step No |
Action(Row
R0 and Column CL are only for demonstration) |
Movement
(Numbers in row R0 and Column CL have numbers which we place diagonally before
they are moved correctly in the magic square) |
|
1 |
Start
with number 1 in middle row, middle column(R1,C2) |
|
|
2 |
Try to
place next number 2 diagonally above, Since there is no row for that number,
but since there is column for that number, place it in the bottom most square
in the same column (R3,C3) Rule 1 else
Condition |
|
|
3 |
Try to
place next number 3 diagonally above, Since there is no column for that number,
place it in the extreme left square in the same row(R2,C1) Rule 3 |
|
|
4 |
Try to
place next number 4 diagonally above(R1,C2), Since that square is already
occupied , place it in the square below in the same column (R3,C1) Rule 2 |
|
|
5 |
Place
next 2 numbers diagonally. 5 in (R2,C2), 6 in (R1,C3) |
|
|
6 |
Try to
place next number 7 diagonally above, Since there is no row and no column for
that number, place it just below 6. (R2,C3) Rule 1 |
|
|
7 |
Try to
place next number 8 diagonally above, Since there is no column for that
number, place it in the extreme left square in the same row(R1,C1) Rule 3 |
|
|
8 |
Try to place
next number 9 diagonally above, Since there is no row for that number, but
since there is column for that number, place it in the bottom most square in
the same column (R3,C2) Rule 1 else
Condition |
|
Thus
with just 3 easy rules and Diagonal, Down, Left Concept we are able to create a 3 by 3 Magic
square
Similarly one can create 5 by 5 and 7 by 7 Magic square an
example of which is given below
In the above examples, 16 and 29 are marked in red to indicate
that only these 2 numbers do not follow the else
part in rule 1.
Now construct a 7 by 7 magic square with odd numbers
starting from 1 using the rules stated above as follows:
Construction of magic square with magic sum of any given
number
In the above figure note that sum of numbers in 4
rows(R,,R4), sum of numbers in 4 columns(C1,,C4) and sum of numbers in 2
diagonals(crossed by red line) are all same and is equal to 45.
How do we construct such magic squares?
Let x be the given Magic sum for which we need to
construct the magic square. We use 9 in 4 different ways to fill the some of
the squares in a 4 by 4 square.
|
Step/Figure |
Action |
Movement |
|
1 |
Note 9= 5+4, use these 2 numbers (5 and 4) to fill
the squares in the squares of last 2 rows(R3 and R4) in the first column(C1) |
|
|
2 |
Note 9=1+8, use these 2 numbers(1 and 8) to fill
the squares in the squares of first 2 rows(R1 and R2) in the second column(C2) |
|
|
3 |
Note 9= 3+6, use these 2 numbers (3 and 6)to fill
the squares in the squares of last 2 rows(R3 and R4) in the third column(C3) |
|
|
4 |
Note 9=7+2, use these 2 numbers(7and 2) to fill
the squares in the squares of first 2 rows(R1 and R2) in the fourth column(C4) |
|
|
5 |
Place 9 in the last row(R4) of last column(C4) |
|
|
6 |
Place 10, 11, 12 in the squares in
(R3,C2),(R2,C1) and (R1,C3) respectively as shown in the figure 6. |
|
|
7 |
If the magic sum required is X, then place X-20,X-19,X-21,X-18
in (R1,C1), (R4,C2),(R2,C3) and (R3,C4) respectively as shown in the figure
7. |
|
Note the following in the figure/step number 7.
·
Sum
of numbers in Row R1= X-20+1+12+7= X
·
Sum
of numbers in Column C4 = 7+2+X-18+9=X
·
Sum
of numbers in the diagonal top to bottom= X-20+8+3+9=X
·
Similarly
one can observe that sum of numbers in other rows, other columns and another
diagonal is X
If we want a magic square whose magic sum required is 45,
then substitute 45 for x in the last figure to get the desired magic square as
in figure 8 which is what we started with as an example
Having seen and understood some of interesting facts about
numbers, let us learn about divisibility tips which are useful in case of
simplification:
|
Divisible
by: |
If: |
Examples: |
|
2 |
The last digit is even (0,2,4,6,8) |
128
Yes 129
No |
|
3 |
The sum of the digits is divisible by
3 |
381
(3+8+1=12, and 12 4)
Yes 217
(2+1+7=10, and 10 3 1/3)
No |
|
4 |
The number formed by last 2 digits is
divisible by 4 |
1312
Yes (12) 7019
No |
|
5 |
The last digit is 0 or 5 |
175 Yes 809
No |
|
6 |
The number is divisible by both 2 and
3 |
114
(it is even, and 1+1+4=6 and 6 2)
Yes 308
(it is even, but 3+0+8=11 and 11 3 2/3)
No |
|
7 |
If you double the last digit and
subtract it from the rest of the number and the answer is: ·
0,
or ·
divisible by 7 (Note: you can apply this rule to
that answer again if you want) |
672 (Double of 2
is 4, 67-4=63, and 63)
Yes 905 (Double of 5
is 10, 90-10=80, and 801 3/7)
No |
|
8 |
The number formed by last three
digits is divisible by 8 |
109816
(81602) Yes 216302
(3027 3/4) No |
|
9 |
The sum of the digits is divisible by
9 |
1629 (1+6+2+9=18, and again, 1+8=9) Yes 2013 (2+0+1+3=6) No |
|
10 |
The number ends in 0 |
220
Yes 221
No |
|
11 |
If (sum of digits in even paces- sum
of digits in odd places)= ·
0,
or ·
divisible by 11 |
1364 ((3+4) - (1+6) = 0) Yes 3729 ((7+9) - (3+2) = 11) Yes 25176
((5+7)
- (2+1+6) = 3) No |
|
12 |
The number is divisible by both 3 and
4 |
648
524
|
1.1 Summary of learning
|
NO |
Points studied |
|
1 |
Meru Prastara Different types of majic squares |
|
2 |
Divisibility
test |
