MULTIPLY NUMBERS FROM 101 THROUGH 109
      In this case the examples would be:-
         101x101
         104x107
         1 05 x105
        106x108
        109x109
And so on----
The total examples will be 45.
Procedure:-
•  The answer will be a five digit number beginning with 1.
•  The next two digits of the answer will equal the sum of ones digits.
•  The last two digits of the answer will equal the product of the ones digits.
Let’s use the trick.
Consider the problem: 102x107.
Step1:- Begin the answer with 1 ---.>   1
Step 2:-Add the ones digits    ------>2+7= 09.
Step 3:- Multiply the ones digits --->2x7=14.
Step 4:-Combine the amount from steps 1,2.and3 writing from left ------>10,914.
Answer 10,914.
Let’s try one more example.
109x105
Step 1:-Begin the answer with 1------------ .>1
Step 2:- Add the ones digits----------- ----->9+5=14.
Step 3:- Multiply the ones digits----------->9x5=45.
Step 4:-Combine the amounts for the steps 1, 2, and 3 .Writing from left to right----->11,445.
Answer is 11,445.
Note :-the sum or the product of ones digits is written in two  ,digits as01,02,03,04,05,06,07,08, and 09.
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SQUARING ANY TWO DIGIT NUMBER ENDING IN ONE OR NINE

Squaring any two digit number ending in 1 or 9 is very interesting and easy method. Two digit numbers ending in 1 are from 11 to 91.Two digit numbers ending in 9 are from 19 to 99.

   The trick is to multiply together the two whole numbers on either side of the number you are squaring, and then add one.
Let’s see how it works.
Problem:-
 31x31 (square of 31)
Step 1:-find the two whole numbers on either of 31 that are 30 and 32
Step 2:-multiply the two whole numbers.30x32=960.
Step 3:- add one to 960 The answer is 961.

Square of 31 is 961.
One more example:
61x61
The two whole numbers on either side of 61 are 60 and 62.
Product of 60 and 62 is 3720.
Adding 1 to 3720 we get 3721.
The answer is 3721.

SQUARE OF ANY TWO DIGIT NUMBER ENDING IN 9..
The method is same as above.
Consider an example of any two digit number ending in 9.

Suppose the number is 19.
Step 1:- find the two whole numbers on either side of 19, that are 18 and 20.
Step 2:- multiply the two whole numbers.
 18x20=360.
Step 3:- adding 1 to 360 ,we get 361.
Square of 19 is 361.

This trick makes the calculation easier by changing one of the numbers into a multiple of 10..

WONDERFUL FOUR

Think of any number.
The number is written in figures and in words.
Suppose we are thinking of 345.
 345 is in figures.
 In words this can be written as ‘three hundred forty five.’
How many letters are in these words?
Count the words.
The letters are 20.
Write 20 in words ‘TWENTY’
Count the letters in twenty.
Number of letters in twenty are 6.
6 is written in words ‘SIX’
Again count the letters in .six.
There are 3 letters in six.
3 is written in words as three.
Three has 5 letters.
5 is written in words as   five.
Five has 4 letters.
4 is written in words a four.
Four has 4 letters.
Try another example.
At last we will come to FOUR.
Does this always happen? What about other languages?

ALWAYS FIVE AND HALF

Think of any digit 1 through 9 in your mind..

Multiply  the number by 11.

Divide the product by its sum of digits.

Answer is always 5.5(five and half)

Example:
Suppose we have chosen the number 8.
8 multiplied by 11.The product is 88.
The sum of digits of 88 is8+8=16.
The product 88 is divided by 16 is 5.5
Answer is 5.5.

Let’s try one more example.
We have chosen the number 7.
7*11=77
Sum of digits =7+7=14.
77/14=5.5.
The answer is five and half.

Triangular Numbers

Pythagoras, the Greek mathematician is known as ‘The father of numbers.’ He explored the mathematical relationship with figurate numbers.
Pythagoras discovered triangular numbers. Triangular numbers are also called Triangle numbers. Pythagoras studied the triangular numbers under figurate numbers in geometry.
First natural number is 1.
Sum of the first two consecutive natural numbers is 1+2 =3
Sum of the first three natural numbers is 1+2+3=6
Sum of the first four natural numbers is 1+2+3+4=10
Sum of the first five natural numbers is 1+2+3+4+5=15
1, 3, 6, 10, 15…are triangular numbers. A    triangle number is the sum of first n natural numbers.
Fredric Gauss discovered the formula for nth triangular number.

Nth triangular number=n (n+1)/2.
For example: 6 th triangular number= 6(6+1)/2=  6(7)/2=  21

Let’s try one more example.
36 th triangular number =36 (36+1) /2
                                         =36 (37) /2= 666.
             666 is called the beast number.

• Every triangular number is a natural number.

• Every triangular number except 1 is divisible by 3 or when divided by 9 the remainder is 1.

• Triangular numbers can never end in 2, 4, 7, or 9.

• Eight times of any triangular number plus 1 is always a perfect square number.

• Nine times of any triangular number plus 1 is always a triangular number.

• Sum of any two consecutive triangular numbers is always a perfect square number.

• Difference between two consecutive squares of any triangular numbers is always a perfect cube of a small triangular number.

• Some triangular numbers are there which reversals are also triangular numbers. The examples are 1,3,6,10,55,66,120,153,171,190 300,595,630,666,820.3003,etc.

Sum Of Ten Successive Fibonacci Numbers

                                     

Sum of any ten successive Fibonacci numbers is eleven times the seventh number.

Observe the following sequence.
1,1,2,3,5,8,13,21,34,55,89,…

In this sequence first two numbers are 1and 1.Third number is the sum of the first two numbers. Fourth number is the sum of the second and the third number. Each subsequent number is the sum of the two numbers before it. This sequence is called Fibonacci number sequence.
 

Fibonacci was an Italian mathematician. He introduced the Hindu-Arabic numeration system
to the west countries. He is well known in the modern world for spreading the Hindu-Arabic
Numeration system. The sequence of numbers is named after him. In Fibonacci sequence the first two numbers are 1 and 1.It is not necessary to take first two numbers be 1and 1.Take any two numbers then form Fibonacci like sequence.

For example
5, 7, 12, 19, 31, 50, 81…

Consider the Fibonacci number sequence of 10 numbers.
1+1+2+3+5+8+13+21+34+55

In this sequence seventh number is 13 and the sum of all ten numbers is 143.143 is the product of 13 and 11.

This rule is applicable for any ten Fibonacci like sequence.

Another example:
5,7,12,19,31,50,81,131,212,343.
In this sequence seventh number is 81.
81x11=891.
Sum of all ten numbers is also 891.

How does this work?
Suppose a and b are any two numbers.
The sequence will be as follows,

a
b
a+b
a+2b
2a+3b
3a+5b
5a+8b
8a+13b
13a+21b
21a+34b

------------
55a+88b

Here seventh number is 5a+8b, when it is multiplied by 11 we get 55a+88b which is the total of all ten numbers.

MAGIC FRACTION OF ODD NUMBER SERIES STARTING FROM ONE

Series of odd numbers starting from 1 is   1+3+5+7+9+11+13+...
Suppose we take the first number of the series as a numerator, we have to take  the Next number in the series as the denominator. Here numerator is 1 and the denominator is 3.  The fraction will be 1/3.

  Now the first two numbers of the series are 1+3 as a numerator.
  Next two numbers are 5+7 .Take this as a denominator.   
  The fraction will be (1+3))/(5+7 )=4 /12.
  Reduced fraction = 1/3.
  In the same way continue the process ( 1+3+5)/(7+9+11)=9/27.
  Reduced fraction = 1/3

  SOME OTHER EXAMPLES:

    (1+3+5+7+9+11)/(13+15+17+19+21+23 )

    =36/108 

Reduced fraction is 1/3.

How many numbers we take from the odd number series starting from 1 as a    numerator, and the next same number of the series as a denominator, the reduced fraction is  always 1/3.

Subtraction Made Easy

Subtraction is one of the basic operations in mathematics. It is opposite of addition. Subtraction means the difference between minuend and subtrahend. If minuend is larger than subtrahend, the difference is positive. If the subtrahend is larger than minuend, the difference is negative.

 If there is no borrowing, the method is very easy. Generally students are bored when there is borrowing or regrouping in subtraction. There is some difficulty in understanding. It requires abstract thinking and drilling.

 There are several methods of subtraction in arithmetic. The subtraction by Vedic mathematics method is one of the easy and simple methods. It is based on the principle, “All from nine and the last from ten”. It is called NIKHILAM. Nikhilam is one of the 16 main sutras of Vedic mathematics.
To apply Nikhilam to a number, subtract each digit from 9, except the last digit. The last digit is to be subtracted from 10.

Example of subtraction
IF SUBTRAHEND IS LESS THAN MINUEND

Example #1
3 6 2 5 - 1 7 8 9
Here the Subtrahend is 1 7 8 9.
Apply the sutra “all from nine and the last from ten” to the subtrahend number.

9-1      9-7      9-8     10-9
8          2         1          1
Now, add the result to the minuend.
3 6 2 5 + 8 2 1 1
= 1 1 3 8 6
Drop the first digit that is 1 from the left.
The result is 1 3 8 6. This is the answer

Example # 2
4 5 2 9 - 2 3 8 0
As per Nikhilam the subtrahend will be
              
9-2    9-3    10-8    0
7        6        2        0
When the last digit at subtrahend is 0, keep 0 as it is.Therfore in the above problem 8 is to be subtracted from 10.
Now add 7  6  2  0 to 4  5  2  9.The result is 1  2  1  4  9.
Drop the first digit to the left the answer is 2 1 4 9.This is the answer.
Therefore  4  5  2  9  -  2  3  8  0  =  2  1  4  9.

Example #3
4  5  8  2  7  -  3  9  8.
Arrange this as , 4  5  8  2  7  -  0  0  3  9  8.
Applying Nikhilam to the subtrahend
9-0     9-0     9-3     9-9     10-8
9         9         6        0         2
The Subtrahend will be 9 9 6 0 2.
Add 9 9 6 0 2 to the minuend.
4  5  8  2  7  +  9  9  6  0  2  =  1  4  5  4  2  9.
Drop the first digit to the left, the answer is 4 5 4 2 9.
Therefore   4  5  8  2  7  -  3  9  8  =  4  5  4  2  9.

IF MINUEND IS GREATER THAN SUBTRAHEND

Example #1
3 5 1 - 4 9 7
Apply Nikhilam to the minuend.
9-3   9-5   10-1
6       4        9
The result is 6 4 9
Add 6 4 9 to the subtrahend.
4  9  7  + 6  4  9 = 1  1  4  6.
Drop the first digit to the left the result is 1 4 6.
Therefore  3  51  -  4  9  7  =  -  1  4  6.

ADVANTAGES.
• The method is easy to understand.
• No need of borrowing and carrying.
• It is a shortcut method.
• No frustration.
• Comfortable.

THINGS YOU SHOULD KNOW

                                                 THINGS YOU SHOULD KNOW
                                              -------------------------------------------------


1. What are addends?
    Numbers in addition are called addends.

2. What are cardinal numbers?
    Numbers used for counting are called cardinal numbers.

3. What is decline?
    When the value of number decreases then that number is called as decline.

4. What is minuend?
    A number, from which another number is subtracted, is called as minuend.

5. What is a numeral?
    A word or symbol used to represent a number is called a numeral.

6. What are ordinal numbers?
     Numerals that show order are called as ordinal numbers.

7. What is a prime number?
    A number that can be built only one way using tiles is called a prime number.

8. What is an odd number?
 An odd number is a whole number that has a 1, 3, 5, 7, and 9 in ones      place.When an odd number is divided by two there will be always a remainder greater than zero.

9. What is a progression?
    A sequence of numbers that has a fixed pattern.

10. What is subtrahend?
      The number that is subtracted is called a subtrahend.

11. What is a protractor?
     A protractor is an instrument in shape of a semicircle used for drawing and    measuring angles in degrees.

12. What is an abundant number?

An abundant number is a number in which the sum of its factors is large than two times the number, example; 12, 18.20…


13. What is a deficient number?
     A number in which the sum of its factors is less than two times the number. Example 5,7,10…

14. What is a perfect number?

A number in which the sum of its factors equals two times the number.The first four perfect numbers are 6, 28 ,496 ,8218.

15. What is a deltoid?

    A deltoid is a kite.

16. What is a complex fraction?

      A complex fraction has a mixed or fractional number for its numerator or denominator,
Example (3/4)/ (2/5.)

Question And Answers

1. What is the name of our numeration system?
       The name of our numeration system is Hindu-Arabic system.

2. When did the system begin?
      The system began shortly before the third century when the Hindus invented the numbers.

3. When and when introduced the Hindu-Arabic system into Europe?
    Leonard Fibonacci introduced the Hindu-Arabic numeration system into Europe in the thirteenth century.

4. What is a numeral?
   A numeral is a symbol that represents a number.

5. What is a number?
   A number is a concept. It exists only in the mind.

6. What are pure numbers?
   Numbers that exist purely as numbers and do not represent amount of quantities are called pure numbers.

7. What is geometry?
   Geometry is a branch of mathematics that includes the study of shape ,size and properties of figures.

8. Who is called a geometer?
   A person who studies geometry is called a geometer or geometrician.

9. Who was the first female mathematician?
  The first known female mathematician was Hypatia of Alexandria.

10. What is a conjecture?
  A preposition that is still unproved is called a conjecture.

11. What is Gold back’s conjecture?
  Every even integer greater than 2 can be expressed as the sum of two primes.

12. What do Americans call a trapezium?
  Americans call a trapezoid to a trapezium.

13. What is analytic geometry?
  Analytic geometry is actually another name for co-ordinate geometry.

14. What are figurate numbers?
  Arrangements of dots to represent numbers as geometrical figures are called figurate numbers.

15. Who is called the father of trigonometry?
   Hipparchus, the Greek astrologer, mathematician is called the father of trigonometry.

Multiply Two Numbers Having Difference of Two

This is a quick way to multiply the numbers that differ by 2. It works best if the first number ends in 9.For example 19x21, 29x31, 39x41, 69x71…etc.

How to do it?

1. Find the middle integer.
2. Square of the middle integer.
3. Subtract 1 from the step2.
For Example,

Multiply 29 by 31.
Middle number is 30.
Square of 30 =900.
900 -1 = 899.

Another example
15 x 17.
Middle number is 16.
Answer to 15 x 17 = 162-1 = 255.

WHY 0! =1

 10! / 9! =10 

 9! / 8!    = 9

 8!  / 7!   = 8

 7! / 6!    =7

 6! / 5!   =6

 5! / 4!  = 5

 4! /3!  = 4

 3 ! / 2! = 3

 2 !/ 1 ! = 2

 1 ! /0 ! =1

 Therefore 0! = 1.

Pythagorean Triple

Mathematics students study the Pythagorean Theorem named after Pythagoras .Pythagoras was a great philosopher and a mathematician. A triple is three sides of a right angle triangle. His name is famous in mathematics due to his theorem. The Pythagorean theorem relates the length of hypotenuse of aright angled triangle to the lengths of the other two sides.

 If you are given one number of a triple, you can construct two other numbers of the triple.

 

How to do this?

 

  

There are two situations, one is the number is odd and another is the number is even.

If the number is odd, follow the following steps.

1. Take an odd number greater than 1.This is the first number of the triple.
2. Square the odd number.
3. Subtract 1and divide by 2, from the step 2.This will be the second number of the triple.
4. Add 1 to the step 3.This will be the third number of the triple.

Example:-

 

Suppose the first number of the triple is 3.

Square of 3 is 9.

According to the step3, by subtracting 1, and dividing by 2 we get (9-1)/2=4.

This will be the second number of the triple.

By adding 1, we get 4+1=5.This is the third number of the triple.

Hence, 3, 4, 5 is a Pythagorean triple.

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If the number is even, follow the following steps.

1.  Take an even number greater than 2.This is the first triple number.
2.  Square the number.
3.  Divide by 4and subtract 1.This is the second number of the triple.
4.  Add 2 to the result of step 3.this will be the third number of the triple.

Example:-

 

Suppose the first number of the triple is 4.

Square of 4 is 16.

Divide it by4, and then subtract 1 we will get 3 .This is the second number of the triple.

Add 2 to the above result.3+2=5.This will be the third number of the triple.

Hence 4, 3, 5 is a Pythagorean triple.

 

 

 Take more examples and find out the triples.

All About One

• The number 1 is the least counting number.

• The number 1 is a cardinal number. The first for 1 is an ordinal number.

• 1 is the smallest triangular number.

• 1 is the smallest Fibonacci number.

• 0! = 1.

• 1 is neither prime nor composite.

• In Latin 1 is called Onus.

• Concurrent lines meet at one point.

• Product of reciprocals is 1.

• The number 1 is the only number that does not change any number by multiplication or when used as a divisor.

MULTIPLICATION OF SERIES OF NINE

Here series of NINE means 9, 99, 999, etc.

Take any digit number as a multiplicand. Multiplier should be the same number of digits of 9.
Example: - 327*999
327 is a multiplicand and multiplier is 999.Both is three digit numbers.
How to find the Multiplication above?

Step 1:- subtract 1 from the multiplicand. This is the first part of the answer. Here 327-1=326.
Step 2:- Take the compliment of the multiplicand. This is the second part of the answer. Here 1000-327=673.
Step 3:- To get an answer, write the result of step 2 after step 1 as 326673.
So the answer of 327*999 is 326673.

Another example: - 47585*99999
Step 1:- subtract 1 from 47585, which will be 47584.
Step 2:- complement of 47585 =100000-47585=52415.
Step3:- To find an answer, write 52415 after 47584.
So the answer of 47585*99999 is 5241547584.

Verify above examples by traditional method and any other alternative method.

Do you Know?

1. 31              is a prime number.
331                is a prime number.
3331              is a prime number.    
33331            is a prime number.
333331          is a prime number.
3333331        is a prime number.
33333331      is a prime number.
333333331    is NOT a prime number. Because 333333331 =17*19607843.
MATHEMATICS ALWAYS NEEDS PROOF.

2. If you break the number 3025 into two parts 30 and 25, the square of 30 + 25 is equal to 3025. Square of   55   = 3025.  In   the same way 2025= (20+25)2.

3. When Square of an odd number is divided by 8, then the remainder is always 1. Exception odd number 1.

4. The sum of two consecutive triangular numbers is a square number. 

5. The product of four consecutive integers is divisible by 24.

6. The product of five consecutive integers is divisible by 120.

7. 153 is an interesting number. The number can be expressed as the sum of cubes of its digits. 153= 1³+5³+3³.

8. Product of two prime numbers is a semi prime number. 4,6. 9,10,14,…are semi primes.

9. Every odd number greater than 7 is the sum of three prime numbers.

10. 1, 2, 3 the three numbers have the same answer when added together and multiplied together.

11. It’s very interesting to know that, there are some numbers whose sum of all the digits of that cubic value is that number.
Examples: 17³ =4913 and 4+9+1+3 = 17
8³ =512 and 5+1+2=8.

12.   3, 4, 5 is a Pythagorean triplet.
33, 44, 55 is also is a Pythagorean triplet.
333,444,555 is also a Pythagorean triplet.
3333, 4444, 5555 is also a Pythagorean triplet.  

13. Sum of 10 Fibonacci numbers is equal to the 11 times the seventh Fibonacci number.
Example: Addition of below 10 Fibonacci number is 143
1, 1, 2, 3, 5, 8, 13, 21, 34, 55
And the 7th Fibonacci number which is 13 multiplied by 11 is also 143.

14. The product of three consecutive integers is always a multiple of 6.

15. nth derivate of  xn is equal to n!

16. If a, b, c, d are four integers then( a2+b2 )( c2+d2)=(ac-b d)2 +(ad +b c)2 or (ac+ bd)2 +(ad-bc)2. This relation is called  Brahmagupta –Fibonacci Identity.

Multiplication Trick for any 3 digit number with 143

Here   I am showing you a mental trick of multiplying any three digit number by 143.


    Step 1:- Take any three digit number.  For example:  587

   Step 2:- Picture the three digit number twice in your mind as 587587.

   Step 3:- divide the above six digit number by seven as 587587/7 = 83,941

83,941 is the answer of 587*143. With little more practice you can do this very easily.

Another example:Let us multiply 253 by 143.

1.  Write the number(253 ) two times 253253
2.  Divide by 7: 253253/7 = 36179
3. Thus 253 multiplied by 143=36179

Verify it. Take more examples and practice it.

FUN WITH ANY THREE DIGIT WHOLE NUMBER.

Here we have an interesting property of any three digit whole number.
Take any three digit whole number.
For example:   935
Repeat the digits and write to the next to the above number.
935935.
The original number became a six digit number.
When such six digit number is divided by  1001 we get the answer     935
Other examples:

738738÷1001=738
135135÷1001=135
777777÷1001=777

This trick is applicable to any three digit whole number.

Note :-Prime factors of 1001 are7,13,and 11.
Any such six digit whole number  is divisible by 7,11,13,77,91,143,and1001.

Amusing Property Of Three Consecutive Numbers.

For any three consecutive numbers product of the first and the last number plus one is equal to the square of the middle number.

Step 1:- Take any three consecutive three numbers.
For example—2, 3, 4

Step 2:- Product of the first and the last number.
Here, 2*4=8

Step 3:- Add one to the product of step 2
8+1=9

Step 4:- Result of Step #3 is the square of the middle number.
that is , Square of three is equal to nine.

Another example,
Suppose three consecutive numbers are 8,9,10.
As  above mentioned,
8*10+1=81.
81 is equal to the square of 9.


Take more examples and think how it works. Enjoy it!

Note for Parents /Teachers:
This way you are introducing your child to formula (a+1)²=a(a+2)+1

Strange Squares

Some pairs of square numbers have a strange property .

For example, Take a pair of 144 and make reverse of it i.e. 441 .
144 is a square of 12 (12 *12 = 144)
441 is a square of 21 (21*21 = 441)

Here 12 and 21 are reverse numbers . and same with their square numbers (144 and 441) .

Another Example is , pair of 169 and 961 .
169 is a square of 13 (i.e. 13 * 13 = 169 )
961 is a square of 31 (i.e. 31 * 31 = 961 )

Here 13 and 31 are reverse numbers . and same with their square numbers (169 and 961) .

If you observe carefully , you will find only two numbers in two digits which has this combinations .

For three digit numbers , 112 and 211 is the combination .

Square of 112 ( i.e. 112 *112) is 12544
Square of 211 ( i.e. 211 *211) is 44521

Can you find any other pairs like this ?