Multiply the numbers close to the power of 10s in just 5 seconds! - Part 2 and Part 3

In my last article "Multiply the numbers close to the power of 10s in just 5 seconds! - Part 1", I have mentioned a simple Vedic mathematics method of how to multiply numbers that are below to the power of 10. In this article, I would mention about - Part 2: Multiplying numbers greater than the power of 10s. and Part 3: Multiplying numbers of which one is greater than power of 10 and the other is less than power of 10.

As mentioned in previous article, in Vedic mathematics this method is known as "Nikhilam Navatashcaramam Dashatah - All from 9 and the last from 10" 

Part 2: Multiplying numbers greater than power of 10s

Example 1: 12 X 15

Step 1: Identify the base (power of 10) to which the number is close. Here, 12 and 15 are close to 10. Therefore, base is 10

Step 2: Identify by how much both the numbers are greater than the base. Here, 12 is more by 2 and 15 is more by 5.

Step 3: Write the original numbers along with the number by which it is greater than 10 as below.

12 + 2

15 + 5

Step 4: Now, Cross add the original number and the surplus number. That is add 12 to 5 or add 15 to 2. So, we get

12 + 2

     X

15 + 5

-----------

 17

-----------

Step 5: Now, vertically multiply the two surplus numbers. Here, the surplus numbers are 2 and 5. Therefore, we have

12 + 2

     X

15 + 5

-----------

 17 / 0 

     1

-----------

Step 6: Now, since the base is 10, the right hand most digit is units and can have only one digit. Therefore, keep the 0 of the 10 on the right hand side and carry the 1 over the left and change the 17 into 18.

12 + 2

     X

15 + 5

-----------

 17 / 0 = 180

     1

-----------

Note: If the number of digits are more than the number of zeroes in the base, the excess digit or digits are to be added to left hand side of the answer.

Therefore we have 12 X 15 = 180

---

Example 2: 101 X 109

Step 1: Identify the base (power of 10) to which the number is close. Here, 101 and 109 are close to 100. Therefore, base is 100

Step 2: Identify by how much both the numbers are greater than the base. Here, 101 is more by 1 and 109 is more by 9.

Step 3: Write the original numbers along with the number by which it is greater than 100 as below.

101 + 1

109 + 9

Step 4: Now, Cross add the original number and the surplus number. That is add 101 to 9 or add 109 to 1. So, we get

101 + 1

      X

109 + 9

-----------

110

-----------

Step 5: Now, vertically multiply the two surplus numbers. Here, the surplus numbers are 9 and 1. Therefore, we have

101 + 1

      X  |

109 + 9

-----------

110 / 9

-----------

Step 6: Now, since the base is 100, the right hand most digit is tens and units and should have two digit. Therefore, fill the vacancies by adding zero. thus

101 + 1

      X  |

109 + 9

-----------

110 / 09 = 11009

-----------

Therefore we have 101 X 109 = 11009

Note: If the Right hand side contains less number of digits than the number of zeros in the base, the remaining digits are filled up by giving zero or zeroes on the left side of the right hand side digit.

---

Example 3: 125 X 101

Step 1: Identify the base (power of 10) to which the number is close. Here, 125 and 107 are close to 100. Therefore, base is 100

Step 2: Identify by how much both the numbers are greater than the base. Here, 125 is more by 25 and 101 is more by 1.

Step 3: Write the original numbers along with the number by which it is greater than 100 as below.

125 + 25

101 + 1

Step 4: Now, Cross add the original number and the surplus number. That is add 125 to 1 or add 101 to 25. So, we get

125 + 25

      X

101 + 1

-----------

126

-----------

Step 5: Now, vertically multiply the two surplus numbers. Here, the surplus numbers are 25 and 1. Therefore, we have

125 + 25

      X

101 + 1

-----------

126 / 25 = 12625

-----------

Therefore we have 125 X 101 = 12625

---

Part 3: Multiplying a number which is greater than power of 10, with a number which is less than power of 10

Example 1: 9 X 15

Step 1: Identify the base (power of 10) to which the number is close. Here, 9 and 15 are close to 10. Therefore, base is 10

Step 2: Identify by how much the numbers are greater than or less than the base. Here, 9 is less by 1 and 15 is more by 5.

Step 3: Write the original numbers along with the number by which it is greater / less than 10 as below.

15 + 5

 9  - 1

Step 4: Now, based on the signs, cross add and cross subtract the numbers accordingly. That is add 9 to 5 or subtract 15 by 1. So, we get

15 + 5

     X

 9  - 1

-----------

 14

-----------

Step 5: Now, vertically multiply the surplus and deficit numbers along with the signs. Here, the numbers are 1 and 5. Therefore, we have

15 + 5

     X

 9  - 1

-----------

 14 / -5

-----------

Step 6: Now, to get rid of the minus sign, subtract the right hand side by the base and less the left hand side by 1. Here, we have base as 10 and right hand side number as 5. therefore we have 10 - 5 =5. Hence, 

15 + 5

     X

 9  - 1

-----------

 14 / -5 = 135

-----------

Therefore we have 9 X 15 = 135

---

Example 2: 101 X 97

Step 1: Identify the base (power of 10) to which the number is close. Here, 101 and 97 are close to 100. Therefore, base is 100

Step 2: Identify by how much both the numbers are greater than and less than the base. Here, 101 is more by 1 and 97 is less by 3.

Step 3: Write the original numbers along with the number by which it is greater than and less than 100 as below.

101 + 1

  97  - 3

Step 4: Now, based on the signs, cross add and cross subtract the numbers accordingly. That is subtract 101 by 3 or add 97 to 1. So, we get

101 + 1

      X

  97  - 3

-----------

 98

-----------

Step 5:  Now, vertically multiply the surplus and deficit numbers along with the signs. Here, the numbers are 3 and 1. Therefore, we have

101 + 1

      X  |

  97  - 3

-----------

 98 / -3

-----------

Step 6: Now, to get rid of the minus sign, subtract the right hand side by the base and less the left hand side by 1. Here, we have base as 100 and right hand side number as 3. therefore we have 100 - 3 =97. Hence, 

101 + 1

      X  |

  97  - 3

-----------

 98 / -3 = 9797

-----------

Therefore we have 101 X 97 = 9797

---

Example 3: 10006 X 9999

10006 + 6

      X

  9999 - 1

-----------

10005 / -6 = (10005 - 1) / (10000 - 6) = 100049994

-----------

Therefore we have 10006 X 9999 = 100049994

---

I am sure with little bit of practice, you can successfully perform multiplications very fast when compared to the traditional method. Also, please note that these methods should be applied accordingly for which it is designed. This method may not give an effective result, if you are trying to multiply 67 and 72. Vedic mathematics defines many methods by which calculations can be made simple, but you need to use them accordingly.

Hope this method was useful for you. If you have any other methods do share and let me know if you have any questions regarding these methods.

1089 pattern and facts.

* Multiplying the number 1089 by the integers from 1 to 9 produces a pattern: multipliers adding up to 10 give products that are the digit reversals of each other. 

1089 x 1 = 1089

1089 x 2 = 2178 

1089 x 3 = 3267 

1089 x 4 = 4356 

1089 x 5 = 5445  (Palindrome)

1089 x 6 = 6534  (reverse of 4356)  

1089 x 7 = 7623  (reverse of 3267) 

1089 x 8 = 8712  (reverse of 2178) 

1089 x 9 = 9801  (reverse of 1089) 

Also note the patterns within each column. First column is increasing order of 1 to 9, second column is increasing order of 0 to 8, third column is reverse of second column and fourth column is reverse of first column.

* Another cool aspects are

-  1/1089 = 0.00 09 18 27 36 45 54 63 72 81 ..... remember the 9 table

-  1/9801 = 0.00 01 02 03 04 05 06 07 08 09 10 11 12 13 14... 

Multiply the numbers close to the power of 10s in just 5 seconds! - Part 1

Multiplication of numbers that are close to the power of 10s (i.e 99999 x 99997 ; 99982 99992) can be easily done in just 5 seconds. Tough to believe it right, but it is possible and all that you need to know is basic subtraction, addition and multiplication tables till 5. Just, follow the below examples with their explanation and by the end of the article you would be all set.

Part 1: Multiplying numbers below to the power of 10s

Example 1: 9 X 6

Step 1: Identify the base (power of 10) to which the number is close. Here, 9 and 6 are close to 10. Therefore, base is 10

Step 2: Identify by how much both the numbers are less than the base. Here, 9 is less by 1 and 6 is less by 4.

Step 3: Write the original numbers along with the number by which it is less than 10 as below.

9 - 1

6 - 4

Step 4: Now, Cross subtract the original number and the deficit number. That is subtract 9 by 4 or subtract 6 by 1. So, we get

9 - 1

X

6 - 4

-----------

5

Step 5: Now, vertically multiply the two deficit numbers. Here, the deficit numbers are 1 and 4. Therefore, we have

9 - 1

X |

6 - 4

-----------

5 / 4 = 54 (since base is 10 we can have 1 digit on the right hand side which is 4)

-----------

Therefore we have 9 X 6 = 54

---

Example 2: 7 X 6

Step 1: Identify the base (power of 10) to which the number is close. Here, 7 and 6 are close to 10. Therefore, base is 10

Step 2: Identify by how much both the numbers are less than the base. Here, 7 is less by 3 and 6 is less by 4.

Step 3: Write the original numbers along with the number by which it is less than 10 as below.

7 - 3

6 - 4

Step 4: Now, Cross subtract the original number and the deficit number. That is subtract 7 by 4 or subtract 6 by 3. So, we get

7 - 3

X

6 - 4

-----------

3

Step 5: Now, vertically multiply the two deficit numbers. Here, the deficiet numbers are 3 and 4. Therefore, we have

7 - 3

X  |

6 - 4

-----------

3 / 2

1

-----------

Step 6: Now, since the base is 10, the right hand most digit is units and can have only one digit. Therefore, keep the 2 of the 12 on the right hand side and carry the 1 over the left and change the 3 into 4.

7 - 3

X |

6 - 4

-----------

3 / 2 = 42

1

-----------

Therefore we have 7 X 6 = 42

Note: If the number of digits are more than the number of zeroes in the base, the excess digit or digits are to be added to left hand side of the answer.

---

Example 3: 91 X 99

Step 1: Identify the base (power of 10) to which the number is close. Here, 91 and 99 are close to 100. Therefore, base is 100

Step 2: Identify by how much both the numbers are less than the base. Here, 91 is less by 9 and 99 is less by 1.

Step 3: Write the original numbers along with the number by which it is less than 100 as below.

91 - 9

99 - 1

Step 4: Now, Cross subtract the original number and the deficient number. That is subtract 99 by 9 or subtract 91 by 1. So, we get

91 - 9

X

99 - 1

-----------

90

Step 5: Now, vertically multiply the two deficit numbers. Here, the deficiet numbers are 9 and 1. Therefore, we have

91 - 9

X  |

99 - 1

-----------

90 / 9

-----------

Step 6: Now, since the base is 100, the right hand most digit is tens and units and should have two digit. Therefore, fill the vacancie by adding zero. thus

91 - 9

X  |

99 - 1

-----------

90 / 09 = 9009

-----------

Therefore we have 91 X 99 = 9009

Note: If the Right hand side contains less number of digits than the number of zeros in the base, the remaining digits are filled up by giving zero or zeroes on the left side of the right hand side digit.

---

Example 4: 93 X 97

Step 1: Identify the base (power of 10) to which the number is close. Here, 93 and 97 are close to 100. Therefore, base is 100

Step 2: Identify by how much both the numbers are less than the base. Here, 93 is less by 7 and 97 is less by 3.

Step 3: Write the original numbers along with the number by which it is less than 100 as below.

93 - 7

97 - 3

Step 4: Now, Cross subtract the original number and the deficient number. That is subtract 93 by 3 or subtract 97 by 7. So, we get

93 - 7

X

97 - 3

-----------

90

Step 5: Now, vertically multiply the two deficit numbers. Here, the deficiet numbers are 7 and 3. Therefore, we have

93 - 7

   X |

97 - 3

-----------

90 / 21 = 9021

-----------

Therefore we have 93 X 97 = 9021

---

Example 5: 883 X 997

Step 1: Identify the base (power of 10) to which the number is close. Here, 883 and 997 are close to 1000. Therefore, base is 1000.

Step 2: Identify by how much both the numbers are less than the base. Here, 883 is less by 117 and 997 is less by 3.

Step 3: Write the original numbers along with the number by which it is less than 100 as below.

883 - 117

997 - 003

Step 4: Now, Cross subtract the original number and the deficient number. That is subtract 883 by 3 or subtract 117 by 7. So, we get

883 - 117

X

997 - 003

---------------

880

Step 5: Now, vertically multiply the two deficit numbers. Here, the deficiet numbers are 117 and 3. Therefore, we have

883 - 117

X   |

997 - 003

---------------

880 / 351 = 880351 (since base is 1000 we can have 3 digits on the right hand side)

-----------

Therefore we have 883 X 997 = 880351

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This method is known as "Nikhilam Navatashcaramam Dashatah - All from 9 and the last from 10" in Vedic Mathematics.

Note:

1. This method can be easily applied to the numbers that are near to the base 10s i.e., 10,100, 1000....

2. If one number is near base of 10s and the other is not, then also it can easily be applied - Ex 88 X 99 or 25 X 98 and above Example 5.

3. The deficit numbers can easily written by following the rule that all the digits (of the original number) are to be subtracted from 9 but the last (right hand most one) one should be subtracted from 10. Ex deficit of 63 is 37 (6-9 =3 / 10-3=7).

With little bit of practice by following the above method, I am sure you can do multiplication of the numbers near to power of 10s in just 5 seconds. Wondering that above method is good for the numbers that are below base 10s but what about numbers which are above it? Don't worry, Vedic mathematics has a solution for it too, I shall discuss the same in my next article, till then practice this one and let me know if you have any questions.

Maths Trick - Guess the number

1. Ask your friend to think of any number between 1 to 10.

    Lets say, s/he had thought of 3. 

2. Ask him/her to double the number.

    6

3. Ask him/her to add 10 to the result (step 2).

    6 + 10 = 16

4. Ask him/her to divide the new number with 2.

    16 / 2 = 8

5. Ask him/her to tell the result and tell them you would tell the number s/he has thought.

    8

Thinking how you would tell him what number s/he has thought. It's very simple, just subtract 5 from the number he has told and you would get the original number he has thought. 

Here it is, 8 - 5 = 3

Simple right! Now want to know how it works, just see the below explanation

In the entire process, you are indirectly adding the original number by 5 and hence by knowing the end result you can guess the original number by just subtracting it by 5

Step 1 - Say the original number what your friend has thought is a.

Step 2 - Double the number - 2a

Step 3 - Add 10 to the new number - 2a + 10

Step 4 - Divide the new number by 2 (2a + 10)/2 = (2a/2) + (10/2) = a + 5

Step 5 - New number - a + 5

Therefore, Original number = New Number - 5 = a + 5 - 5 = a

Note: You can also use 2 and 3 digit numbers to make it harder! 

Have fun playing this trick on your friends.

Maths Trick - Guess the result - 1089.

1. Ask your friend to think of or write down any 3 digit number such that the digits are in ascending order (highest to lowest) and that they are not repetitive. 

    Lets say, s/he has written 765. While your friend is writing down the number, you write down 1089 and don't show it to your friend.

2. Ask him/her to reverse the number and subtract it from the original number (step1).

    765 - 567 = 198

3. Ask him/her how many digits the result has. If it has 3 digits, ask him to reverse the result and if it has only 2 digits, then ask him to add a zero in front of it and then reverse the digit. Ex Number - 099  reverse number - 990

    Here it is 3 digits, and hence reverse number is 891

4. Ask him/her to add this number (from step 3) to its previous number (from step 2 - obtained by doing subtraction) and ask him/her not to tell you the result.

    891 + 198 = 1089

5. Now you can reveal the number (written by you in step 1) which is an exact match of their result (step 4)

Curious to know how it works? It is simple, just read the below explanation.

By asking your friend to subtract the original number from the reversed number, you are making the number indirectly multiple of 99. 

Say the Original number is xyz = 100x + 10y + z

Therefore, Reversed number is zyx = 100z + 10y +x

Subtracting these both, xyz - zyx = (100x + 10y + z) - (100z + 10y +x) = 99x -99y = 99 (x-y)

Now, if you see the below 99 table you can see that (A.) Products are digit reversals of each other, that is 99 x 6 = 594 which is reverse of 495 = 99 x5. Such is the case with others. Therefore (B.) the result of adding the number with its reversed number is always 1089.

99 x 2 = 198 + 891 = 1089

99 x 3 = 297 + 792 = 1089

99 x 4 = 396 + 693 = 1089

99 x 5 = 495 + 594 = 1089

------------------------------------------

99 x 6 = 594 + 495 = 1089

99 x 7 = 693 + 396 = 1089

99 x 8 = 792 + 297 = 1089

99 x 9 = 891 + 198 = 1089

99 x 10 = 990 + 099 = 1089

Have fun playing the trick on your friends.