Method 1

Product of two digit numbers

This method is based on the sutra 'Urdhva Triyagbhyam' of the respectable Vedas . It means vertically and cross wise . It is applicable to all cases of multiplication of a large number by another large number . It is applicable in division also .
Explaining the method by example :-

Example 1 : - Find the product of 41 * 23 ?

Answer : -

Step 1 :-

First multiply the right most digit of the multiplicand and the multiplier.
i.e.
1 * 3 = 3
This givess the right most digit of the answer .
If a two digit number is generated then retain the right mast digit for the answer and rest are used as a carry to the next step .

Obtained digit : - 3
Step 2 : -

Now cross multiply .
Multiply first digit of the multiplicand (41) with the second digit of multiplier(23)
i.e.
4 * 3 = 12
and
Multiply second digit of the multiplicand (41) with the first digit of the multiplier (23)
i.e.
1 * 2 = 2
Add the above two digts + carry from the the first step( if any )
i.e.
12 + 2 + 0 = 14
4 is placed to the left of digit of step 1 and 2 is retained for carry.

Obtained digit : - 43

Step 3 : -

Now multiply the left most digit of multiplier and multiplicand
i.e.
4 * 2 = 8
Add carry from step 2
i.e.
8 + 1 = 9
Place it to the left of obtained digit .
i.e.
943
which is the answer.

Hence 41 * 23 = 943

The above proceure is very simple and can be completed in one single line only as shown .

     4          1
2 3
8  :    12 : 1 * 3 = 3
    :      2 :
   1:       0 :     
9         4      3


Example 2 : - Find the product of 59 * 45 .

Answer : -

    5          9
4 5
20 : 36 : 9 * 5 = 45
     : 25 :
  6 : 4 :
26   5      5

Answer 2655.


Example 3 : - Find the product of 33 * 24 .

Answer : -

    3          3
2 4
6 : 6 : 3 * 4 = 12
   :12 :
1 : 1 :
7   9   2
Answer is 792.

Algebraic Proof : -

Let the two 2 digit numbers be (ax+b) and (cx+d). Note that x = 10. Now
consider the product
(ax + b) (cx + d) = ac.x2 + adx + bcx + b.d
= ac.x2 + (ad + bc)x + b.d

You can observe that if x = 10 then cofficient of 10 ⁰ i.e. unit's place is according to step 1 and cofficient of 10¹ is step 2 and cofficient of 10² is step 3.

Practice Exercises : -

Find the product of following numbers : -
1.) 29 * 13
2.) 34 * 98
3.) 43 * 56
4.) 55 * 87
5.) 23 * 79