Multiplying multidigital decimal integers

We start doing this exercises by multiplying a few digits number by a one digit number.

We do it by multiplying all concecutive digits of the biger number by that digits. We star with the last one then tens and so on. units. The results are written below the bottom line under the corresponding columns of the bigger number. When a single result of multiplication is bigger then ten this generate a decimal similarely as in additions and this carry is then added to the next column multiplication result and so on.



The procedure is repeated until the last column is multiplied and the result is written.

Let's make an example:     6127 * 3   =   (  7     +      20      +     100      +      6000  ) * 3
wykonujemy kolejno                  7 * 3  +  20 * 3  +  100 * 3  +  6000 * 3
otrzymując:                                    21    +     60     +     300     +     18000     =     18381

We can show this like follows:
 

multiplications		products
7*3					2	1
20*3					6	0
100*3				3	0	0
6000*3		1	8	0	0	0
Total:		1	8	3	8	1

If we give up writting the last zeros but remember to shift left concecutive results we get this:

multiplication		products
7*3					2	1
2*3					6
1*3				3
6*3		1	8
total:		1	8	3	8	1

the product of 7*3 is a two digit numbers and 2 is a decimal carry to added in the next column. Similarely the multiplication 6*3 gives a carry equal 1. Carries may be memorized and added i the following columns or may be written. If there is no more columns like after multiplication 6*3 and ther is a decimal cary both figures can be written at the beginning of the result.

The schematic of the multiplication can be farther simplify as follows:

6 1 2 7 *       3       2 1       6       3     1 8       1 8 3 8 1

2     6 1 2 7 *       3         1       6       3     1 8       1 8 3 8 1

2     6 1 2 7 *       3 1 8 3 6 1 1 8 3 8 1

2     6 1 2 7 *       3 1 8 3 8 1 first column 7 * 3 = 21, write 1 carry 2 seconde column 2 * 3 = 6, add 2

After you have mastered this one digit multiplication procedure you can increace the number of digits in the second number by clicking the left button.

All the procedures are the same exept that we have to find result for every digit of the second number. We start with its units then tens and so on. The results are written one under un other but we shift every next result by one digit left. Then we adding them. Off cours we have to handle decimal carries as well.

The method will be explaine on the following example:
1030 * 431 = 1030 * ( 1+30+400)    wykonujemy kolejno   1*1030  +  30*1030  +  400*1030
We got: 1030 + 30900 + 412000 = 443930 which is the final result.

Let's show this this way:

multiplications		products
1*1030				1	0	3	0
30*1030			3	0	9	0	0
400*1030		4	1	2	0	0	0
total:		4	4	3	9	3	0

This method differ by that that we shifted the results instead of writing final zeros. Let's have a look:

multiplications		products
1*1030				1	0	3	0
3*1030			3	0	9	0
4*1030		4	1	2	0
total:		4	4	3	9	3	0

Good luck and enjoy!

The exercise is available in Math/Physics section.

designed by Witold Wojcik

---