## Multiply Using Lines

Notice that a multiple digit number can be rewritten as a sum of the digits in that number each of which is multiplied by a power of ten depending on its position in the number (units, tens, hundreds place etc...):

For example 23 is 2*10^1 + 3*10^0 and 32 can be written 3*10+2*1 (where 10^1=10 and 10^0=1).

Using the distributive property of multiplication over addition you could treat each multi-digit number as a polynomial and then just multiple the polynomials together using the distributive property. Here is what the distributive property looks like for a two binomials

(a + b)(x + y)
ax + ay+bx + by

This is to notice that a is multiplied by both y and x while b is also, in other words both a and be are distributed over x and y each in turn and all the results added to gether.

So to multiple two two digits numbers (say 23 times 32) we get:

(20 + 3)(30+2)
20x30 + 20x2+3x30 + 3x2 (following the rule above)
600   +   40+90   + 6   (doing the multiplication)
600   +     130   + 6   (adding the two middle terms)
600   +  100+30   + 6   (separating out the amount over 100 (i.e. 30) from 100 in the middle term)
700   +      30   + 6   (adding the 100 to the 600)

The reason we do it this way is to stress the "powers of 10" part, keeping the 100's and 10's and units all together.  If the last term had been bigger than 10 then we would have taken the 10's part of it and added it to the middle term first and then if the middle term was bigger than 100 we would have taken the 100's part of it and added it to the first term (i.e. the 100's term).  That's why we split up 130 into 100+30 and grouped the 100 over with the 600 to get 700.

This seems like a strange way to multiply until you see it done with lines.  Here is the original video from YouTube.COM that illustrates this: http://www.youtube.com/watch?v=hdBadkRz3vE.

Here is a link to an animation I created to multiple 23x32 using the same method except it is prefixed by an animation that uses the distributive property of multiplication over addition to do the same thing.  What this shows is that the 'multiply-with-lines' is a way of multiplying using the distributive property of multiplication over addition: http://www.youtube.com/watch?v=HeG2hCe5eJg.

Here is the final frame of the graphic: 