## 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)