Multiplication with positional notation and the distributive property

In this post we will be learning how to use Positional Notation to perform multiplication.

If you need a review on Positional notation please refer to my previous post on this subject.
Positional Notation

To refresh on what positional notation looks like we will write the number 235 with positional notation.

 2 (10 2) + 3 (10 1) + 5(10 0)

With a smaller example we can see if this allows us to learn something new about multiplication.
So to multiply 32 and 24 we will first write these two numbers into positional notation.

 3 (10 1) + 2 ( 10 0) X 2 ( 10 1) + 4 (10 0)

Does this look familiar? If we replace the 10 with x and since 100 = 1 we will replace it to and then we have:.
 3 x 1 + 2 x 2 x 1 + 4

To put into a more familiar form:

(3x + 2)(2x + 4)

With our number in this form we can now use algebra to simplify.

From distributive property.
 a(b+c) = (ab) + (ac)

We can now multiply this out using what is sometimes referred to as the foil method *(First Outer Inner Last)
 (3x + 2) (2x + 4)

 6(x2) + 12x + 4x + 8

 6(x2) + 16(x) + 8

Now if we substitute the 10 back in for x

 6 (10 2) + 16(10) + 8 6(100) + 160 + 8 600 + 160 + 8

 768

Too see how this works scaled up we will multiply 3259 and 1564

 3 (10 3 ) + 2(102) + 5(10) + 9 1(103) + 5(102) + 6(10) + 4
Now again to replace the 10 with x and since 1(x) is just x we remove the 1 in the second number:
 3(x3) + 2(x2) + 5x + 9 x3 + 5(x2) + 6x + 4
And again we can put this into the familiar form:
 (3 x3 + 2x2 + 5x +9) (x 3 + 5x2+ 6x + 4)

 3 x6 + 15 x5 + 18 x4 + 12 x3 + 2x5 + 10x4 + 12x3 + 8x2 + 5x4 + 25x3 + 30x2 + 20x + 9x3 + 45x2 + 54x + 36

Now we gather like terms:
 3 x 6 + 17 x5 + 33x4 + 58x3 + 83x2 + 74x + 36

If we put the 10 back in for x then multiply we will get
 3,000,000 + 1,700,000 + 330,000 + 58,000 + 8,300 + 740 + 36

 4,700,000 + 388,000 + 9,040 + 36

 5,088,000 + 9,076

And here is our answer using positional notation and distributive property to multiply two numbers.
 5,097,076         This post first appeared on Math-Journal:Understanding Math, please read the originial post: here

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Multiplication with positional notation and the distributive property

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