Multiplication in a Vertical Fashion

I was doing some multiplications 'in my head' the other day. No big deal, just some easy multiplications. In school, elementary or middle school, I was taught how to multiply in a vertical representation. I'll have to go on Google Docs or something for illustrations. However, today I realized that doing math this way is less intuitive than doing math in a horizontal fashion.

In the vertical fashion of performing the multiplication of big numbers, you are required to place a zero after a certain row is completed at the end of some numbers. I don't remember ever being explained the logical background of placing a zero after the the first row of this 'large' multiplication is completed. I think the horizontal approach to solving a multiplication is more intuitive.

Say you have to multiply 34 by 42. There's only one way that you can perform with the vertical method or fashion where you stack these two numbers and multiply the bottom right number with the top right and then the top left numbers placing the product in that order under the stack. I'm sure anyone who has read this far knows enough about multiplication to know the next steps.

But, if you think of the multiplication horizontally, there are many ways that this multiplication can be performed by taking advantage of the commutative, associative, and distributive properties of operations. You could see the multiplication as 34(40 + 2) = (30 + 4)(40 + 2) = 1200 + 60 + 160 + 8 You don't even have to think about 'placing a zero' or how many zeros you should place.