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So I'm reading Spivak Calculus and it makes sense. But, what I can't understand is, what is the purpose of the following:

$$\begin{align}13\cdot4 &= (1\cdot10 + 3)\cdot4\\&= 1\cdot10\cdot4 + 3\cdot4\\&= 4\cdot10 + 12\\&= 4\cdot10 + 1\cdot10 + 2\\&= (4 + 1)\cdot10 + 2\\&= 5\cdot10 + 2\\&= 52\end{align}$$

Could anyone please explain?

Sab ಠ_ಠ
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  • That is to break up how the multiplication really works in decimal notation. Those are the operations you do when you multiply. The only thing that you learned them very early in your life and accepted them as such. Here he is showing that the algorithm to multiply numbers in decimal notation (that you know) follows from the rules distributivity, associativity, and commutativity, plus $1$ being the neutral. –  Apr 01 '14 at 21:53
  • I think he wants to illustrate the way we multiply numbers. –  Apr 01 '14 at 21:53
  • @PiOverTwo Why not (1.12+1).4 ? – Sab ಠ_ಠ Apr 01 '14 at 22:04
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    @Sab He is trying to write numbers in decimal notation. Assume that you don't know who to multiply numbers in decimal notation. It is true that you could write $13=1\cdot 12+1$ (assuming you do know how to sum in decimal notation). But then latter you will have to multiply $12\cdot 4$. How to do it? –  Apr 01 '14 at 22:08
  • I get it now. Thanks :) – Sab ಠ_ಠ Apr 01 '14 at 22:11
  • @PiOverTwo - Let's say I had to do 14.5, here is how I would proceed. Is that the correct method of doing it? 14.5 = (1.10+4).5 = (1.10.5) + (4.5) = (1.10.5) + 20 = 1.10.5 + 2.10 = 5.10 + 2.10 = 10.(5+2) = 70 – Sab ಠ_ಠ Apr 01 '14 at 22:16
  • Yes. That is the way. –  Apr 01 '14 at 22:37

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