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Spivak state's that one of the consequence of the distributive property is being able to solve multiplication of Arabic numerals.

    1 3
  x 2 4
  _______
   3 1 2

is arranged as:

$13 . 24$

$=13 . (2 . 10 + 4)$

$= 13 . 2 . 10 + 13 . 4$

$= 26 . 10 + 52$

$= 312$

and

  1 3
  x 4
 _____
  5 2

is arranged as:

$=13 . 4$

$=(1 . 10 + 3) . 4$

$=1 . 10 . 4 + 3 . 4$

$=4 . 10 + 12$

$=4 . 10 + 1 . 10 + 2$

$=(4 + 1) . 10 + 2$

$=5 . 10 + 2$

$=52$

so why isn't $4 . 10 + 12$ directly written as $52$ like in the first problem?

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  • $\begingroup$ To save additional writing effort? With the first illustration, Spivak clearly outlines how the distributive property operates- and that is the entire point. $\endgroup$
    – Manan
    May 23 '20 at 7:23
  • $\begingroup$ @Manan how does writing additional lines save effort? it increases it $\endgroup$
    – Richard
    May 23 '20 at 8:21
  • $\begingroup$ Call it a typo, or an overlooked error. I think the author has made his point about the distributive property clear. $\endgroup$
    – Manan
    May 23 '20 at 8:59
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He is showing us how did he obtain $13.4=52$ in the first working using $P9$ property explicitly.

\begin{align}13\cdot 24 &= 13 \cdot (2\cdot 10 + 4) \\ &=13 \cdot 2 \cdot 10 + \color{blue}{13 \cdot 4} \\ &= 26 \cdot 10 + \color{blue}{52} \end{align} The multiplication $13\cdot 4=52$ uses $P9$ also:

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  • $\begingroup$ Thanks for pointing out the mistake $\endgroup$ May 23 '20 at 18:29

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