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Why is it that $$ab=ba$$ for positive integers $a,b$? Is there an intuitive explanation or we just need to accept it as a given fact?

Aryabhata
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Smith
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    It'sa religious tenet. we don't discuss it in public. – Will Jagy Jan 08 '15 at 02:33
  • Is "An $a\times b$ rectangle has the same area as a $b\times a$ rectangle as they are related by reflection" an intuitive explanation? How are you defining multiplication? – Milo Brandt Jan 08 '15 at 02:34
  • Try to think abstractly. Sometimes an algebraic relation has geometric interpretation, sometimes no. – Yes Jan 08 '15 at 02:44
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    It's actually a legitimate question, because after all, it isn't true that $a\div b= b\div a$ for all integers $a,b$, nor is it true that $a-b=b-a$. To the layman, who typically has no exposure to the concept of binary relations, this may seem weird. And pursuing the question will open a door to a magnificent world of mathematical exploration. – MPW Jan 08 '15 at 02:48
  • Will Jagy is joking in case you don't realize. You don't have to accept anything as fact if it cannot be justified. Without any justification the most you can say is "It is an assumption." – user21820 Jan 08 '15 at 02:52
  • @user21820: Surely OP realizes this. If he doesn't, it's probably best that he accept the proposed tenet and refrain from ever mentioning it publicly again. We of the cloth will forgive this one transgression. ;) – MPW Jan 08 '15 at 03:09
  • @MPW: Hahaha.. But maybe, just maybe, $a \times 2 > 2 \times a$ for sufficiently large $a$ (ordinarily?). Or maybe $ab = 0$ for some large $a$ and $b$ (if things come full circle). =) – user21820 Jan 08 '15 at 03:20
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    I think a better duplicate would be How to explain the commutativity of multiplication to middle school students? – MJD Jan 08 '15 at 03:42

4 Answers4

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The number $ab$ represents the number of unit squares in an $a\times b$ rectangle. The number $ba$ represents the number of unidt squares in a $b\times a$ rectangle. Can you see that they have the same number of $1\times 1$ squares in them?

ncmathsadist
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Imagine if you have a rectangle with $a$ rows and $b$ columns. If you want to calculate the area of the rectangle, then you can calculate it as rows x columns, which gives $ab$. You can also calculate it as columns x rows, which gives $ba$. Since the area of the rectangle didn't change at all here, you have $ab=ba$.

onetoinfinity
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It comes from the associativity & commutativity of addition, and the Peano definition/construction of the positive integers.

You can prove that $ab=ba$ for all positive integers $a,b$ by induction on $a$.

$P(a=1)$ is just the definition of $b$, and the inductive step is easy.

N. S.
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suppose you have a grid of points with $a$ rows and $b$ columns. The total number of points will be $ab$.

Now if you count it row-wise: each row has $b$ points and there are $a$ such rows. Thus the number of points is $ba$.

Now if you count it column-wise: each column has $a$ points and there are $b$ such colums. Thus the number of points is $ab$.

But both should be equal because you are counting the same set of points.

Anurag A
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