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Imagine $a \times b$ was not defined and we need to come up with something.

Here is the justification I could come up with:

Suppose, I have to measure this thing called area, I have to come with a notion of a unit area just as there is a unit value for a number. The reason is that if we have to measure something, there has to be some fundamental building block and we express everything in terms of. So the easiest way to define a unit area is to take a 1x1 rectangle. Now, if I just represent the number $a$ on the x-axis of the Cartesian coordinate system, I still don't have an area because it has only one dimension. So I just go to 1 on the y-axis and draw a line so that I can come up with something! Now I can see that there are $a$ unit squares in there. Extending this to $b$, I would have $a \times b$ unit areas. Is that good way to think of why the area is defined as $a \times b$?

I think the key was to idea build this fundamental notion of a unit area!

I would like to know if there is a better way to convince that $a \times b$ makes the most sense to measure area!

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    $\begingroup$ Yes, that makes perfect sense -- actually, just reading the title made me come up with the exact same justification that you wrote down. $\endgroup$ – Mees de Vries Oct 12 '18 at 13:28
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    $\begingroup$ Presupposing that a 1x1 square has a 1*1=1 area pretty much begs the question, does it not? $\endgroup$ – bukwyrm Oct 12 '18 at 13:30
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    $\begingroup$ @bukwyrm No - you're simply defining your unit of measurement. $\endgroup$ – Jack M Oct 12 '18 at 15:13
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    $\begingroup$ I think it's because the only measure that we can define that has all of those nice properties are constant times Lebesgue measure (Lebesgue measure it's the usual area). If we additionally say that unit square has area 1, we have Lebesgue measure, and then the area of the rectangle becomes ab. Look up measure theory, it's a very interesting subject $\endgroup$ – Jakobian Oct 12 '18 at 15:29
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    $\begingroup$ @Nishant : if you come to measuring area completely naive, why go with a 1x1 rectangle? - the idea of reducing area to plots defined by measures of length with right angles to each other is the actual quantum leap - why is the area unit not 42 heart shaped dots with circumference 1? after you define the unit of area as a rectangle 1x1 the question 'why is the area of rectangle axb' is kinda pre-answered. $\endgroup$ – bukwyrm Oct 15 '18 at 7:36
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Yes, exactly. The way area works is that you pick a single shape called your unit - for example a particular square - and define the area of any shape to mean the number of copies of the unit required to exactly cover the shape, even if this requires rotating or even cutting the unit into pieces. From this it follows that integer-length rectangles have the expected area formula.

This is the most fundamental definition of area since it's the only one which explains why area is an important concept. Geometry was, so the legend goes, originally developed by Egyptians in order to work out how much plots of land should be worth (for the purposes of taxation, etc). Imagine an ancient farmer is trying to decide which of two rectangular plots of land is the best, and you, a time traveler from the future, are trying to convince him to use the concept of area to decide. If you were to say:

"Plot A is better because it has higher area, where area is by definition the product of the lengths of the sides!"

Then he would rightly be very skeptical of your claim, because what does the product of the side lengths have to do with anything? However, if you were to say

"Plot A is better because it consists of 20 unit squares, whereas plot B consists of only 10 unit squares."

Then the farmer should be convinced, because any given unit square can hold the same amount of crops, no matter how the unit squares are arranged.

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Think of the properties you'd like your measure to have:

  • it should be a function of the lengths of the sides, i.e. $f(a,b)$
  • it should be symmetric, because if you flip the rectangle its area doesn't change, so $f(a,b)=f(b,a)$
  • it should be linear in its arguments. Think of a rectangle with sides $2a$ and $b$: it's basically the juxtaposition of two rectangles with sides $a$ and $b$ and the total area is the same. Now think of a rectangle with sides $a+c$ and $b$: it's just two rectangles with sides $(a,b)$ and $(c,b)$. So, your measure should satisfy $f(\lambda a+c,b)=\lambda f(a,b)+f(c,b)$
  • it should be null if one of the sides has length zero: $f(a,0)=f(0,b)=0$
  • ...there are other properties you'd want, but they follow from previous ones

If you want all of the above properties for your measure, then you're actually defining a multiplication. And that's why the area of a rectangle is defined as the product of the length of its sides.

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    $\begingroup$ It's not obvious that you "want" any of these properties. If you define area as just some measure satisfying these axioms, then there's no good reason to care about area. For example, how would you justify that the price of a plot of farmland should be proportional to its area? $\endgroup$ – Jack M Oct 12 '18 at 14:51

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