I've been wondering about the theory behind the distributive property lately.
For example: 2(pi * r^2) is just 2 * pi * r^2.
However, when you add a positive number like +3. You get 2(PI*r^2 +3). But, that isn't just 2*PI*r^2+3.
Its: 2pi*r^2 + 2*3.
So I was wondering why that is. Why do you only have to multiply once with the whole multiplication part(with pi^r2), instead of having to multiply 2 by both pi and r^2.
So then I thought isn't it just: (pi * r^2 + 3) + (pi * r^2 + 3)? Then, I tried to simplify that, thinking it would help me understand why...but all that did was make me more confused than when I started. Could someone help me understand please?
Answer
The distributive property,
and most basic properties of real numbers,
comes from geometry.
A non-negative value
corresponds to the length of a line segment.
Adding values corresponds to placing
two segments together
and measuring their length.
Since the order that the segments are placed
does not change the total length,
addition is commutative.
Looking at three segments,
the length is the same if
the first two are placed and then the third,
or the last two are placed and then the first.
Therefore
addition is associative.
Multiplying two segments
corresponds to getting the area
of a rectangle with sides the lengths
of the two segments.
Since swapping the two segments
just rotates the rectangle by 90 degrees,
which does not change the area,
multiplication is commutative.
Consider two rectangles
with a common height $h$
and bases $a$ and $b$.
Their areas are
$ah$ and $bh$,
and the sum of the areas is
$ah+bh$.
Place these two rectangles together
so their common height lines up.
They now form a single rectangle
with base $a+b$ and
height $h$,
and the area of this rectangle
is $(a+b)h$.
This means that
$ah+bh = (a+b)h$,
which is the distributive law.
These laws were, of course,
extended to other type of numbers
(real, complex, fields, ...),
but they all started with geometry.
Off topic but interesting:
Try to prove that
$\sqrt{2}$ is irrational
using only geometric concepts
and proofs.
No algebra is allowed.
(I think I'll even propose this as a question.)
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