What is the relationship between area and dimensions?

[physio]

I was just wondering the other day about the concept of area...Area to me is the space occupied in 2d by a bounded figure... I wanted to find out WHY the area of a square is s^2 or why area of a rectangle is lxb...Consider the dimensions of a rectangle 7x5. The area can be expressed as 5 strips of length 7 i.e. 7+7+7+7+7=35, but now the strips are each 1 unit wide and hence the formula works (width of 5 units is divided into 5 parts of 1 unit), what if the strips are made smaller and smaller such that the strips are infinitesimally small then the formula doesn't make any sense because the width of 5 units is being divided into infinite parts and hence the area is coming out to be zero, thus making me dumbfounded as to the approach adopted by me earlier.

Thus, why is the area of a rectangle or square- lxb or sxs?

 

[Number Nine]

what if the strips are made smaller and smaller such that the strips are infinitesimally small then the formula doesn't make any sense

Obviously the formula will not make sense if you compute it completely differently.
Of course, it's entirely possible to add together all of those infinitesimally small strips, which will give you the right answer; you just have to use calculus.

The approach you described (using strips of elements unit 1 wide) seems perfectly intuitive to me and clearly describes why the area of the square is computed as it is, so I'm not sure what else you want. You could use calculus and integrate over a square to the find the area, I suppose, which would give you the formula s². Let's do that (if you don't have any experience with calculus, just watch how a bunch of incomprehensible stuff happens that gives you the right answer):

We want to compute the area of a square with side length a. We can do this by integrating the function f(x) = a over the interval 0 < x < a...

$$\int_0^a\! a \, \mathrm{d} x \: = \: a \cdot a - a \cdot 0\: = \: a^2$$

What we've done here is exactly what you described; we took each of those slices and shrunk them down until they were infinitesimally small, and then added them up. This gives us the right answer.

 

[Kraflyn]

Hi.

Hm.

If width of single strip is identically zero, then there is nothing that could be done. However, if strip width is identically zero, then width is not infinitesimal. So Your reasoning got astray at the moment You passed from "infinitesimal" to "identically zero". You might have not noticed this transition. If infinitesimal, then not zero. Yes, as close to zero as You like. Never zero though. Use limits. Or integral, of course. I guess question was of speculative nature.

This reminds me of Zeno paradox.

Cheers.

 

[physio]

thanks i understood...i guess i have to learn calculus to understand the realm of mathematics which deals with infinitesimally small quantites...how is this book by michael spivak? Everybody told me it's great. What do you suggest?

 

[Kraflyn]

Hi.

I don't know about Spivak's Calculus. I do know about a ton of other books I've read on calculus, though. There are 2 types of calculus books. First type are books for students of mathematics and theoretical physics. They have to know it rigorously because it's in the curriculum and profs lecture it that way. Second type of calculus books are books written for engineers. Those written for engineers go straight for the head: they aim at calculating things. Engineer books are not concerned with purely theoretical aspects of calculus. So, one might go for second type books first, and when differential details become clear, one is advised to take a byte at the real thing. Otherwise, if not informed on the subject at all, one is easily lost in all the details of theoretical math. And calculus is not easy one way or another. It's a huge area and one never gets to master it entirely. Ever. Finally, in my opinion, reading only one book on calculus is not enough. The beast is too huge for only one weapon.

Cheers.

 

[physio]

Thanks...! I will look into these things.

 

[Vorde]

Spivak's a pretty serious Calculus textbook. I've heard it's one of the best, but from what I've gathered unless you're very confident about your mathematic skills (which you may well be), I'd start with a simpler book.

 

[Dickfore]

In Euclidean geometry, the area of a square with side $a$ is postulated to be $a^2$ (hence, the name squared). To justify this claim, imagine you increase the side n times. How many small squares fill the large square?

Then, to prove the formula for the area of a rectangle:
http://optimizingke.com/wp-content/uploads/2010/05/Inner-Area-to-Outer-Area.bmp
The side of the outer square is $a + b$, and the side of the inner square is $a - b$ (assuming $a > b$). Then, the area of the outer square is the sum of the areas of the inner square and the four identical rectangles:
$$(a + b)^2 = (a - b)^2 + 4 A$$
$$A = \frac{(a + b)^2 - (a - b)^2}{2}$$
$$A = \frac{(a^2 + 2 a b + b^2) - (a^2 - 2 a b + b^2)}{4}$$
$$A = a \, b$$

 

[physio]

Which is that simpler book for calculs? I don't think my math skills are that great. I would surely start with that book where everything is simple enough to completely grasp a particular idea and a concept. Please suggest some simpler book...!

Thanks for your replies...!

 

[clm222]

Are you asking why a square with a side length of say, 2cm, has an area of 4cm²?
If so, you can think of it as taking the 2cm along one side and multiplying it by how many of those there are. one side is 2cm wide, and stretches over 2cm. If you are asking how you get 4cm² from multiplying 2cm by 2cm, you can make a small dimensional analysis.

substitute cm into "d" you then have ${(2d)^2}={2^2}{d^2}=4{d^2}$

plug our dimension back into d's place, you get:

$${2cm{\cdot}2cm}=4{cm^2}$$

Another example is acceleration. In algebriac terms acceleration is:

$\frac{{v_f}-{v_i}}{t}=a$ where:

v_f=final velocity
v_i=initial velocity
t=time
a=acceleration.

there are fundamental units of quantity; here is the wiki page for the SI Units http://en.wikipedia.org/wiki/International_System_of_Units#Units_and_prefixes
We take these fundamental units to make others, such as these: http://en.wikipedia.org/wiki/List_of_physical_quantities

notice the velocity is $\frac{d}{t}$, time is (fundemntal)=t, and acceleration is $\frac{d}{t^2}$.

You may ask, "why is the time squared for acceleration?" treat the dimensions algebriaclly.
$$\frac{(\frac{d}{t})}{t}=\frac{d}{tt}=\frac{d}{t^2}$$
The quantities' dimensions treat each other algebriaclly.