Intuitive problem about the square of fractions If I think about squaring as working out the area of an actual square then the following statement seems highly intuitive to me
The area of the square will always be bigger than the length of any individual side.
1)  $2*2 = 4$ follows this rule
2)  $\frac{1}{2} * \frac{1}{2} = \frac{1}{4}$ doesn't
When you make each side of the second square 3 times longer, the statement holds again. So there is a specific subset of small squares for which my statement doesn't work.
I'm hoping this is logical to someone, who can push me into the intuitive insight behind this. I understand it in pure numbers. But is my intuition so absurd given a drawing of a square?
 A: It doesn't really make sense to compare an area to a length. We measure both with the same kind of numbers, but that is mostly just because it's the numbers we have. So your statement is not quite right to begin with.
What we mean by the length of a line is the number of times a certain, chosen unit length goes into it. And what we mean by the area of a shape is the number of times the unit area goes into it. For convenience (because it makes computations relatively easy and is not obviously less practical than any other choice) there's a convention that the unit area is that of a square whose side is a unit length.
But that doesn't really mean that the unit square is "the same as" the unit length, even though we represent both by the number $1$.
If we have a line with the length $1/2$, it is shorter than the unit length. The canonical shape with the area $1/2$ is a rectangle with side lengths $1/2$ by $1$. If we compare that to a square of length $1/2$, we need to make one of the sides of the rectangle shorter while the other stays the same. Therefore the number for the area of the square must be less than the number for the length of its side.
On the other hand, if the length if the side is $2$, then the canonical shape with area $2$ is a rectangle with sides $1$ by $2$. Here we have to make one of the sides longer in order to get a square of side length $2$, so the number for the sqare's area is larger than the number for its side's length.
A: The correct insight here is that area and length, or more generally any two different types of quantities, cannot be compared. How much "bigger" is 10 meters than 1 square meter? Such questions don't make sense. You have been leaving off the units in your computations, which almost always leads one astray. The correct statements would be
$$2\text{ units }\times 2\text{ units } = 4\text{ square units}$$
$$\frac{1}{2}\text{ units }\times \frac{1}{2}\text{ units } = \frac{1}{4}\text{ square units}$$
where the units could be whatever you want (feet, meters, inches, etc.) It certainly makes sense to say that $$\frac{1}{2}\text{unit}<2\text{ units}$$ and that $$\frac{1}{4}\text{square units}<4\text{ square units},$$ but it doesn't make sense to talk about whether $2\text{ units}$ is bigger or smaller than $4\text{ square units}$. They're apples and oranges.
