WHY HALF TIMES HALF IS QUARTER?

Let us understand the fractional multiplication visually. You may feel it is elementary. But it gives new perspective to fractional multiplication.
First, let us take 4*3. Let us take a vertical line equally divided into 4 parts and horizontal line into 3 parts.

The area enclosed is the result of 4*3. There are 12 unit squares. Hence the answer is 12 units.

Now, how to visualize (1/2)*(1/2). Take a vertical line, divide into 2 equal parts and name it 1/2 and 1. Repeat for horizontal line.
Here 1/2 and 1/2 bounds one square which is one fourth (1/4) area of the whole unit square. So 1/2*1/2=1/4.

In multiplication, the answer need not increase but diminish also.
Now let us prove 1/3*6/2=1.
Draw vertical line and graduate it in 1/3. Mark horizontal line in terms of 1/2.

1/3 and 6/2 bounds 6 squares. The area of each square 1/6 (since unit square contains 6 squares).
Hence the total area of six squares is 6 *1/6 =1. Therefor 1/3*6/2=1. Now you can realize why 6 in 1/6 is called Denominator because it indicates denomination.

We got the answer as 'one' in the above case. That can also be proved visually. Move the last four shaded squares to the unit square and fill it up -as illustrated below. Now the shaded region occupies one unit area.

Hence 4/7 means 4 times 1/7 (denomination). If the fraction is exactly divisible like 1/2 =0.5, it is rational number. If it is not divisible like 1/3 =0.333... It is irrational number. The value of pi = 22/7 is an irrational number. (We know that the circle of circumference 22 units has the diameter of 7 units). People still find more digits in pi using super computers. By the way, all natural constants are irrational numbers like e = 2.71....root 2 =1.414...
Dealing with the fractions is a tricky business. Another article tells why we add the fractions the way we add.
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