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.
-----------------------------------------------------------------

Comments

THE EARTH, A SUPER ORGANISM

JOIN MY COURSE: "Become a programmer in a day with python" A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess. It says, "Earth is a self-regulating organism like a human being. The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment". Hence the earth is still suitable for the life to thrive. Imagine, in a particular place, there are lot of flowers. Some flowers are white and some are darkly coloured. We know, white reflects light and heat while dark absorbs the same. White flowers can thrive in hot climate. But dark flowers requires cold climate. The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist. This is the essence of "Gaia" theory. On our earth, ...

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. ...

DISORDER IS THE "ORDER OF THE DAY"

Imagine a balloon full of air. The air molecules are moving randomly inside the balloon. Let us pierce the balloon with a pin. The air rushes out. Why should not the air molecules stay inside the balloon safely and ignore the little hole? That is not the way the world works. The molecules always "want to occupy as many states as possible". Hence the air goes out in the open to occupy more volume. The things always goes into disorder (entropy) and the disorder increases with time. The above statement is what we call "second law of thermodynamics". Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze. Freezing means bringing things to order and arrangement. It violates the second law. Hence it will never happen. Hence heat must flow from high ...

sciencecapsules

Search This Blog