Degree

The degree system of measuring angles is a gift to us from the Babylonians (5000-500 BC residents of the present day Iraq). They used a base of 60 for their number system (just like we use a base of 10 for the decimal system and a base of 2 for the hbinary system). They extended this base 60 system to measuring time and angles as well (and we still use both of them).

Corresponding to HMS (Hour-Minutes-Seconds) system of measuring time, there is DMS (Degree- Minutes-Seconds) system of measuring angles. Hence,

1 degree = 60 minutes ('), and
1 minute = 60 seconds (").

Additioally,

1 complete rotation = 360 degrees (probably because Babylonians had 360 days in their calendar year, 30 days in each of 12 months)


Radian

One radian is the angle between two radii of a circle that subscribes an arc of length one radius on the circumference. How do we know how many radians a particular angle is equal to? We simply ask, "How many arcs, each of length one radius, does the angle subscribes on the circumference?" If an angle subscribes an arc of length 2 radii on the circumference, then the angle must be 2 radians.

How many arcs, each one radius long, are there in the whole circumsnference ( a complete rotation)? 2*pi, right? Yeap. So there are 2*pi radians in a complete circle. This gives us:

1 complete rotation = 2*pi radians= 360 degrees
=> pi radians = 180 degrees
=> 1 radian = 57.3 degrees (approximately).


Gradian (gon or grade)

This system of measuring angles divides a complete rotation into 400 gradians or gons or grades. Hence,

1 complete rotation = 400 gradians
=> 1 right angle = 100 gradians
=> 90 degrees = 100 gradians
=> 1 degree = 10/9 gradian.

Equivalently,
1 gradian = 0.9 degree.

Similarly,
2*pi radians = 400 gradians
=> pi radians = 200 gradians
=> 1 radian = 63.662 gradians (approximately).

A few days ago I woke up at around 2am to visit the rest room. I had left my MSN messenger on. There I had a message asking if I could prove the irrationality of sqrt(3). One of my college friends had sent the message. I tried to send her an outline of a proof by contradiction that I thought should work. She initially thought I was wrong, but later when I expanded each step in detail, she was convinced. Here is the proof:

Assume sqrt(3) is rational. Now since it is rational by assumption, it can be written as p/q (in its most reduced form). So we have,

sqrt(3)=p/q
or, 3=p^2/q^2
or, 3*q^2=p^2 ----(i)
q is even --> q^2 is even (even multiplied by even)
q^2 even -->3*q^2= p^2 is even (odd multiplied by even is even)
P^2 even --> p is even

So q is even --> p is even. But this would imply p/q is not in its most reduced form, since both p and q have a common factor of 2. So we cannot assume that q is even. Now let us take the other possibility: that q is odd.

q is odd--> q^2 is odd (odd multiplied by odd is odd) --> 3*q^2= p^2 is odd --> p is odd.

Hence, we must assume that both p and q are odd. So we can write p and q as:

p=2*m+1 and q=2*n+1, where m and n are any integers.

Now let us substitute this new expressions of p & q in eqn. (i).

3*(2*n+1)^2=(2*m+1)^2
or, 3*(4*n^2 + 4*n +1)= 4*m^2 + 4*m +1
or, 12*n^2 + 12*n + 3 = 4*m^2 + 4*m +1
Now divide both sides by 2. So we have,
6*n^2 + 6*n + 1 + 1/2 = 2*m^2 + 2*m +1/2
or, 6*n^2 + 6*n +1 = 2*(m^2 + m)

Now notice that on the left hand side the first term is even (since 6 which is an even is multiplying n^2). Similarly 6*n is even. But the last term is odd. So on the left hand side we have even + even + odd = odd. On the right hand side, the terms in the parentheses are multiplied by 2 (an even), and hence is even. So we have,

odd=even.

This is a contradiction that we arrive at if assume that sqrt(3) is rational. Hence, it cannot be rational, and hence must be irrational (since these two are mutually exhaustive possibilities: a number must either be rational or irrational). Q.E.D.

I saw this search tool bar on google.com site which is set as my default homepage (since I use it the most). I got attracted to the toolbar not because it would let me search from any page I am on, but because it had the pop-up blocking feature.

As I read about other features of this little toolbar, I happened to come across the "blogThis" botton and get introduced to blogging for the first time. So here I am giving it a try, and see if I can really keep up to posting something new every now and then.

Btw, today is Valentine's day. So let me wish you all a very HAPPY VALENTINE'S DAY. May you all find the true love, and be happy ever after.