POWERFUL MATH IN CLOCK'S FACE

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The clock's face has 12 elements. Let us assume that it forms a group G.
1. Add any two numbers in the clock's face.
2'0clock +3 hours =5'o clock
2+3=5, 5+4=9,
But 9+5=14 =2.
Adding any two numbers will always give another number in group G. Hence the 'group' satisfies 'closure' property.

2. 3'oclock +12 hours =3'oclock
9'oclock+12 hours =9'oclock.
so A*12=A
If you add any element 'A' in the group to 12, the result is always 'A'
The '12' is called the 'identity' of the group.

3. Take 5, add 12-5 (7) to it.
5+7 =12 we get the identity. 7 or 12-5 is said to be the inverse of '5'. Hence, for every element 'a' of the group, there is another element b, so that a+b =identity. Element b is called the 'inverse' of a. In our case b=12-a.

4. Group elements also exhibit 'associativity' property.
(a+b)+c =a+(b+c).

Now we can define the 'group'
"A group is a set of elements equipped with binary operation addition or multiplication and has the above four properties".

We will create one problem with the clock's face. You are allowed to move the 'hour's needle' by 2-step or 2 hours only. You have to move 4 times and you have to reach 3'o clock exactly. The question is 'where to start' initially?
4times *2hours =8 hours we have to move hour-needle by 8 hours and reach 3'oclock.
Start+8hours =3'oclock.
x+8 =3
x=3-8
Instead of subtracting 8, we can add 'inverse of 8=4 to 3.
Hence x=3+4 =7
You have to start from 7'oclock to reach 3 in 4 steps.

Rotations, transformations always form the group. Rubik cube solution algorithm is based on group theory. We know, Rubik cube mainly involves rotations.

Weekdays also form a group.
SUN MON TUE WED THU FRI SAT
1 2 3 4 5 6 7

They are similar to the clock's numbers. But they have identity 7.
Here also, we will create a problem and solve it using the group's properties.

Today is Tuesday(3). I want to find immediate 'Sunday' after a month. How many days I have to wait for that Sunday?
If you add 7(4times) to 3, you will again reach (3) after 28 days.
3+(4*7) = 3 (7 being identity)
We know, 3(Tue) + 5 days =8 =1(sun)
If you again add 5 days to 3, you will reach Sunday. That is, you have to wait 28+5 =33 days. to reach Sunday after a month.
Groups are another wonder of mathematics. They are applied in all fields of science.

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