Funny Vedic Partial Fraction
$latex E = \frac {2x+1}{x^2 -5x +6} = \frac {2x+1}{(x-3)(x-2)}$
$latex E = \frac {A}{x-3}+ \frac{B}{x-2}$
By quick LOOK:
x =3 => $latex A =\frac {2x+1}{x-2} = 7$
x =2 => $latex B = \frac {2x+1}{x-3}= -5$
=> $latex E = \frac {7}{x-3} - \frac{5}{x-2}$
Monday, 1 April 2013
3P in Research
By Prof C.N. Yang (Nobel Prize Physics)
Perception - grasp the problem with intuition and good vision.
Persistent - keep trying, never give up.
Power - to overcome difficulties.
Perception - grasp the problem with intuition and good vision.
Persistent - keep trying, never give up.
Power - to overcome difficulties.
Hairy Ball Theorem
'Hairy Ball Theorem' (Topology)
1. "You can't comb the hairs on a coconut."
2. "There exists a calm point on earth where there is no wind."
Both 1 & 2 are 'Hairy Ball Theorem'.
1. "You can't comb the hairs on a coconut."
2. "There exists a calm point on earth where there is no wind."
Both 1 & 2 are 'Hairy Ball Theorem'.
If we have a ball with hairs sticking out from each point on it, then impossible to comb the hairs flat with a continuous movement without leaving at least one hair sticking up vertically (typically, at the center of a swirl). Not applicable to doughnut.
=> Wind pattern:
Ball = Earth
Combed down hairs = wind
Length of hairs = wind's force
=> there must be a place on Earth where there is no wind at all.
Vedic (Factorize)
Vedic Sutras:
[s1]: proportionally
[s2]: first by first and last by last
Example 1: E= 2x² + 7x +6
Split 7x = 3x+4x
First ratio of coefficient (2x²+3x) -> 2:3
Last ratio of coefficient (4x+6) -> 4:6=2:3
=> 1st factor = (2x+3)
2nd factor:
2x²/(2x) +6/(3)= (x+2)
=> E = (2x+3).(x+2)
Example 2: Factorize E(x, y, z) = x²+xy-2y²+2xz -5yz-3z²
1. Let z = 0
E'= x²+xy-2y² = (x+2y)(x-y)
2. Let y=0
E'= x²+2xz-3z² = (x+3z)(x-z)
=> E(x, y, z) = (x+2y+3z)(x-y-z)
Example 3: P(x, y, z) = 3x² + 7xy + 2y² +11xz + 7yz + 6z² + 14x + 8y + 14z + 8
1. Eliminate y=z=0, retain x:
P = 3x²+14x+8= (x+4)(3x+2)
2. Eliminate x=z=0, retain y:
P = 2y²+8y+8 = (2y+4)(y+2)
3. Eliminate x=y=0, retain z:
P = 6z²+14z+8 =(3z+4)(2z+2)
=> P =(x+2y+3z+4).(3x+y+2z+2)
[s1]: proportionally
[s2]: first by first and last by last
Example 1: E= 2x² + 7x +6
Split 7x = 3x+4x
First ratio of coefficient (2x²+3x) -> 2:3
Last ratio of coefficient (4x+6) -> 4:6=2:3
=> 1st factor = (2x+3)
2nd factor:
2x²/(2x) +6/(3)= (x+2)
=> E = (2x+3).(x+2)
Example 2: Factorize E(x, y, z) = x²+xy-2y²+2xz -5yz-3z²
1. Let z = 0
E'= x²+xy-2y² = (x+2y)(x-y)
2. Let y=0
E'= x²+2xz-3z² = (x+3z)(x-z)
=> E(x, y, z) = (x+2y+3z)(x-y-z)
Example 3: P(x, y, z) = 3x² + 7xy + 2y² +11xz + 7yz + 6z² + 14x + 8y + 14z + 8
1. Eliminate y=z=0, retain x:
P = 3x²+14x+8= (x+4)(3x+2)
2. Eliminate x=z=0, retain y:
P = 2y²+8y+8 = (2y+4)(y+2)
3. Eliminate x=y=0, retain z:
P = 6z²+14z+8 =(3z+4)(2z+2)
=> P =(x+2y+3z+4).(3x+y+2z+2)
Euler: V- E + R = 2
1. Euler wrote to Goldbach @1750, "it astonishes me these properties have not been noticed by anyone else."
Euler's Polyhedron formula:
V- E + R = 2
Remember trick: “VERsion 2”
V= Vertices, E= Edges, R= Regions (or Faces)
Note: = 0 (instead of 2) for torus doughnut
2. Euler proved it 1 year later: intuition leads to discovery, then prove it by logic.
Example:
Football: Vertices=60, Edges E=90, Region R=32 faces (12 pentagons, 20 hexagons):
V- E + R = 60 - 90 + 32 = 2
[caption id="" align="alignright" width="300"]
A football (or soccer ball) icon. (Photo credit: Wikipedia)[/caption]
Euler's Polyhedron formula:
V- E + R = 2
Remember trick: “VERsion 2”
V= Vertices, E= Edges, R= Regions (or Faces)
Note: = 0 (instead of 2) for torus doughnut
2. Euler proved it 1 year later: intuition leads to discovery, then prove it by logic.
Example:
Football: Vertices=60, Edges E=90, Region R=32 faces (12 pentagons, 20 hexagons):
V- E + R = 60 - 90 + 32 = 2
[caption id="" align="alignright" width="300"]
A football (or soccer ball) icon. (Photo credit: Wikipedia)[/caption]
Vedic (Multiply)
Vedic Math & 16 Sutras
[s2]: All from 9 and the last from 10
[s3a]: Vertically and
[s3b]: Cross-wise
Example: 872 x 997 = Y ?
Apply [s2]: (8-9) =-1 , (7-9)= -2 , last (2-10) = -8
872 -> [-128]
[s2]: (9-9) = 0 & (9-9)=0 & last (7-10)=-3
997 -> [-003]
Arrange in 2 vertical columns as:
872 -> [-128]
997 -> [-003]
[s3a]: (Vertically):
[-128] x [-003] =384
[s3b]: (Cross-wise):
872 + [-003] = 869
=> Y = 869,384
Now, Quick Demo : Calculate 892,763 x 999,998 = Y
892,763 [-107,267]
999,998 [-2]
=> Y= 892,761,214,534
[s2]: All from 9 and the last from 10
[s3a]: Vertically and
[s3b]: Cross-wise
Example: 872 x 997 = Y ?
Apply [s2]: (8-9) =-1 , (7-9)= -2 , last (2-10) = -8
872 -> [-128]
[s2]: (9-9) = 0 & (9-9)=0 & last (7-10)=-3
997 -> [-003]
Arrange in 2 vertical columns as:
872 -> [-128]
997 -> [-003]
[s3a]: (Vertically):
[-128] x [-003] =384
[s3b]: (Cross-wise):
872 + [-003] = 869
=> Y = 869,384
Now, Quick Demo : Calculate 892,763 x 999,998 = Y
892,763 [-107,267]
999,998 [-2]
=> Y= 892,761,214,534
Hypotenuse
Hypotenuse
[Greek: Stretch Against]
Pythagoras was a visiting scholar in Egypt where he saw the pyramid workers used a rope construction tool. The closed rope had 3 knots separated at distance apart in ratio 3:4:5. When it was 'stretch against' on 3 poles formed a right-angled triangle.
[Greek: Stretch Against]
Pythagoras was a visiting scholar in Egypt where he saw the pyramid workers used a rope construction tool. The closed rope had 3 knots separated at distance apart in ratio 3:4:5. When it was 'stretch against' on 3 poles formed a right-angled triangle.
Subscribe to:
Posts (Atom)