Vedic Maths - Squares n Cubes

1. Squares:

1.1. Squares ending in 5:

Separate >> a | 5

Answer : a(a+1) | 25

Eg 35² >> 3 | 5 >> (3 X 4) | 25 >>1225

Sly, 45² >> 4 | 5 >> 20|25 >> 2025

1.2. General case:

Use Cross-multiply method of multiplication explained in previous post.

2. Cubes: (2 digit no >> ab³)

{a + b)³ = a³ + 3a²b + 3ab² + b³

= a³ + a³(b/a)¹ + a³(b/a)² + a³(b/a)³ ……… (1)

+ 2 a²b + 2 ab² …….. (2)

Modus Operandi:

i. Split no as a | b

ii. Find a³ and ratio b/a

iii. From this we get eq (1) by

We get next no by multiplying current no by the ratio b/a.

Left most no is a³.

iv. We get eq (2) by multiplying middle two terms of eq (1) by 2

v. Add only corresponding terms.

Eg. 27³

Here a = 2, b = 3

Therefore, a³ = 8 , b/a = 7/2 (=3.5)

Therefore eq (1) = 8 | 28 | 98 | 343

(8 X 3.5 = 28, 28 X 3.5 = 98, 98 X 3.5 = 343)

eq (2) = | 56 | 196 |

(28 X 2 = 56, 98 X 2 = 196)

Sum = 8 | 84 | 294 | 343

Since only 1 digit can be inside on bar, carry forward

8 | ₈4 | ₂₉4 | ₃₄3

8 | ₈4 | ₃₂8 | 3

8 | ₁₁6 | 8 | 3

19 | 6 | 8 | 3

Therefore, 27³ = 19683

Vedic Arithmetic Part 2: Multiplication

1. Multiplier is 9, 99, 999, 9999, ….

17 X 9 = 153, 17 X 999 = 16983

Here, we treat 17 as m (multiplicand) and 9, 999,…. (multiplier) as n

1.1. no of digits of n >= no of digits of m

Answer : (m – 1) | (n - LHS)

eg. 17 X 999

Answer: (17 - 1) | (999 - LHS) >> 16|(999 - 16) >>> 16983

Sly 17 X 99 >> 16|(99-16) >> 1683

1.2. else

Therefore we split m into p|q where no of digits of q = no of 9’s in n

Answer : (m - 1 - p) | (n + 1 - q)

Eg. 177 X 99

Split >> 1 | 77

Answer: (177 – 1 - 1) | (99 + 1 - 77) >> 175|23 >> 17523

Sly, 243 X 9 >> 24 | 3 >> (243-1-24)|(9+1-3) >> 218|7 >>2187

2. Multiplier = ca, multiplicand = cb & a+b = 10, 100, 1000, ….

ie total of last digits is 10, 100, 1000, … & whose prev part is exactly same

Answer: c(c+1) | ab

Eg. 237 X 233

Here c = 23, a = 7, b = 3

Answer: 23 X 24 | 7 X 3 >> 552 | 21 >>55221

Sly, 54 X 56 >> 30|24>>3024

3. Multiplicands & multiplier are near base

Find 2 small multipliers($) with signs

3.1. Base is 10, 100, 1000, ….

Answer: (base+ sum of 2 small multipliers) | (product of 2 small multipliers : Note base)

For, 97 X 105,

Base is 100, and $ are -3, +5

Answer: (100 + (-3+5) ) | -15 >> 102 | -15

To remove –ve, take carry :: 101 | (100 -15) >> 101|85 >> 10185

Sly, 102 X 103 >> 105|06 (Since base is 100, therefore 2 digits after |)

3.2. Else (working base (wb))

Multiply or divide only LHS of answer

(if wb is greater,multiply else divide)

I always prefer to keep wb greater.

Consider 31 X 33

Here wb is 30 (& base is 10), $ are +1, +3

Therefore, (30+1+3) | 3 >> 34 | 3

Since wb greater, multiply by 30/10 = 3

Answer: 34X3 | 3 >> 102 | 3 >> 1023

4. General case : Cross Multiplication

4.1. 2 digit multiplication

ab X cd >>> ac | ad+bc | bd

Eg. 23 X 12 >> 2 | 4+3 | 6 >> 276

4.2. 3 digit multiplication

abc X def >>> ad | ae+bd | af+dc+be | bf+ce | cf

Eg. 103 X 143 >> 1 | 4 | 6 | 12 | 9 >> 14729 (1 carried forward)

Overview of Mathematics

Click Below Mindmap to see overview of Mathematics. Most of the details are from Wikipedia.


Above mindmap classifies Maths into four field:
1. Quantity
Here we identify what number is. ie. we are dealing with real axis
This branch of Maths has to do with calculation. eg. If I buy a cake worth of Rs. 45 and give you Rs 50 note , you will have to give me Rs. 5 back .

2. Structure
This branch deals with relation between number. eg. If x=1, y=3; then x+y=4
(Assume we have 2 or more nos on same real axis interacting with each other)

3. Space
Here we bring into another axis. This branch deals with measuring land. Therefore, here basic unit is a point which is ordered pair of 2 nos.

4. Change
This relates with function that shows change.

(Please see Wikipedia for more detail)

If you want clickable imagemap-html of above mindmap, please leave your email id in comments.

Vedic Arithmetic Part 1

Sometimes in exam after a long calculation, we worry if our answer is right or wrong. Also, going back and calculating again is waste of time. Here is faster way to verify your answer (of addition, subtraction and multiplication).

1. Digital Root

How to evaluate it --

a. Adding digits
eg. 2351068
Digital root is 2+3+5+1+0+6+8 = 25, 2+5=7

b. Casting Out nines
eg. 2351068
Cast out 3,6 ; 1,8 ; 0 (since their sum is 9) (Here 9 ~ 0)
Therefore digital root is 2+5 = 7

Use :
Used to verify addition , Subtraction and multiplication

Rule: Digital root is preserved during above operation.

eg:

Addition
981723 + 54281 = 1036004
Digital root::
of 981723 is 3
of 54281 is 2
of 1036004 is 5 (which is 3+2, therefore our answer is correct)


Multiplication
213 X 121 = 25773
Digital root ::
of 213 is 6
of 121 is 4
of 25773 is 6

6X4=24 (whose digital root is 2+4=6 , which is same as that of 25773, therefore our answer is correct)

My Curriculum Vitae in form of Mindmap

Click on image

Mind Map -- Yoga

The eight limbs of yoga :
Yama - ethical restraints
Niyama - ethical observances
Asana - postures
Pranayama - breath and prana control.
Pratyahara - control of the senses by the withdrawal of prana from them
Dharana - single-point concentration
Dhyana - unbroken concentration (meditation)
Samadhi - superconsciousness

Mindmap
(Click on image)





Please see Wikipedia for more detail.

Remembering Periodic Table

During my 12th, I was fascinated by Periodic Table. I spent 3 days to remember it and came up with this mnemonic (It is partly created by my IIT Sir).

You need to know Hindi to remember it. (Sorry for my non-Hindi speaking friends - u have to come up with your own mnemonic).

Also it has been 3 years that I have not revised it yet, so I don't remember La-Ac series. I should have wrote it down :-(


(Click on Image)



Columnwise --

Hritik aur LiNa KaRtab dekhene Circus me Fhire.
(H Li Na K Rb Cs Fr)

Beta Mange Car Scooter Baap Roaye.
(Be Mg Ca Sr Ba Ra)

Scooter Yamaha Lagao Ac.
(Sc Y La Ac)

Tiwari Zara Half Rose-falooda Lana.
(Ti Zr Hf Rf)

Valia Nimboo ne Tapad kHaya.
(V Nb Ta Ha)

Crow....Ulta M by Sugrive.
(Cr Mo Sg)

Mange Tc Railwe ticket uNse.
(Mn Tc Re Ns)

FeRuOs -- Harsh (Chemistry class -- what is iron called ???)
(Fe Ru Os Hs)

(Then Niketan went on expedition to find Copper)
In village CoRhIri (corhiri),
(Co Rh Ir)

Niketan ne Ped pe bethe Popat se pucha ???
(Ni Pd Pt )

He said Copper(Cu) Aage Aur hai.
(Cu Ag Au)

But there he only got Zinc Cds Heather graham (known for her bold scenes).
(Zn Cd Hg)

Bola Alladin Gana Inka Talo.
(B Al Ga In Tl)

Captain Siraj Gela Sanki Pabal (respelling of pagal).
(C Si Ge Sn Pb)

Nana Patekar Agnisakshi Se bana Billionaire.
(N P As Sb Bi)

O Style Se phanta Tee-shirt Polo.
(O S Se Te Po)

For Cluch and Break Install At&T. (v famous company - At&T Bell Labs)
(F Cl Br I At)

Helen Ne Arbaaz aur Kareena ka Xe-ray Roentgen(Rn) se karaya.
(He Ne Ar Kr Xe Rn)

Actually there is idiotic story associated with each line or groups of line
eg -

A son ask Dad for Car and Scooter. After hearing this Dad starts to cry. Not only this Son insist for putting AC in Yamaha Scooter :-) On this Dad ask his servant Tiwari (he was my Physics prof during college -No offense intended) to bring half Rosefalooda (since he could not afford full due to ever increasing demands of his son) . Finally Dad gets really frustrated and slaps his son (Nimboo Valia - My friend Harsh Valia, boisterous boy. His pet name is Iboo but we used to tease him as Nimboo)

Village teacher had an unusual way of teaching his student. He used to teach alphabet using names. eg A for Atmaram,.. So when he reaches alphabet O, he writes W and ask "What is this?" He didnot expect any answer since he had not taught that letter. But one person Sugrive answers "Its Mangalram but is tied upside down" :-)

So on ....

This was Mnemonic for Columns.
To remember sequence of columns (ie rows) :

Lisa and Beny Both Came to Nine instead Of Five
(Li Be B C N O F)
Sodium(Na) and Magnesium(Mg) All Sick People Sat Closely)
(Na Mg Al Si P S Cl)
KC college of Arts and Science took Tiwari Valia who was Crying for Money and Fame to teach Conic aur Zn
(K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn)


My point here is almost anything can be learnt it we have childish attitude (by this I mean Inquisitiveness, Simplicity, Playfulness,...).


Thats all folks !!!

Learning Process for Students

We can classify entire learning process into 3 parts:

1. I/P (from Books or Classes to Brain)
2. Remembering things
3. O/P (from Mindmap to appropriate format)

1. I/P (from Books or Classes to Brain)

This part deals with how to read n understand. Since most of subjects we learn are deductive, we are trained to study every thing logically or vertically.

If you are a research student, I would suggest :

1. Prepare Knowledge book/base (using techniques described in part 2).

2. Read Edward DeBono's Lateral Thinking and use techniques mentioned in that book to think creatively. Yes, Creative works are not always accidental but they can be developed if assumptions are challenged.

Remember, It takes an unusual mind to analyze the obvious. Eg, I consider Einstein to be great not because he could solve complex equations (infact in that period, there were Mathematicians and Physicist capable of solving more complex equations). I think he is great because he challenged the obvious assumption of time being non-relative wrt frame of reference.

2. Remembering things

Use following techniques:

1. Mindmap

Use Association(with real life example), Exaggeration, Sex (or probably your GF name/attribute - It difficult to not remember your spouse, and if you have associated an idea/concept with her, I think you will never forget that concept), colors, images/sketches etc... (Read Tony Buzan's book for more detail).

But most important fact is arrangement of ideas in Mindmap (It should reflect your thinking process).

2. Mnemonics

(Eg. Remembering 2nd/3rd row of Periodic table::

Lisa and Beny Both Came to Nine instead Of Five

Sodium(Na) and Magnesium(Mg) All Sick People Sat Closely)

(Neglect Ne and Ar)

IMP:: Simplicity of above two concept is its power.

3. O/P (from Mindmap to appropriate format)

Remember not to take Lisa as Lithium (Sounds funny!! But if above process was fed in Computers, Machine Learning Algorithms would do that). They are two seperate entities in real world but are associated to provide us an aid to remember them.

If you are communicating your ideas to audience/person, you might consider reading NLP books (Frankly I have not studied it in detail. I saw a NLP book in my friend's mom's Library and I read few pages while I was waiting). It suggest that your communication is successful if at the end of communication, mind map in your head is similar to mindmap in receiver's head. Therefore we use PPT during presentation (Actually, most ppts are hierarchical minmaps).

Note, I have not discussed concentration and relaxation techniques. (Pranayam, Meditation are useful) (See Megaliving by Robin Sharma)

I have assumed that most of the readers of this post are Students (especially from Rationalist/Deductive branch of studies eg. Physics, Maths, Biology, CS etc... and not Arts or Literature students).