ALGORITHM FOR ADDTION OF TWO NUMBERS

Type of addition


1) Integer addition.

a) Positive integer addition. e.g. 2+9=11

b) Negative integer addition. e.g. -2+9=7

2) Float addition

a) Positive float addition e.g. 2.1+9.1=11.2

b) Negative float addition. e.g. -2.2+9.2=7.0

3) Fractional number addition.

a) Positive fractional number addition. e.g. 2 1/2+9 1/2=12.

b) Negative fractional number addition.

e.g. -2 1/2+9 1/2=7.

4) Complex number addition.

a) Positive complex number addition. (2+3i)+(3+3i)=5+6i.

b) Negative complex number addition -(2+3i)+(3+3i)=1

I am dealing with positive integer addition.

A) PROBLEM DIFINITION

1) We want addition of two numbers of n digits. Which is in present defined up to some limit? Our aim is to extent up to n desired digits.

2) We can go for addition of n number of n digits.

E.g 24+24+25+36+14+15+25………………+35=

For base100 system.

E.g 145+2125+369+158+789+………………+245=

For base 1000 system.

As we see the addition n number of two digits and three digits. We can go for addition n number of n digits long.

AXIOM: (satish) Addition of two n digits number i.e. (a + b) gives the result in n digits or (n+1) digits.

Ex (1):-11+11=22. EXPLANATION:-Two digit no. add by two digit number should give the result in n i.e.2,for n+1 i.e.3 above result in two digit number.

Ex(2):-999+999=1998. EXPLANATION:-Three digit no. add by three digit number should give the result in n i.e. 3, for n+1 i.e. 4 above result in three digit number.

DISCRIPTION OF BASE:- We generally dill with base-10 number system in day today life. It deals with ten symbols each of one digit long. If we go for base100 system then we will deals with 100 symbols each of two digits long. Same manner we can go for base 1000, 10000……..up to n base n is multiple of 10.

Ex- In 100 number base system symbol will be 00, 01, 02, 03, 04…….99.

In 1000 number base system symbol will be 000, 001,002,003,004…….999.

In 10000 number base system symbol will be 0000, 0001,0002,0003,0004…….9999.

In 10^5 number base system symbol will be 00000, 00001,00002,00003,00004…….99999.



How we decide the base of any problem?

a):-We can choose the base as our desire (according to calculating limit of computer)

e.g.123456 in 100 base system 12│34│56 . In 1000 base system 123│456 .In 10,000 base systems 0012│3456 and so on.

E.g.57123458 in 100 base system 05│71 │23│45│58 .

In 1000 base system. 057│123│458.

In 10, 00 base systems 0005│7123│4558 and so on.

We make a partition according to base from left hand side

GENERATION OF GENERAL FORMULA:-BY APPLICATION OF ALGEBRA IN DECIMAL NUMBER SYSTEM:-

In decimal number:

1234

9999

In symbolic form:

1 2 3 4

↓ ↓ ↓ ↓

A1 A2 A3 A4

9 9 9 9

↓ ↓ ↓ ↓

B1 B2 B3 B4

(A1*10^3+A2*10^2+A3*10^1+A4*10^0)

+

(B1*10^3+B2*10^2+B3*10^1+B4*10^0

(A1+B1)10^3+(A2+B2)10^2+(A3+B3)10^1+(A4+B4)10^0

GENERAL ALGORITHM

In slot division result always found in slot of equal number of digits i.e. equal to base.

Base10

E.g. 1234+1234=2│4│6│8 e.g. 4321*4321=8│6│4│2

As we se in result each slot contain one element not more then that or less.

Base100

e.g. 1234*1234=24│68. e.g. 4321*4321=86│42.

As we se in result each slot contain two element not more then that or less.

Base1000

e.g. 1234+1234 = 002│468 e.g. 4321+4321= 008│642

As we se in result each slot contain three element not more then that or less.

Base10000

e.g. 1234+1234=0000│2468. e.g. 4321+4321=0000│8642.

As we se in result each slot contain four element not more then that or less.

Base100000

e.g. 1234+1234=00000│02468. e.g. 4321+4321=00000│08642.

As we se in result each slot contain four element not more then that or less.

Base n

General rule for number of element in slot in any base-10 number system.

Each slot will contain n element not more then that or less.

A):- Deeside the base of given number according to our wish or desire in slot

Ex:-base 10

1 0 1 1 1 0 1 1

≡

3 4 1 1 3 4 1 1

Ex:-base 100

13567891113 & 34789410101 01 35 67 89 11 13

≡

03 47 89 41 01 01

Ex:-base 10000

0001365789111131 0001 3657 8911 1131

≡

100147894110101 1001 4789 4110 1011

B):Name each slot from left or right of no. as we wish, let take left to right for first no.

Ex:-from left to right

12345 23456 A1 A2

In symbolic form

23456 34567 B2 B1

C):-Apply formulae as per slot division

for

A1 A2

B2 B1

1):-(A1+B1)

2):-(A2+B2)

D1):-Step for base 100000

12345 23456 A1 A2

In symbolic form

23456 94567 B1 B2



1234523456+2345694567=3580218023…….(1)

FIND:-

a):-int (A1+B1/base)=cy1

E.g. int(23456+94567/10000)=1=cy1

b):-int [(A2+B2+CY1)/BASE] =cy2

E.g. int(12345+23456/10000)=0=cy2

D2):-FIND

a):-mod (A1+B1, base)=18023

b):-mod [(A2+B2++cy1), base] =35802.

c):-mod (cy2, base) =0

Print the result from bottom to top

35802│18023 ………………………………………….(2).

As (1) and (2) are same so our algorithm gives correct result.















Written By

Satish kumar singh ,

Number theory(prime numer)

B.E (Comp.Tech), R.F Engg,

GTL PVT

Mo no +91-9199516727

Sat.wnz@gmail.com

MULTIDIMENSIONAL NUMBER SYSTEM

MULTIDIMENSIONAL NUMBER SYSTEM




BACK GROUND

Multiplication is one of the best fundamental operations which is in used to present world. No doubt all basic fundamental operation is need of any subject. In every era man were doing to calculate as their need, but every time limitation of digit bind them. We have introduced a new concept of multidimensional number system. Which is able to do basic fundamental operation as our desire i.e. we can do +, - , *, /, GCD, LCM and many more operation according to our need of digit limitation. In present time this operation can be performed in 10 to 16 digit number on modern computer. But by application of multidimensional number system we can go for any number of digits as our desire. First time concept of multidimension was introduce by NEWTON & broadly used by EINSTIEN. It is concept of physics & mathematics. But till now it is dream of every one to work with multidimension system. In computer we use three dimension systems; generally two dimensional systems are in used. We have introduced a concept of multidimension in basic fundamental operation of mathematics (number theory).



OBJECTIVE

1) To give a new environment to computer, so that computer can increase their calculating capacity. Without adding complicated hardware feature or combination of several computer.

2) To end up the limitation of digit in any language in mathematical function.



USE

1) Multidimensional number system will increase the range of every mathematical function.

Ex. In +, -, *, /, we can perform the operation up to our need what ever it is?

2) From C coding of newly introduce method of multiplication we can design fastest hardware for multiplication to computer.

3) We can make a calculator in several digit as our need.

4) Helpful in higher scientific calculation can be done by p4 computer; in present day highly upgraded computer is in use.

5) Multidimensional number system can extended to binary number, hexadecimal number system and for any other system.

6) If applied for figure it will give better view.

7) It will enrich many concepts of physics and chemistry.



WHAT OTHER DO FOR CALCULATION IN HIGHER DIGIT?

1) Upgraded computer are used

Ex. Super computer series

2) Combination of several computers performs the task.

Ex. Recently American student made a combination of 50,000 computers for calculating the million digit long 43rd prime number.



WHAT WE ARE DOING.

1) We have simply introduced a concept of multidimensional number system.

2) Simple way to perform the multiplications which take minimum time in calculation.

3) We have introduce the correlation between number system representation & algebra, And get a positive result.



IMPLIMENTATION

1) This process gives us 100% correct result tested by pen & paper. Tested by higher calculating calculator.

2) We have made a certain algorithm for it & it gives positive result in MS-EXCEL & C programming for 1 to 120 digit long result.