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Difference between revisions of "Division of Zero by Zero"

(→‎Jyotiraditya Jadhav Proof for Zero by Zero)
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== Detailed proof ==
 
== Detailed proof ==
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We will form two solution sets namely set(A) and set(B)
+
We will form two solution sets (namely set(A) and set(B))
  
  
 
Solution set(A):  
 
Solution set(A):  
  
−
Now if we divide zero by zero then
+
If we divide zero by zero then
  
 
<math>0/0</math>  
 
<math>0/0</math>  
  
−
we can write the 0 in the numerator as <math>(1-1) </math> and again other in the denominator as <math>(1-1)</math>,  
+
We can write the 0 in the numerator as <math>(1-1) </math> and in the denominator as <math>(1-1)</math>,  
  
  
−
=<math>(1-1)/(1-1)</math> and that equals to be <math>1</math>
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=<math>(1-1)/(1-1)</math> equaling <math>1</math>
  
  
−
Now we can write the 0 in the numerator as <math>(2-2) </math> and again other in the denominator as <math>(1-1)</math>,  
+
We can then write the 0 in the numerator as <math>(2-2) </math> and in the denominator as <math>(1-1)</math>,  
  
  
Line 35: Line 35:
  
  
−
Now we can write the 0 in the numerator as <math>( infinity- infinity) </math> and again other in the denominator as <math>(1-1)</math>,  
+
We can even write the 0 in the numerator as <math>( \infty- \infty) </math> and in the denominator as <math>(1-1)</math>,  
  
  
−
=<math>( infinity-infinity)/(1-1) </math>
+
=<math>( \infty-\infty)/(1-1) </math>
  
−
= <math> infinity (1-1)/(1-1) </math>                                                            [Taking <math> infinity  </math> as common]
+
= <math> \infty(1-1)/(1-1) </math>                                                            [Taking <math> \infty</math> as common]
  
−
= <math> infinity  </math>
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= <math> \infty</math>
  
  
−
So, the solution set(A) comprises of all the values of real numbers.  
+
So, the solution set(A) comprises of all real numbers.  
  
  
−
set(A) = <math>\{- infinity.....-3,-2,-1,0,1,2,3.... infinity \}  </math>
+
set(A) = <math>\{- \infty.....-3,-2,-1,0,1,2,3.... \infty\}  </math>
  
  
 
Solution set(B):
 
Solution set(B):
  
−
Now if we divide zero by zero then
+
If we divide zero by zero then
  
 
<math>0/0</math>  
 
<math>0/0</math>  
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=<math>0^1/0^1 </math>
 
=<math>0^1/0^1 </math>
  
−
= 0^(1-1)                                                                              [Laws of Indices, <math>a^m/a^n = a^m-n </math>]
+
= 0^(1-1)                                                                              [Laws of Indices, <math>a^m/a^n = a^{m-n} </math>]
  
 
= <math>0^0 </math>
 
= <math>0^0 </math>
  
−
=<math>1 </math>                                                                                        [Already proven<ref>https://brilliant.org/wiki/what-is-00/</ref>]
+
=<math>1 </math>                                                                                        [https://brilliant.org/wiki/what-is-00| Already proven]
  
  
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−
Now we can get a finite value to division of <math>0/0 </math> by taking intersection of both the solution sets.
+
Now we can get a finite value to division of <math>0/0 </math> by taking the intersection of both the solution sets.
  
 
Let the final solution set be <math>F </math>
 
Let the final solution set be <math>F </math>
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<math>A\bigcap B </math> = <math>F </math>
 
<math>A\bigcap B </math> = <math>F </math>
  
−
<math>\{- infinity.....-3,-2,-1,0,1,2,3....infinity \}  </math> <math>\bigcap </math> <math>\{1\} </math>
+
<math>\{- \infty.....-3,-2,-1,0,1,2,3....\infty\}  </math> <math>\bigcap </math> <math>\{1\} </math>
  
 
<math>F </math> = <math>\{1\} </math>
 
<math>F </math> = <math>\{1\} </math>
  
  
−
Hence proving and deriving value of  <math>0/0 =1 </math>
+
Hence proving <math>0/0 =1 </math>

Latest revision as of 19:07, 4 February 2023

Division of Zero by Zero, is an unexplained mystery, since decades in field of Mathematics and is refereed as undefined. This is been a great mystery to solve for any mathematician and rather to use limits to set value of Zero by Zero in differential calculus one of the Indian-Mathematical-Scientist Jyotiraditya Jadhav has got correct solution set for the process with a proof.

About Zero and it's Operators

Discovery

The first recorded zero appeared in Mesopotamia around 3 B.C. The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth

Operators

"Zero and its operation are first defined by [Hindu astronomer and mathematician] Brahmagupta in 628," said Gobets. He developed a symbol for zero: a dot underneath numbers.

Detailed proof

We will form two solution sets (namely set(A) and set(B))


Solution set(A):

If we divide zero by zero then

$0/0$

We can write the 0 in the numerator as $(1-1)$ and in the denominator as $(1-1)$,


=$(1-1)/(1-1)$ equaling $1$


We can then write the 0 in the numerator as $(2-2)$ and in the denominator as $(1-1)$,


=$(2-2)/(1-1)$

= $2 (1-1)/(1-1)$ [Taking 2 as common]

= $2$


We can even write the 0 in the numerator as $( \infty- \infty)$ and in the denominator as $(1-1)$,


=$( \infty-\infty)/(1-1)$

= $\infty(1-1)/(1-1)$ [Taking $\infty$ as common]

= $\infty$


So, the solution set(A) comprises of all real numbers.


set(A) = $\{- \infty.....-3,-2,-1,0,1,2,3.... \infty\}$


Solution set(B):

If we divide zero by zero then

$0/0$

We know that the actual equation is $0^1/0^1$


=$0^1/0^1$

= 0^(1-1) [Laws of Indices, $a^m/a^n = a^{m-n}$]

= $0^0$

=$1$ Already proven


So, the solution set(B) is a singleton set


set(B) =$\{1\}$


Now we can get a finite value to division of $0/0$ by taking the intersection of both the solution sets.

Let the final solution set be $F$


$A\bigcap B$ = $F$

$\{- \infty.....-3,-2,-1,0,1,2,3....\infty\}$ $\bigcap$ $\{1\}$

$F$ = $\{1\}$


Hence proving $0/0 =1$

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