Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Division of Zero by Zero"

m (latexed infinities)
m (Grammar and readability edits, lmk if it's not needed or wanted)
Line 10: Line 10:
  
 
== Detailed proof ==
 
== Detailed proof ==
−
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>
+
=<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>( \infty- \infty) </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>,  
  
  
Line 45: Line 45:
  
  
−
So, the solution set(A) comprises of all the values of real numbers.  
+
So, the solution set(A) comprises of all real numbers.  
  
  
Line 53: Line 53:
 
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>  
Line 75: Line 75:
  
  
−
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>
Line 87: Line 87:
  
  
−
Hence proving and deriving value of  <math>0/0 =1 </math>
+
Hence proving <math>0/0 =1 </math>

Revision as of 17:03, 2 November 2021

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<ref>https://brilliant.org/wiki/what-is-00/</ref>]


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 $0/0 =1$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Division_of_Zero_by_Zero&oldid=164485"