Difference between "Not Defined" and "Infinity" [Division by zero]

Replies

• 

  Anil Jain

  The_Big_K

  Okay, let's see if we have confusion about this concept: -
  
  What is the result that you get when you divide a number by zero? Is it "infinity" or is it "not defined"?
  
  Is there any difference between the two?
  
  Or Infinity is not defined? 😁

  There are two concepts...
  
  Let me try to explain this...
  
  Something divided by 0 would always be 'undefined', but my question is is 'Infinity' is defined. We assume that countless figure is 'Infinity', but isn't that undefined again.
  
  Anways in Layman's language why something divided by 0 is undefined. I found beautiful excerpt as follows:
  
  twelve divided by three:
  
  You have twelve cookies to share equally among three children, how many cookies does each child get?
  
  five divided by one:
  
  You have five cookies to share equally among one child, how many cookies does that child get?
  
  zero divided by one:
  
  You have zero cookies to share equally among one child, how many cookies does that child get?
  
  Now comes the philosophical part:
  
  one divided by zero:
  
  You have one cookie to share equally among zero children, how many cookies does each child get?
  
  Claude
  
  It is good to 'make sense' out of the choices so that you don't have to rely on memory. It is even better if the kids can make sense out of it! (Surrendered K... 😁)
  
  -CB
• 

  Kaustubh Katdare

  One cookie given to zero children. Each gets zero cookies , woa! So one divided by zero is zero
• 

  Raviteja.g

  yes as crazyboy said that infinity is a countless number and undefined is a number we know but we can not define which is correct
  right?
  but as per above explanation number divided with zero is zero
  what a logical explanation!
• 

  pooja sehgal

  dividing by zero we get infinity in mathematics
  actually it is x/0=approaches to infinity
  and in maths infinity is ( largest no of items that can b defined)
  it is different from other real numbers, not exactly defined
  so in terms of maths it is not defined
  but in other philosiphical, logical terms one can reach to different meaning as explained above
• 

  Anil Jain

  The_Big_K

  One cookie given to zero children. Each gets zero cookies , woa! So one divided by zero is zero

  
  Wow Biggi... 😁
  
  you Got it 😁 that, I was talking about invisible man who do not eat cookies... 😁, thats why cookies= 0
  
  
• 

  vipandeep

  Lets try it this way.
  
  x = a/b or xb=a we need to find x right?
  
  now if b = 0 x.0=a 0=a
  
  now two cases arise
  
  1. if a = 0
  
  if a = 0 then we can have any number as a solution of xb=a because no matter what value is x it doesn't matter since x.0 = a where a = 0
  
  2. if a not = 0
  
  then we have problems. x.0=a where a is not 0. we need to find a number x for which the x.0 becomes a. but there is not way we can get x so that x.0 equals a non-zero number.
  
  such number does not exist.
  
  so it is undefined.
• 

  Ankita Katdare

  @Vipandeep: That's a good explanation. So, it is clear that undefined and infinity are two different things, right?
• 

  silverscorpion

  Hmm.. Let me try something..
  
  Let's say we have a number 5. Divide it by 10. You get 0.5.
  If you divide it by 0.1, you get 50. Going on, we have,
  
  5/0.1 = 50; 5/.01 = 500; 5/.001 = 5000 etc..
  
  We see that as the denominator approaches zero, the value of the division becomes larger and larger.
  So, when the denominator actually reaches zero, then the division will give you a very very large number.
  We call that very large number by the name infinity. The exact value of infinity is not defined. But infinity itself is very well defined.
  
  Now, undefined is something else.
  Take the expression 0/0. What will it give? You have,
  
  1) 0 divided by any number is 0
  2) any number divided by 0 is infinity
  3) any number divided by the same number is 1
  
  Now, 0/0 is 0 or infinity or 1?? This is undefined.
  So, expressions like 0/0. 0/infinity, infinity/0, 0^0, 0^infinity etc.. are undefined.
  But infinity is defined. 😀😀
• 

  Ankita Katdare

  silverscorpion

  We see that as the denominator approaches zero, the value of the division becomes larger and larger.
  So, when the denominator actually reaches zero, then the division will give you a very very large number.
  We call that very large number by the name infinity. The exact value of infinity is not defined. But infinity itself is very well defined.
  
  Now, 0/0 is 0 or infinity or 1?? This is undefined.
  So, expressions like 0/0. 0/infinity, infinity/0, 0^0, 0^infinity etc.. are undefined.
  But infinity is defined. 😀😀

  Very clearly stated. Thank you. It is very clear now. 😀
• 

  ISHAN TOPRE

  Yeah.You have explained the meaning of life. lol :razz:
• 

  silverscorpion

  ishutopre

  Yeah.You have explained the meaning of life. lol :razz:

  Hmm.. All hail me!!
  
  By the way, on an unrelated note, the answer to life, the universe and everything is 42. 😎
• 

  Deepika Bansal

  SS: That's a well stated answer. Thank you for clearing the difference between infinity and undefined.
• 

  ajayk2

  Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
  
  (i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
  
  (ii) a/0 is undefined (a is a non-zero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
  
  Now I've a question here.
  
  We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0.
• 

  ajayk2

  Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
  
  (i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
  
  (ii) a/0 is undefined (a is a non-zero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
  
  Now I've a question here.
  
  We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0.
• 

  silverscorpion

  ajayk2

  Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
  
  (i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
  
  (ii) a/0 is undefined (a is a non-zero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
  
  Now I've a question here.
  
  We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0.

  
  Hi Ajay,
  Refer to my post above.
  a/0 is not undefined. It's what we call infinity. 0/0 is undefined.
  And tan 90 is infinity precisely because cos 90 is zero..
• 

  ajayk2

  Yes I read your explanation but my point is that way the denominator would be very close to zero but it will never be equal to zero. You are talking about the concept of limit. It's okay to write
  
  lim (tan x) = inf. but tan 90 = inf is wrong I think. so lim a/x has got meaning but
  x-> 90 x->0
  
  a/0 is undefined and hence meaningless. Tell me what you think.
• 

  Nobody_

  If n is any number.
  
  n/0 = undefined
  (going by the cookie story..if u have n cookies and u give them to no one, then you simply cannot determine how many each will receive..it's sort of meaningless, hence undefined)
  
  0/0
  this has 2 theories:
  a) undefined (following the same cookie story; the division being pointless, and thus undefined)
  b) Infinity! (because 0 multiplied by anything is 0, following rules of multiplication where both the dividend and divisor are 0)
  
  0/n = 0
  (any number multiplied with 0 gives 0)
• 

  Ramani Aswath

  I missed this post. Since school days I have been fascinated by this. Actually there are many infinities some of which are demonstrably larger than others.
  This article sheds some (darkish!) light on this:
  Strange but True: Infinity Comes in Different Sizes - Scientific American
  
  In his entertaining book:'One, Two, Three Infinity, George Gamov has a chapter on this. I strongly recommend this book to all. According to him there are three infinities, each of a different size.
• 

  Shashank Moghe

  Anil Jain

  one divided by zero:
  
  You have one cookie to share equally among zero children, how many cookies does each child get?

  Wrong question. There is no child to begin with. It is like being asked to write one of those essays "If I were not born". If I were not born, I would not be sitting in this school listening to a teacher asking me to write an essay.
• 

  Shashank Moghe

  ajayk2

  Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
  
  (i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
  
  (ii) a/0 is undefined (a is a non-zero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
  
  Now I've a question here.
  
  We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0.

  If you try that on a scientific calculator, it should give you NaN (Not a Number). It is not "infinity". Tan(90) is just "not defined".
• 

  Shashank Moghe

  silverscorpion

  Hi Ajay,
  Refer to my post above.
  a/0 is not undefined. It's what we call infinity. 0/0 is undefined.
  And tan 90 is infinity precisely because cos 90 is zero..

  Division by zero is not defined. Any hypotheses based on the presumption of the kind x/b = a where b = 0, are making a fundamental mistake of assuming there exists the RHS 'a'. It does not, because the operation is not defined!