0 / 0 and 0^0 ARE defined!

[brand1130]

Can someone explain the logic to me as to why these two quantities are undefined? ("division by zero is impossible" won't cut it, as I will prove.)<br /><br />First of all, let's take a simple division example. 6 / 3 is the same as saying let's take 6 dots and turn it into three equal piles. How many dots will be in each pile? 2 ... Which is the answer. The same thing goes for 15 / 5 ... If you take 15 dots and turn them into 5 equal piles, you'll have 3 dots in each pile.<br /><br />Okay, with that in mind, let's look at division by zero. You have 1 / 0, which is the same as saying you have 1 dot and you're going to turn it into zero piles. IMPOSSIB LE... This, I understand ... That is not a balanced equation because you have 1 dot on one side and none on the other. Okay, I get it.<br /><br />HOWEVER!!!! When you have 0 / 0, you are taking no dots on one side and turning them into no piles on the other. On both sides, you have nothing, which is balanced and is therefore DEFINED AS 0! 0 / 0 is 0!<br /><br />With that in mind, proving 0^0 = 0 is easy: 0 / 0 = 0^1/0^1 = 0^(1-1) = 0^0 = 0<br /><br />And so, I believe that division by zero in this case alone is possible. Can someone give me a legitimate refute?

 

[brand1130]

Can someone explain the logic to me as to why these two quantities are undefined? ("division by zero is impossible" won't cut it, as I will prove.) <br /><br />First of all, let's take a simple division example. 6 / 3 is the same as saying let's take 6 dots and turn it into three equal piles. How many dots will be in each pile? 2 ... Which is the answer. The same thing goes for 15 / 5 ... If you take 15 dots and turn them into 5 equal piles, you'll have 3 dots in each pile. <br /><br />Okay, with that in mind, let's look at division by zero. You have 1 / 0, which is the same as saying you have 1 dot and you're going to turn it into zero piles. IMPOSSIB LE... This, I understand ... That is not a balanced equation because you have 1 dot on one side and none on the other. Okay, I get it. <br /><br />HOWEVER!!!! When you have 0 / 0, you are taking no dots on one side and turning them into no piles on the other. On both sides, you have nothing, which is balanced and is therefore DEFINED AS 0! 0 / 0 is 0! <br /><br />With that in mind, proving 0^0 = 0 is easy: 0 / 0 = 0^1/0^1 = 0^(1-1) = 0^0 = 0 <br /><br />And so, I believe that division by zero in this case alone is possible. Can someone give me a legitimate refute?

 

[telfrow]

Duplicate post, isn't it? <div class="Discussion_UserSignature"> <strong><font color="#3366ff">Made weak by time and fate, but strong in will to strive, to seek, to find and not to yeild.</font> - <font color="#3366ff"><em>Tennyson</em></font></strong> </div>

 

[kmarinas86]

<font color="yellow">With that in mind, proving 0^0 = 0 is easy: 0 / 0 = 0^1/0^1 = 0^(1-1) = 0^0 = 0</font><br /><br />what does x equal?<br />x^2=4<br />x=4^(1/2)<br />x=2<br /><br />what does 0 equal?<br />0^0=x<br />0=x^(1/0)<br />0=something to the ?undefined?<br /><br />This is why you can't have 0^0. <br /><br />Saying that a^b=c is like saying e^(ln(a)*b)=e^(ln(c)) which leads us to ln(a)*b=ln(c)<br /><br />If a and c are 0 then what is ln a and ln c? Both are negative infinity.<br /><br />0^0 actually is a indeterminant.<br /><br />An example of an indeterminant in calculus is where you have<br /><br />lim [(x-1)/(x-1)] = 1<br />x - /> 1<br /><br />equally, you could have this<br /><br />lim [0*(x-1)/(x-1)] = 0<br />x - /> 1<br /><br />or this<br /><br />lim [3*(x-1)/(x-1)] = 3<br />x - /> 1<br /><br />Indeterminants are mathmatically relevant when x is actually changing and is a constituent of a limit.<br /><br />I suggest you take a calculus class. I took one last year and it got me over the mystery of "undefined".<br /><br />You'll need it if you want to get into higher mathematics.

 

[najab]

http://mathforum.org/library/drmath/view/63315.html

 

[kmarinas86]

<font color="yellow">With that in mind, proving 0^0 = 0 is easy: 0 / 0 = 0^1/0^1 = 0^(1-1) = 0^0 = 0</font><br /><br />what does x equal?<br />x^2=4<br />x=4^(1/2)<br />x=2<br /><br />what does 0 equal?<br />0^0=x<br />0=x^(1/0)<br />0=something to the ?undefined?<br /><br />This is why you can't have 0^0. <br /><br />Saying that a^b=c is like saying e^(ln(a)*b)=e^(ln(c)) which leads us to ln(a)*b=ln(c)<br /><br />If a and c are 0 then what is ln a and ln c? Both are negative infinity.<br /><br />0^0 actually is a indeterminant.<br /><br />An example of an indeterminant in calculus is where you have<br /><br />lim [(x-1)/(x-1)] = 1<br />x - /> 1<br /><br />equally, you could have this<br /><br />lim [0*(x-1)/(x-1)] = 0<br />x - /> 1<br /><br />or this<br /><br />lim [3*(x-1)/(x-1)] = 3<br />x - /> 1<br /><br />Indeterminants are mathmatically relevant when x is actually changing and is a constituent of a limit.<br /><br />I suggest you take a calculus class. I took one last year and it got me over the mystery of "undefined".<br /><br />You'll need it if you want to get into higher mathematics.

 

[telfrow]

Duplicate response, isn't it? <div class="Discussion_UserSignature"> <strong><font color="#3366ff">Made weak by time and fate, but strong in will to strive, to seek, to find and not to yeild.</font> - <font color="#3366ff"><em>Tennyson</em></font></strong> </div>

 

[kmarinas86]

ya...

 

[raghara2]

Actually in division a definable quantity divided by itself equals, by definition of division, one.<br /><br />Also I seen 2^0 = 1<br />3^0 = 1<br />Thus 0^0 = 1 if these two zeroes are equal.<br /><br />In fact there is some possibility that 0^0 is very close to 1. <br /><br />If you have problems to visualize division, the best way is to use rectangle. Area is dividend, one side is divisor, and the side at the right angle to the divisor is a result.<br /><br />Of course there is that pesky saying about definitions in Math. Something like Math can't completely define itself. <br /><br />The only question is why is this in science and astronomy board.

 

[newtonian]

brand1130 - 0/0 is similar to infinitely small divided by infinitely small, with the specification that both infinities are identical in value.<br /><br />Normally, identical quantities divided by itself = 1.<br /><br />Now, factor 0 as 1-1, or more exactly +1+-1.<br /><br />Now, that makes the fraction: 1-1/1-1<br /><br />And the equation: 1-1/1-1=X<br /><br />Now, multiply both sides of the equation by 1-1<br /><br />Now, 1-1=x<br /><br />Therefore x=0<br /><br />Hence 0/0 = 0<br /><br />However, both factors of zero = 1!<br /><br />i.e. +1/+1= 1<br /><br />-1/-1 = 1<br /><br />Now, make y=-1<br /><br />Now 1+y/1+y=x<br /><br />Now subtract y from top and bottom (numerator, denominator?) of the fraction.<br /><br />Now we have 1/1=x, hence x=1.<br /><br />Since 1+y= 0 (remember, y=-1), we have now proven that 0/0=1!<br /><br />OK, I just proved two things: <br /><br />1. 0/0 = 0<br /><br />2. 0/0 = 1<br /><br />What did I do wrong?<br /><br />Now, 10^0 = 1?<br /><br />How does one define 10 to the zeroth power?

 

[kmarinas86]

<font color="yellow">brand1130 - 0/0 is similar to infinitely small divided by infinitely small, with the specification that both infinities are identical in value. <br /><br />Normally, identical quantities divided by itself = 1. <br /><br />Now, factor 0 as 1-1, or more exactly +1+-1. <br /><br />Now, that makes the fraction: (1-1)/(1-1) <br /><br />And the equation: (1-1)/(1-1)=X <br /><br />Now, multiply both sides of the equation by 1-1</font><br /><br />if<br />(1-1)/(1-1)=X <br />then<br />(1-1)=X*(1-1)<br /><br />Then what is X?<br /><br />X can be anything you like.<br /><br />If it can be anything you like, we call that undefined.<br />

 

[newtonian]

Kmarinas86 - OK, that is 0 x 0 not 0/0.<br /><br />Perhaps that is why the Romans elimated zero from their math - i.e. Roman numerals have no zero!<br /><br />When things start getting way out, I try to get down to earth.<br /><br />So, say zero apples divided by zero apples in a fruit stand.<br /><br />You do end up with zero apples, not one apple or anything else we like.<br /><br />Otherwise, that fruit stand might end up with many apples just because the shopkeeper decided to divide his zero apple inventory by zero!!!!<br /><br />Frankly, I like your idea - it could solve world hunger!<br /><br />BTW - what did I do wrong in my above calculation???

 

[kmarinas86]

<font color="yellow">So, say zero apples divided by zero apples in a fruit stand. <br /><br />You do end up with zero apples, not one apple or anything else we like. <br /><br />Otherwise, that fruit stand might end up with many apples just because the shopkeeper decided to divide his zero apple inventory by zero!!!! <br /><br />Frankly, I like your idea - it could solve world hunger! <br /><br />BTW - what did I do wrong in my above calculation???</font><br /><br />"Zero apples" actually means "no apples"<br /><br /><font color="yellow">So, say no apples divided by no apples in a fruit stand.</font><br /><br />You can't divide no apples.<br /><br />Same thing as...<br /><br />"I'm not dividing, for I have nothing."<br /><br />When you don't divide, you have the number you started with.<br /><br />x/0=a<br />x=a*0<br />x must be zero for a to equal anything - indeterminate<br />If x equals 1, then there is no finite solution for a<br /><br />1/0 defies the purpose of the "/" sign<br /><br />But did you know the factorial of 0 is 1<br />and the factorial of 1 is 1 ?<br /><br />Factorials are used to determine "e" which is used for natural log functions.<br /><br />http://mathforum.org/dr.math/faq/faq.0factorial.html<br />http://mathforum.org/library/drmath/view/53562.html<br /><br />Factorials are the closest thing we have to physical (existing) properties.<br /><br />To get any number you just take one factorial x! and divide it by another (x-1)!.<br /><br />1!/0!=1<br /><br />http://en.wikipedia.org/wiki/Gamma_function<br /><font color="yellow">In mathematics, the Gamma function extends the factorial function to the complex numbers. The factorial function of an integer n is written n! and is equal to the product n! = 1 × 2 × 3 × ... × n. The Gamma function "fi</font>

 

[newtonian]

kmarinas86 - On that link on theories - interesting but way off thread theme Do you want to start a thread on this?<br /><br />There are some problems with the article - very interesting problems.<br /><br />The most obvious is semantics - notably the variable definitions of the word "theory."<br /><br />Semantics really complicates matters - hence my thread theme. Note the word "things." That would include theories, models, hypothesis, interpretations, etc., etc.<br /><br />There are other problems with the article, and these also involve definitions - notably - how does one define science? or Creationism? Or evolution?<br /><br />Was historical science pseudoscience simply because the scientific theories (or teachings, pick your word) of the past have proven to be false (the greek prefix psuedo means false in English)?<br /><br />Was Darwin a creationist because he believed in a Creator who breathed life into either one or a few species (see Origin of the Species, by Charles Darwin)?<br /><br />Are evolution and creation antonyms such that one cannot believe in the evolution of creation or in the creation of evolution?<br /><br />Note also the definition of mutations by evolutionists - have you noticed that over time the definiton has come to include not only random changes (change=mutation?) but also informational mutations such as antibiotic resistance, etc.? And by clearly non-random processes including cross-species gene sharing?<br /><br />Most older quotes of scientists concerning mutations are implying random or chance changes, btw. <br /><br />Getting back to thread theme, one must agree on definitons before coming to logical agreement or solid conclusions (as in fact vs. theory).<br /><br />How does one define zero. How does one define 10 to the zeroth power? And, related to this, what are the variable definitions of infinities ( e.g. 1/0 vs. the expansion (e.g. the diameter of our universe during infinite future time [a time-dependent infinity if our universe is fini

 

[kmarinas86]

<font color="yellow">kmarinas86 - On that link on theories - interesting but way off thread theme Do you want to start a thread on this? <br /><br />There are some problems with the article - very interesting problems. <br /><br />The most obvious is semantics - notably the variable definitions of the word "theory." <br /><br />Semantics really complicates matters - hence my thread theme. Note the word "things." That would include theories, models, hypothesis, interpretations, etc., etc. <br /><br />There are other problems with the article, and these also involve definitions - notably - how does one define science? or Creationism? Or evolution?</font><br /><br />Evolution, Intelligent Design, and Creationism have been called many things:<br />Theory<br />Fact<br />Historical Account<br />Hypothesis<br /><br />But rarely are they called <b>theses</b>. The word thesis is a purely objective and invariable term which refers to a made proposition.<br /><br /><font color="yellow">Was historical science pseudoscience simply because the scientific theories (or teachings, pick your word) of the past have proven to be false (the greek prefix psuedo means false in English)?</font><br /><br />History is dead. Today is alive. For science to be alive, it should focus on the present. For me, the most scientific science is the science which deals with today.<br /><br /><font color="yellow">Are evolution and creation antonyms such that one cannot believe in the evolution of creation or in the creation of evolution?</font><br /><br />Depending on the definition of course.<br /><br />In my definition, there are not mutually exclusive.<br /><br />For instance, one can believe in God and Evolution and know Microevolution.<br /><br />Another way of saying it is to say:<br />"One can believe in God and know Evolution and Microevolution."<br /><br /><font color="yellow">Note also the definition of mutations by evolutionists - have you noticed that over time the definiton has come to inc</font>

 

[newtonian]

kmarinas86 - I think we agree on most of the points you covered. However, to clarify:<br /><br />In astronomy, past is very relevant - we are viewing past history of our universe through telescopes!<br /><br />Also, Biblical astronomy, while written long in the past, is accurate and much of it has been proven by now.<br /><br />On mutations: scientists knew about gene sharing across species barriers, such as sharing antibiotic resistance with other species of bacteria, was known from when you were born???<br /><br />Er, how young are you? [you don't have to answer that!)<br /><br />By random, evolutionists generally mean by chance, not requiring informational (one definition of intelligence vs. chance) direction or input.<br /><br />I should warn you of a specific and common logical fallacy - as in this argument:<br /><br />Evolution by chance not requiring intelligent design is proven by the formation of new beneficial traits by mutation (the listener assumes these are chance mutations).[Plus mechanisms, including natural selection]<br /><br />Do you see the problem with definitons? (the argument will point to mutations which are more or less non-random, involving informational matrixes, etc.)

 

[frobozz]

"How does one define zero. How does one define 10 to the zeroth power? And, related to this, what are the variable definitions of infinities ( e.g. 1/0 vs. the expansion (e.g. the diameter of our universe during infinite future time [a time-dependent infinity if our universe is finite in size at this time]."<br /><br />The definitions of the so called "inifinites" are simple. Begin with countable infinity - any set which we can put in a one to one corrospondance with the collection of natural numbers . The "second type" of infinity - in a loose sense of the term as their are many "infinities" is the uncountable infinity. This is simply any infinite set which CANNOT be placed in a one to one corrospondance with the natural numbers. I suppose examples are only fair here, although it's CAN be fun to find them on your own. <br /><br />(a) The set of all even numbers is countable.<br /><br />(b) The set of all rational numbers is uncountable.<br /><br />(c) The set of all irrational numbers is uncountable.<br /><br />(d) The set of all real numbers is uncountable.<br /><br />In the time honored tradition of mathematicians everywhere, I leave the proofs as excercises to the reader. ( you can easily find them only - Search for set theory, cantor, diagonal arguments, countable, etc).

 

[frobozz]

"And the equation: 1-1/1-1=X<br /><br />Now, multiply both sides of the equation by 1-1<br /><br />Now, 1-1=x "<br /><br />Your mistake, assuming I read you correctly happens here. <br /><br />1-1/1-1 = x multiplying both sides by 1-1 gives you<br /><br />1-1 = x (1-1)<br />0 = 0.<br /><br />You do not get the desired conclusion.

 

[ag30476]

0/0 is not defined because of the way rational numbers are defined. The rational numbers is the quotient field of the integers and by construction you can't define division by zero. <br /><br />Essentially a/b=c means that there is a unique number c such that bxc=a. But 0xc=0 for all numbers c by definition so there is no unique c such that 0/0=c. Thus 0/0 is not defined.<br /><br />In fact dividing by zero is a problem and will give wrong answers. <br /><br />This has a good explanation:<br />http://en.wikipedia.org/wiki/Division_by_zero<br /><br />For what division means look at:<br />http://en.wikipedia.org/wiki/Division_%28mathematics%29<br /><br />For the definition of a field, the construction of a quotient field and the field of Rational number see:<br />http://en.wikipedia.org/wiki/Field_%28algebra%29<br />http://en.wikipedia.org/wiki/Quotient_field<br />http://en.wikipedia.org/wiki/Rational_numbers<br />