A few days ago, I was having dinner with my family whilst absorbed in a discussion with my parents and baby sister about the subject of Mathematics. After a heated discussion on Euclidean geometry and its relevance in the modern age, my sister asked a most interesting question:
"Bhaiya (meaning brother in our native language, Hindi), what is zero divided by zero?"
I was pretty surprised at the question, given that we never divide by zero in mathematics class, but at the same time, I was very intrigued. I promised that I would come up with a proper answer after doing the requisite research, which I did. This article contains my own initial theorizing, as well as what I (and many mathematicians) believe to be the correct answer.
But yes, you may ask, why can't I divide by zero?
Let us leave 0/0 to the side for a while, and take the example of 1/0. We know that any real number, when divided by another real number, gives a real number as the answer. Say 1/0 = x.
But, if one multiplies both sides of the equation with zero, we get the following:
x * 0 = 1
which is simply not possible.
Therefore, we might say that 1/0 is simply an exception to this rule, and try to work it out anyway considering unreal numbers as the answer as well. By comparing this problem to that of cutting a cake into zero pieces (with the cake itself being one piece), we will end up with ∞ as the answer. Even on the Riemann sphere, 1/0 = ∞ is a bijection with very nice properties. (A bijection is a case of two different sets, in this case 1/0 and ∞, having the same number of elements.) So, we may say that any real number divided by 0, including 0, is ∞.
However, going forward with 1/0 = ∞, we would end up with a similar equation of ∞ * 0 = 1. Considering this to be true (as we are considering 1/0 = ∞ as true) would mean that infinity simply doesn't exist, as no number that gives 1 when multiplied with 0 exists. This also means, of course, that the hypothesis of dividing a real number by zero to yield ∞ is also not true, which also means, sadly, that 0/0 and 1/0 are NOT ∞ (which, somewhat ironically, gives credit to the rule of dividing a real number by another real number giving a real number).
Going forward with the rule of dividing a real number by another real number giving a real number, we may say, thus, that 0/0 = 0 or 1, right? However, haven't we just proven that dividing a number by zero is not allowed?
Well... not really. If we say that 0/0 = x (for example), we can also say that x * 0 = 0, which is perfectly true for any real or unreal number.
So, the possible values for 0/0 are now 1, 0 or any number x.
Now, let us prove, by method of contradiction, why 0/0 is neither 1 nor 0.
Say 0/0 = 1; then (0/0)*2 = 1 * 2
which means 0/0 = 2
but 0/0 is 1 also and 1 is not equal to 2.
Thus, 0/0 is not 1.
Say 0/0 = 0; then 0/0 = 0/1
which means that (0/0) + 1 = 1
Now, 0/0 + 1/1
= (0*1 + 0*1)/0*1
= (0 + 0)/0
= 0/0 which we are assuming to be 0
but that means 0 = 1
which is not true.
So, we say that 0/0 is neither 1 nor 0; and due to the properties of this equation, 0/0 could have only been 0 or 1! We CAN say that 0/0 is some number x, but due to the number 0 itself, whose properties, again, we created, 'x' has no meaning, as we decreed that dividing by 0 is not allowed in mathematics (and we can see why; it is such a God-awful conundrum!)
Hence the final answer-
0/0 is undefined.
QED.
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"Bhaiya (meaning brother in our native language, Hindi), what is zero divided by zero?"
I was pretty surprised at the question, given that we never divide by zero in mathematics class, but at the same time, I was very intrigued. I promised that I would come up with a proper answer after doing the requisite research, which I did. This article contains my own initial theorizing, as well as what I (and many mathematicians) believe to be the correct answer.
Now, there are several answers to this question. One might say 0/0=1, as any real number 'r' gives 1 when divided by itself, and 0 is definitely a real number. Yet others may say 0/0=0, as 0 divided by any real number yields 0. Some may say 0/0 is ∞, citing the Riemann sphere and calculus to say that, since 1/0 is ∞, 0/0 must be ∞ also. To properly understand and realize the correct answer to 0/0, we must go back to when mathematics was first conceived, and when it evolved into what we know it as, today.
Mathematics, as well as numbers themselves, are concepts completely original, completely created by us. The sciences work with plants, animals, chemicals, energy, mass, time and such other units to understand the world around us, but math isn't like that. Math is a discipline we created for our own convenience, to make our lives easier. We represent units such as speed, time and distance in numerical values not because it is impossible to understand them any other way, but because it is much easier to compute their value when representing it numerically.
Addition, subtraction, multiplication and division are phenomena that we not only named, but created for our own convenience, quite unlike refraction of light and the law of energy conservation (for example), which are simply phenomena we have observed, named, and since today, refer to by those names. If we didn't exist, these happenings would have existed anyway, unlike addition and multiplication, and math as a discipline!
Again: we created mathematics. Therefore, it wouldn't be an unreasonable claim to say that we also created the rules defining math, right? One of those rules is that we are not allowed to divide by zero, in any shape or form.
Mathematics, as well as numbers themselves, are concepts completely original, completely created by us. The sciences work with plants, animals, chemicals, energy, mass, time and such other units to understand the world around us, but math isn't like that. Math is a discipline we created for our own convenience, to make our lives easier. We represent units such as speed, time and distance in numerical values not because it is impossible to understand them any other way, but because it is much easier to compute their value when representing it numerically.
Addition, subtraction, multiplication and division are phenomena that we not only named, but created for our own convenience, quite unlike refraction of light and the law of energy conservation (for example), which are simply phenomena we have observed, named, and since today, refer to by those names. If we didn't exist, these happenings would have existed anyway, unlike addition and multiplication, and math as a discipline!
Again: we created mathematics. Therefore, it wouldn't be an unreasonable claim to say that we also created the rules defining math, right? One of those rules is that we are not allowed to divide by zero, in any shape or form.
But yes, you may ask, why can't I divide by zero?
Let us leave 0/0 to the side for a while, and take the example of 1/0. We know that any real number, when divided by another real number, gives a real number as the answer. Say 1/0 = x.
But, if one multiplies both sides of the equation with zero, we get the following:
x * 0 = 1
which is simply not possible.
Therefore, we might say that 1/0 is simply an exception to this rule, and try to work it out anyway considering unreal numbers as the answer as well. By comparing this problem to that of cutting a cake into zero pieces (with the cake itself being one piece), we will end up with ∞ as the answer. Even on the Riemann sphere, 1/0 = ∞ is a bijection with very nice properties. (A bijection is a case of two different sets, in this case 1/0 and ∞, having the same number of elements.) So, we may say that any real number divided by 0, including 0, is ∞.
However, going forward with 1/0 = ∞, we would end up with a similar equation of ∞ * 0 = 1. Considering this to be true (as we are considering 1/0 = ∞ as true) would mean that infinity simply doesn't exist, as no number that gives 1 when multiplied with 0 exists. This also means, of course, that the hypothesis of dividing a real number by zero to yield ∞ is also not true, which also means, sadly, that 0/0 and 1/0 are NOT ∞ (which, somewhat ironically, gives credit to the rule of dividing a real number by another real number giving a real number).
Going forward with the rule of dividing a real number by another real number giving a real number, we may say, thus, that 0/0 = 0 or 1, right? However, haven't we just proven that dividing a number by zero is not allowed?
Well... not really. If we say that 0/0 = x (for example), we can also say that x * 0 = 0, which is perfectly true for any real or unreal number.
So, the possible values for 0/0 are now 1, 0 or any number x.
Now, let us prove, by method of contradiction, why 0/0 is neither 1 nor 0.
Say 0/0 = 1; then (0/0)*2 = 1 * 2
which means 0/0 = 2
but 0/0 is 1 also and 1 is not equal to 2.
Thus, 0/0 is not 1.
Say 0/0 = 0; then 0/0 = 0/1
which means that (0/0) + 1 = 1
Now, 0/0 + 1/1
= (0*1 + 0*1)/0*1
= (0 + 0)/0
= 0/0 which we are assuming to be 0
but that means 0 = 1
which is not true.
So, we say that 0/0 is neither 1 nor 0; and due to the properties of this equation, 0/0 could have only been 0 or 1! We CAN say that 0/0 is some number x, but due to the number 0 itself, whose properties, again, we created, 'x' has no meaning, as we decreed that dividing by 0 is not allowed in mathematics (and we can see why; it is such a God-awful conundrum!)
Hence the final answer-
0/0 is undefined.
QED.
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Wow!!.. this refreshed one of my subject called 'Theory of Mathematics" during my graduation. We used to prove such equations like 1+0=1, 1*X=X, X*0=0.... Genius writing. Keep it up.
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