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When Professor John Horton Conway entered the lecture hall one morning at the MathPath 2003 summer program, a student asked him what 0/0 is.
Conway said "it is a symbol, " and continued. "Symbols don't mean
anything until someone gives them a meaning. A more appropriate question then is this: What does 0/0 mean? Generally, what does a/b mean?"
What does "a/b" mean?
Here a and b are the names of numbers.
Answer: a/b is the number c such that cb = a.
What does 6/2 mean?
Answer: The number c such that 2c = 6. In fact, there is just one such number, namely 3. So "6/2" means 3.
What does "0/0" mean?
0/0 means the the number c such that 0×c = 0. What value of c will make this equation true?
How about 1? or 2, or -26/31? Yes! c can be any number and still satisfy 0×c = 0. Therefore, 0/0 does not mean any particular number - or even anything until we give it some new meaning.
What does "1/0" mean?
1/0 is the number c such that 0×c = 1. But there is no such number. So 1/0 has no meaning. It follows that 0 cannot divide any number except itself.
What does "0/1" mean?
0/1 is the number c such that 0 = 1×c. So c = 0. Hence 0/1 = 0. Similarly, 0/b = 0 for any b ≠ 0. This is interesting. zero divided by any non-zero number is zero. This is a unique property of zero; this is equivalent to the statement that zero multiplied by any non-zero number is zero.
What does "1/¥" mean?
This symbol,1/¥, does not make sense, because ¥ is not a number - it is only an idea. But suppose it is a number.
Then, 1/¥ would be the number c such that ¥×c = 1. So, there is no such number.
What does "¥/¥" mean?
While it seems that any positive value of c will satisfy ¥ = c × ¥, the equation has no meaning because ¥ is not a number. Therefore, ¥/¥ has no meaning.
What does "00" mean?
In trying to answer this, we could first ask what "0n" means for n ≠ 0, n an integer.
For instance, "03" = 0×0×0 = 0.
01 = 0.
So 0n is the value of the string where 0 occurs n times.
Then, 00 is the value of the string where 0 occurs 0 times. But if 0 occurs 0 times, the string has no value and meaning.
What does "a0" mean?
Here again it is the value of the string where a occurs zero times. However, a0 = an-n = an/an = 1 for n ≠ 0, a ≠ 0.
Thus we assign a0 the value 1 when a ≠ 0.
What does "¥0" mean?
Since ¥ is not a number, this symbol has no meaning."
It is popular to call 0/0 indeterminate, which is justified because any value will satisfy c.
¥/¥ is also a candidate to be called indeterminate and it is often so called, but it is not even indeterminate because
¥ is not a number and so ¥/¥ has no meaning.
It is also popular to consider 1/0 as infinity; this is the result of thinking of 1/0 as the limit of n = 1/(1/n). However, we found that 1/0 has no value. The popular notion of giving 1/¥ the value 0 is in thinking of it as the limit of 1/n. However, we found that 1/¥ is not 0.
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A NOTE ON 1/0
Bhaskara II (1114 - 1185 A.D.), was one of the most accomplished of all India’s great mathematicians. He is credited [1] with explaining the previously misunderstood operation of division by zero. He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3. So, dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. Ultimately, therefore, dividing one into pieces of zero size would yield infinitely many pieces, indicating that 1 ÷ 0 = ∞ (the symbol for infinity).
[1] Baskara II, Lilavati
(G.R.T.)
Ü
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Last updated - January 23, 2005
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