A summer program and resource for middle school students showing high promise in mathematics

 MATHEMATICS CONCEPTS 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. ******************** 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  BACK MathPath - "BRIGHT AND EARLY" Send suggestions to webmaster@mathpath.org Copyright © 2001- MathPath Last updated - January 23, 2005