Why are Operations of Zero so Strange?Date: 03/17/97 at 21:01:38 From: Jonah Knobler Subject: Why are Operations of Zero so Strange? I am a student in an Algebra II class, and I'm still somewhat naive about stuff like division by zero. 1) Why do we say things like 1/0 are undefined? Can't you call 1/0 infinity and -1/0 negative infinity? Why not? 2) What is 0 * (1/0)? Would it be zero, since whatever (1/0) is, we're taking it NO times? Or would it be 1, since a/b * b = a? Is this why we say it's undefined? 3) What is the quantity 0^0 (zero to the zeroth power)? Would it be 0, since 0 times anything is 0, or would it be 1, since anything to the 0 power is 1? Or is this undefined, too? 4) If it is possible to raise something to the zeroth power, can you find the zeroth root of something? Is it one? Is it ever done? It just seems so odd to me that zero, the very center of our entire concept of mathematics (something long held to be the purest, most perfect thing in existence) should have such gaping flaws. - Jonah Knobler Date: 03/18/97 at 13:06:33 From: Doctor Rob Subject: Re: Why are Operations of Zero so Strange? >1) Why do we say things like 1/0 are undefined? Can't you call 1/0 >infinity and -1/0 negative infinity? Why not? 1/0 is said to be undefined because division is defined in terms of multiplication. a/b = x is defined to mean that b*x = a. There is no x such that 0*x = 1, since 0*x = 0 for all x. Thus 1/0 does not exist, or is not defined, or is undefined. You wish to introduce a new element (or maybe two elements) infinity which you wish to append to the real number system. That is not prohibited. After all, that is how we got from natural numbers to integers (appending negative integers and zero), and from integers to rationals (appending ratios of integers), and from rationals to reals (appending limits of convergent sequences), and from reals to complexes (appending the square root of -1). What you end up with is not the real number system, however. Furthermore, if you wish to define the four operations + - * and / for this new system, you probably want them to be the same on real numbers, and just add on the definitions of things like infinity + r and r/infinity, for any real number r. Some of these work fine. It makes sense to define: infinity + r = r + infinity = infinity (-infinity) + r = r + (-infinity) = -infinity infinity + infinity = infinity (-infinity) + (-infinity) = -infinity infinity - r = infinity (-infinity) - r = -infinity r - infinity = -infinity r - (-infinity) = infinity infinity - (-infinity) = infinity (-infinity) - infinity = -infinity infinity * r = r * infinity = infinity for r > 0 (-infinity) * r = r * (-infinity) = -infinity for r > 0 infinity * r = r * infinity = -infinity for r < 0 (-infinity) * r = r * (-infinity) = infinity for r < 0 infinity * infinity = (-infinity) * (-infinity) = infinity infinity * (-infinity) = (-infinity) * infinity = -infinity infinity / r = infinity for r > 0 (-infinity) / r = -infinity for r > 0 infinity / r = -infinity for r < 0 (-infinity) / r = infinity for r < 0 r / infinity = 0 r / (-infinity) = 0 Where we get into trouble is with defining the following: infinity + (-infinity) (-infinity) + infinity infinity - infinity (-infinity) - (-infinity) 0 * infinity infinity * 0 0 * (-infinity) (-infinity) * 0 infinity / infinity infinity / (-infinity) (-infinity) / infinity (-infinity) / (-infinity) infinity / 0 = infinity (-infinity) / 0 = -infinity These expressions are called "indeterminate forms." These can all have a large range of different values, depending on exactly where the "infinity" parts came from. As a result, the system you construct is not closed under addition, subtraction, multiplication, or division. Other indeterminate forms are 0^0, 1^infinity. You will encounter them again when you take calculus. >2) What is 0 * (1/0)? Would it be zero, since whatever (1/0) is, >we're taking it NO times? Or would it be 1, since a/b * b = a? Is >this why we say it's undefined? This is one of the indeterminate forms, 0 * infinity, mentioned above. >3) What is the quantity 0^0 (zero to the zeroth power)? Would it be >0, since 0 times anything is 0, or would it be 1, since anything to >the 0 power is 1? Or is this undefined, too? This, too, is an indeterminate form. Its logarithm is 0 * infinity. >4) If it is possible to raise something to the zeroth power, can you >find the zeroth root of something? Is it one? Is it ever done? Since x^0 = 1 for any nonzero x, only 1 could possibly have a zeroth root, but, again, which x are we going to use? This means that the zeroth root of 1 is again indeterminate. It is never done. >It just seems so odd to me that zero, the very center of our entire >concept of mathematics (something long held to be the purest, most >perfect thing in existence) should have such gaping flaws. These are not considered flaws. Zero just has the property that you can't divide by it. It's a feature, not a bug! :-) When you learn higher abstract algebra, you will find that in some rings there are objects called zero-divisors which, like zero here, have no multiplicative inverse. Stuff happens! We have to learn to live with this situation. As an example of the situation in the last paragraph, consider the arithmetic of the clock. The set of elements are the hours {1, 2, ..., 12}. Addition is performed by adding the integers, and if the result is bigger than 12, subtract 12. Subtraction is performed by subtracting the integers, and if the result is less than 1, add 12. Multiplication is performed by multiplying the integers, and if the result is bigger than 12, subtract 12 repeatedly until it is not. These operations have sensible interpretations with respect to real clocks and time measures. The additive identity (or zero) is 12. The additive inverse of 12 is 12, and that of any other hour h is 12-h. The multiplicative identity is 1. The hours 1, 5, 7, and 11 have multiplicative inverses: themselves! The other hours are zero- divisors, and have no multiplicative inverses. You can verify that 2*h can never give the answer 1 by trying all h's. They are called zero-divisors because of equations like 3*8 = 12, where 12 acts as the zero element, so 3 and 8 divide zero. This example may seem strange, but it is a perfectly good example of what is called a ring in higher abstract algebra. Keep up the good questions! -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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