Equations with InfinityDate: 07/29/98 at 03:37:47 From: Robert Rundle Subject: Infinity Given that infinity is a concept which has not been proven, how is it possible to use it in maths equations? Is it just used as a concept number, and does that mean that any equation which uses infinity is also a concept and cannot be proved or disproved? Date: 07/29/98 at 15:08:46 From: Doctor Rob Subject: Re: Infinity Equations such as 1/infinity = 0 are not equations in the real number (or complex number) system. They are shorthand for some equations involving limits. The "infinity" in the above expression is short for: lim g(x) x->infinity where g(x) is any function which grows without bound as x does. In this case, the equation is shorthand for: 1/[ lim g(x)] = lim 1/g(x) = 0 x->infinity x->infinity This limit also uses the symbol infinity, but the limit has a precise definition: lim f(x) = L x->infinity is defined to mean that for any e > 0, no matter how small, there exists a B (depending on e and f), such that for all x > B, |f(x)-L| < e. In other words, as x grows without bound, f(x) gets arbitrarily close to L. Such equations involving limits are, indeed, provable (or disprovable), and so the shorthand versions are provable (or disprovable). One must, however, be very careful when writing down such shorthand "equations" to be sure the full version is true. For example, infinity - infinity = 0 is not true, and this is one of the ways you can see that infinity does not behave like a real number. To see the correctness of this last statement, write: infinity - infinity = lim (x+c) - lim x = lim (x+c-x) = lim c = c where c is any real number, and the limits are taken as x->infinity, or write: infinity - infinity = lim (2*x) - lim x = lim (2*x-x) = lim x = infinity This means that infinity - infinity is not a meaningful shorthand for any limit equation, since it has no particular fixed value. Such expressions are called "indeterminate" expressions, because you cannot determine their values. - Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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