Education Technology

Solution 11818: Algebra Problem Identities for 00 using the TI-89 Family, TI-92 Family, Voyage™ 200 Graphing Calculators, or Derive™ 5 And 6.

Why does 0^0 =1 using the TI-89 family, TI-92 family, Voyage 200, or Derive 5 and 6?

While this was considered a domain error on TI-85, TI-83, and TI-82, there are a number of reasons that algebra problems are handled reasonably with the identity 00=1.

1. Apply the binomial theorem:

00 = (1-1)0 = binomial(0,0) * 10 * (-1)0 = 1.

2. Apply the definition from set theory. If a and b are nonnegative integers, let A and B be sets having cardinality a and b, respectively. Then the value of ab is the cardinality of the set of all mappings from B to A.

In particular, if a = b = 0, then A = B = the empty set, and there is exactly one function that maps the empty set to itself (namely, the empty function). Therefore, 00 = 1.

3. Apply the algebraic notion of an empty product, which makes sense in any semigroup. The empty product is the unit element. When you are multiplying an empty collection of numbers together, it doesn't matter whether the missing numbers would have been zero or not, had they been present. An empty collection of zeros has the same value as an empty collection of ones, or an empty collection of twos, etc. Therefore, 00 = 10 = 20 = x0 = 1.

Finally, note that limits are not relevant to the discussion. Since the function f(x,y) = xy has an essential discontinuity at (0,0), no choice of f(0,0) will make the function continuous there. That does not, however, mean that the function is necessarily undefined. Because of reasons 1-3 above, 00 = 1 is the value that makes sense.

There are some who like to distinguish between the integer 0 and the real number 0, claiming that 00 = 1 when viewed as an operation on integers, but 00 is undefined when viewed as an operation on reals. Others prefer to let 00 = 1 regardless of the type of the operands.