What is zero to the zeroth power

Another post on numbers for today. I was reading a post on the question of what 0⁰ is. If you remember your math you'll know that any number a raised to the zeroth power power is 1, namely A⁰=1. There are a couple ways of explaining the logic of this.

One is to say that Aⁿ = 1 x A x A x A ... A_n. In other words, if you want to find the exponent of 25ⁿ then you take 1 and multiply it times 25, n times. So, if n is 0, then it's just 1 by itself, so 25⁰=1.

A second way to look at it, since A^-n = 1 / Aⁿ, that would mean A⁰ equals A^n-n, which is another way of saying Aⁿ / Aⁿ. Since a number divided by itself is always 1, then this would equal 1.

Another way is to think of it like a limit using fractions of exponents. A^1/2 is another way of saying the square root of A, and A^1/3 the cube root and A^1/4 the fourth root. Now as you take bigger and bigger roots of a number, the value gets smaller and smaller. But the value doesn't approach zero, it approaches one. For example, you'll notice if you calculate, say 5^{1/100,000,000} you'll get a number very close to 1 (my calculator gave 1.0000000160943792), and it only gets closer as you increase the value. This makes sense, since the root is just an exponent in reverse, and if you think of multiplying any number times itself that's less than 1, it'll just get smaller and smaller, approaching 0. So, the root of any number greater than 1 will always have to be greater than one, so that when it multiplies by itself it gets bigger. So, the n root of a number greater than 1, as n gets bigger and bigger will get closer to 1. Meanwhile that value of 1 / 100,000,000 is only getting closer to 0. In other words, if you have a formula Aⁿ = x, as n gets smaller and smaller (approaches 0), the value approaches of x approaches 1.

All of this makes perfect sense, until you ask what happens when A = 0, then it all just breaks down. If we think of it the first way, then we'd say that 0ⁿ = 1 x 0 x 0 x 0 ... 0_n; thus 0⁰ = 1. If we look at it the second way, we'd say that 0^n-n = 0ⁿ / 0ⁿ; but that would mean we'd be dividing by 0, which is undefined. For example, since, 0^2-2 would equal 0² / 0², this would equal 0 / 0, which doesn't mean anything. Thus, by the second way, 0⁰ is undefined. But then if we look at it the third way, then we would say, since 0^1/2 = 0 and 0^1/3 = 0 and 0^1/4 = 0, and so on, then it would seem 0⁰ = 0.

Wikipedia has an extensive entry on this question. It's a debated question, apparently with three different positions: either it must equal 1, it is undefined, or it varies based on the circumstance. One fellow named Benson, who falls into the third camp, says, "The choice whether to define 0^0 is based on convenience, not on correctness."

Over at the Measure of Doubt blog, Julia Galef, took this idea and concluded that math is something that is invented for the purpose of being useful. This doesn't mean it's untrue, it might just mean that math is more like a language or a pattern of relations. Any particularly mathematical theorem is true because it's just describing relations between various mathematical elements. But a mathematic formula that describes a phenomenon out in the world would be different; it would be a description, using mathematical symbols instead of words, and would be true to the degree to which it is an accurate description. To me, the idea that a mathematical definition would vary based on convenience seems wrong, since the virtue of math is its precision and internal consistency, and it also seems like the decision to make 0^0 equal 1 is also based on arguments from convenience. So, I'd have to go with the undefined camp.

Another post on numbers for today. I was reading a post on the question of what 0⁰ is. If you remember your math you'll know that any number a raised to the zeroth power power is 1, namely A⁰=1. There are a couple ways of explaining the logic of this.

One is to say that Aⁿ = 1 x A x A x A ... A_n. In other words, if you want to find the exponent of 25ⁿ then you take 1 and multiply it times 25, n times. So, if n is 0, then it's just 1 by itself, so 25⁰=1.

A second way to look at it, since A^-n = 1 / Aⁿ, that would mean A⁰ equals A^n-n, which is another way of saying Aⁿ / Aⁿ. Since a number divided by itself is always 1, then this would equal 1.

Another way is to think of it like a limit using fractions of exponents. A^1/2 is another way of saying the square root of A, and A^1/3 the cube root and A^1/4 the fourth root. Now as you take bigger and bigger roots of a number, the value gets smaller and smaller. But the value doesn't approach zero, it approaches one. For example, you'll notice if you calculate, say 5^{1/100,000,000} you'll get a number very close to 1 (my calculator gave 1.0000000160943792), and it only gets closer as you increase the value. This makes sense, since the root is just an exponent in reverse, and if you think of multiplying any number times itself that's less than 1, it'll just get smaller and smaller, approaching 0. So, the root of any number greater than 1 will always have to be greater than one, so that when it multiplies by itself it gets bigger. So, the n root of a number greater than 1, as n gets bigger and bigger will get closer to 1. Meanwhile that value of 1 / 100,000,000 is only getting closer to 0. In other words, if you have a formula Aⁿ = x, as n gets smaller and smaller (approaches 0), the value approaches of x approaches 1.

All of this makes perfect sense, until you ask what happens when A = 0, then it all just breaks down. If we think of it the first way, then we'd say that 0ⁿ = 1 x 0 x 0 x 0 ... 0_n; thus 0⁰ = 1. If we look at it the second way, we'd say that 0^n-n = 0ⁿ / 0ⁿ; but that would mean we'd be dividing by 0, which is undefined. For example, since, 0^2-2 would equal 0² / 0², this would equal 0 / 0, which doesn't mean anything. Thus, by the second way, 0⁰ is undefined. But then if we look at it the third way, then we would say, since 0^1/2 = 0 and 0^1/3 = 0 and 0^1/4 = 0, and so on, then it would seem 0⁰ = 0.

Wikipedia has an extensive entry on this question. It's a debated question, apparently with three different positions: either it must equal 1, it is undefined, or it varies based on the circumstance. One fellow named Benson, who falls into the third camp, says, "The choice whether to define 0^0 is based on convenience, not on correctness."

Over at the Measure of Doubt blog, Julia Galef, took this idea and concluded that math is something that is invented for the purpose of being useful. This doesn't mean it's untrue, it might just mean that math is more like a language or a pattern of relations. Any particularly mathematical theorem is true because it's just describing relations between various mathematical elements. But a mathematic formula that describes a phenomenon out in the world would be different; it would be a description, using mathematical symbols instead of words, and would be true to the degree to which it is an accurate description. To me, the idea that a mathematical definition would vary based on convenience seems wrong, since the virtue of math is its precision and internal consistency, and it also seems like the decision to make 0^0 equal 1 is also based on arguments from convenience. So, I'd have to go with the undefined camp.

What is zero to the zeroth power