If you only have zero-questions left

September 18, 2018

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Whether you are an econometrician, a mathematician or a hobbyist with some interest in numbers, there is a high probability that you once racked your brain over the question why you cannot divide by zero. Or the equality 0!=1, why is that not just equal to zero? The number zero has some interesting properties, and in this article, we will try to find some logical reasoning behind these.
Let us start off with some exponential calculations involving zero. First of all, we find that 0^x=0 for any x>0. This makes perfect sense as 0^x = 0*0*0*...*0 for x times. As any number multiplied by zero is zero, this adds up to zero. However, what about x^0? You could say that x^a is x multiplied for a times after each other. Hence, x^3 is three times x: x*x*x. Similarly, x^2 gives x*x and x^1 gives x multiplied just once, which is x. In this sense, x^0 would give x zero times, which gives kind of… nothing? However, as is widely known, we have that x^0=1 for any x \neq 0. What could be a reasonable explanation for that?
What if we take a look at it from another perspective, and start with 2^4? This gives us:

  •        2^4=2*2*2*2=16
  •        2^3=2*2*2=8
  •        2^2=2*2=4
  •        2^1=2

As you can see, the answer is divided by our base number, which is 2, in every step. This is of course clear, as when the exponent decreases by one, you multiply the base one time fewer by itself. From this point of view, we can see why it is clear that 2^0 must equal 1:

  •        2^3=2^4/2=16/2=8
  •        2^2=2^3/2=8/4=4
  •        2^1=2^2/2=4/2=2
  •        2^0=2^1/2=2/2=1

This adds up for all base numbers, as we will find x^0 = \frac{x^1}{x} = \frac{x}{x} = 1, for all x \neq 0.

Does 0^0 equal something or nothing?

An explanation for x^0=1 may be a clarification that you have been waiting for a long time, however it also brings up another, fairly more difficult question: what about 0^0? We have 0^x=0 and x^0=1, but does that mean that 0^0=0 or 0^0=1?
Using limits, we can approach 0^0 in different ways. Firstly, let us try the limit of x^0 where x approaches zero. As we saw previously, x^0 =1 for all x \neq 0. This gives

    \[\lim_{x\to0} x^0 = 1\]

as well, so one might argue that 0^0 = 1. However, we can also take the limit of 0^x, where x can approach zero from both the negative and the positive direction. As zero to the power of any negative number gives a division by zero,

    \[\lim_{x\to0^-} 0^x = undefined\]

Later in this article, we will elaborate on why exactly a division by zero turns out to be undefined. Approaching zero from the positive direction gives

    \[\lim_{x\to0^+} 0^x = 0\]

As limits can bring us to 0^0 approaching 1, 0 and undefined, it seems as if the value of 0^0 can best be said to be undefined. However, most people stick to the opinion that 0^0 = 1, and some that 0^0 = 0. What turns out, is that setting 0^0=1 works out best in most calculations. Take for example the Binomial Theorem, and let us calculate the value of 1^0. The Binomial Theorem is given by

    \[(x+a)^n = \sum_{k=0}^{n} \binom{n}{k} x^k a^{n-k}\]

with the Binomial Coefficient

\binom{n}{k} = \frac{n!}{k!(n-k)!}

Now set x=0, a=1 and n=0. This gives: (0+1)^0 = 1^0, which is, as we have found previously, equal to 1. The Binomial Theorem in this case gives us:

    \[(0+1)^0 = \sum_{k=0}^{0} \binom{0}{k} 0^k 1^{0-k} = \binom{0}{0} 0^0 1^0\]

= \frac{0!}{0!(0-0)!}* 0^0 *1^0 = 1 *0^0*1

As will later in this article be explained, 0!=1. With this equation, we find that using the Binomial Theorem, we can only say that 1^0=1 if we set 0^0=1. Although limits show us different outcomes, and it can therefore be quite a sensitive subject for some mathematicians, it is in most cases the best, or the least complex choice to say that 0^0=1.

The impossibility of dividing by nothing

Another question regarding the number zero is why we in general cannot divide by zero. Why is any division by zero undefined and will a calculator always give an error message? Before we dive into the impossibility of dividing by zero, we will first look at some intuition of why people might argue that a number divided by zero is not undefined but rather approaches infinity.

“Multiplication is just glorified adding and division is just glorified subtraction.” – James Grime

In the words of the passionate mathematician [expand title=”James Grime,”]Next to a passionate mathematician, James Grime is also member of the team behind the YouTube channel Numberphile. This quote and some of the examples in this article are from their video Problems with Zero with Matt Parker.[/expand] “Multiplication is just glorified adding and division is just glorified subtraction.” To illustrate this, take the multiplication 6*5. This is just adding the number 6 for 5 times: 6+6+6+6+6=30. Similarly, 30/6 is subtracting 6 of 30 until this is not possible anymore: 30-6-6-6-6-6=0. As 6 is subtracted 5 times, 30/6=5. In this sense, 30/0 would mean that 0 is subtracted from 30 until the number 0 is reached. However, we find 30-0-0-0-...-0=30. This continues infinitely many times, which gives reason to believe that 30, or any other number, divided by 0 approaches infinity.
Our next step is to use limits in finding a value for 1/0. We find that 

    \[\lim_{x\to0^+} 1/x = \infty\]

where the plus sign again means that we are approaching zero from positive direction. However, approaching zero from the negative direction, we find

    \[\lim_{x\to0^-} 1/x = -\infty\]

 [laread_preformat]For a lot of mathematicians, infinity is not really a number but more of an idea. Hence, we cannot assign the value ‘infinity’ to a function or equality. For this reason, it is not said that the limit of 1/x equals infinity, where x approaches 0 from the positive side, but that it approaches infinity. Otherwise, one could say that the limit of 2/x equals infinity, and that the limit of 1/x equals infinity as well. This would imply that 2 equals 1, which is obviously not true.[/laread_preformat]
Hence, the value of 1/0 can be argued to approach both infinity and minus infinity. As these are two different answers, we say that 1/0 is undefined and that dividing by zero in general is undefined. Therefore, any calculator will give an error message whenever it is given a calculation where dividing by zero is involved.
A real life example of why a number divided by zero can lead to complicated situations, is given in the image below. At first glance, this calculation seems quite right, although the outcome is really odd. Taking a closer look, one sees that after the second last equation, both sides are divided by (a-b). As can be found in the first equation, a equals b and therefore (a-b)=0. Hence, after the second last equation, a division by zero is done on both sides. This gives the strange answer that 2 equals 1, and shows us why division by zero cannot give a proper answer and hence should be seen as undefined.

Source: www.knowyourmeme.com


Last but not least, what about the factorial? If one writes 4 factorial, or 4!, it is equal to 4! = 4*3*2*1 = 24. Similarly, 3! = 3*2*1 = 6, 2! = 2*1 = 2, 1! = 1 and hence, 0! = ... 0? False, 0! equals 1. But why?
Well, let us approach it in the same way we approached x^0. We see that 3!=4!/4, 2!=3!/3 and 1!=2!/2. With this in mind, it is quite logical to say that n!=\frac{(n+1)!}{n+1}, with n \geq 0, and we find 0! = 1!/1 = 1/1 = 1. However, we can also explain this more intuitively. As you might know, 3! describes the number of ways you can order three objects, which is six. Similarly, you can order two subjects in 2! = 2 ways and 1 subject in just 1! = 1 way. It may sound as a rather philosophical question, but in how many ways could zero objects be sorted? Taking into account [expand title=”vacuous truths,”] Vacuous truth is a term associated with both mathematics and logic, and regards the truth of statements about an empty set. As the set of objects in this case is empty, it is said to be vacuously true that the set can be ordered in one way.[/expand] the most logical answer to this question seems to be that the zero objects can be ordered in just one single way, with an empty table as outcome.
Although zero can be quite a tricky number, we have seen that most calculations and equalities around it do make a lot of sense. And this can come in handy, as things generally make more sense and are easier to remember once you have found the intuitively logical reasoning behind it. Luckily, for the most detail- and discussion loving mathematicians, there is always room left for discussions in ambiguous, but highly interesting subjects as calculations with zero.

Dit artikel is geschreven door Marleen Schumacher.

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