Think about any two numbers you add together, either positive or negative, rational or irrational, real or imaginary. Regardless of the non-zero real numbers that you add together or subtract from, the number will change. For example:

2 + 2 = 4
1 ⁄ 2 - 2 = -3 ⁄ 2
Π + e ≈ 5.85987448205...

However, have you truly thought about how the only case where this is not true is with the number 0? This is a unique property that does not occur with any other real number. When we add anything to 0, we get itself. This contradicts of the most basic principles of numbers called the axiom of Archimedes, which says “that if you add something to itself enough times, it will exceed any other number in magnitude” (Seife 20). This can never occur with the number zero, so this is one way that it misbehaves. Regardless of whether we add 0 to itself, or if you add it to something else, it is impossible for it to increase. This is such an abstract concept, that in addition to videos being made about common rules of addition and subtraction, they have to specifically make videos to help children understand these ideas, just like the ones made by Jack Hartmann.

Multiply by Zero:

When it comes to multiplying two numbers, it can be thought of as a rubber band (Seife 20). When we multiply by a number greater than one, then the rubber band stretches bigger by that factor. When we multiply by a number between one and zero, then the rubber band shrinks or gets smaller by that factor. However, think about multiplying by the number zero. When we multiply anything by zero, the solution is always zero. Think about what this does to that rubber band. Suddenly the whole rubber band is at zero, and it seems so abnormal (Seife 21). This idea of anything being multiplied by zero is an interesting one. Take for example the distributive property. This is the idea that a (0+0) =a(0)+a(0). We know that (0+0)=0, so this means that a(0)=a(0)+a(0). However, this means that there was a value that when added to itself, it is still itself. This only occurs when we have the number 0. Therefore, everything multiplied by itself is 0 (Seife 23).

Dividing Zero:

"Division involving zero seems to cause more than ordinary difficulty for students of today as it did for mathematicians of the eighteenth and nineteenth century" (Lappan, Wheeler 42). There are a variety of cases when the digit 0 appears in mathematics. Because of the connotations of the number meaning nothing, often times young children will look at the number 0 as meaning nothing. This especially comes from the abnormal outcomes that occur with this number 0 depending on what is happening, especially because there is a unique whole number solution to 0 ÷ 6, no solution exists for 6 ÷ 0, and infinite solutions exist for 0 ÷ 0 (Lappan, Wheeler 43). To start, let us address the idea of dividing 0 by another number. In school we discuss division as taking the dividend as the number of something, such as buttons, and the divisor as how many groups we are dividing the dividend into evenly. Therefore, if we had 6 buttons and wanted to divide them between 3 people, we could divide the buttons evenly so each person gets 2.

However, what happens when we divide 0? Well, we are taking 0 buttons, or no buttons and dividing them among however many people. How many will each person get? Well if I have nothing and I take all that I have and split it, each person doesn’t get any buttons, because there were no buttons to begin with. This is an abstract concept, but one that students can generally understand until we introduce more division with 0.

Divide by Zero:

We have been told our entire lives that we are not supposed to divide by zero. However, why is this the case? Let us look at what it means when we divide something.

3 ⁄ 0 = ?

When we divide two numbers, we get out a solution that can be multiplied by the denominator to get the number in the numerator. Therefore, let’s look at the example above.

3 ⁄ 0 = ?
3 = ? ∙ 0

This means that in order for this statement to be true, we need to find a number that we can multiply by 0 to get the number three. However, as discussed above, any number multiplied by the number 0 is always going to be 0. Therefore, there are no solutions to this expression above. This means that the solution is undefined, or we cannot find a solution to make this true.

Zero Divided by Zero:

This is one that is even harder for us to wrap our minds around, because it is something that breaks all the rules that we have ever learned when it comes to the number 0. This contradicts the idea that we cannot find a solution to anything divided by 0, however, we also know that anything divided by itself should be 1. Therefore, what is the solution to 0 ⁄ 0? Let us look at it as we did above.

0 ⁄ 0 = ?

We do not know what 0 ⁄ 0 is. However, we know that it must equal something. Therefore, let’s look for what this solution could be. Because of the concept of division, we can rewrite this idea in the following way:

0 ⁄ 0 = ?
0 = 0 ∙ ?

Now, with this equation in mind, we need to solve for the question mark. The solution is the number that we can multiply by 0 to get 0. One solution could be 1, because 1 ∙ 0 = 0. However, the solution could also be 2, 3, 4, Π, or really any number. This is because any number multiplied by 0 is 0. Because of that, there are infinite solutions. Therefore, 0 ⁄ 0 could be equal to 1, however, it can also be equal to many other things. Because we do not only have one solution, we cannot determine the solution to 0 ⁄ 0. Although some people say the solution is infinity, because we know infinity is not at number, so it cannot be a solution. Therefore, we say that the answer is indeterminate because there are infinitely many solutions to the answer above.

This is where the true confusion comes in for young mathematicians. Division with 0 can produce one answer, no answers, or infinitely many answers, so it can be difficult for students to understand when each is the case. However, with these explanations, it can help our students better understand it.

Power of Zero:

The concept of an exponent or the power of a number in general is described as the number of times we are multiplying a number by itself. For example, 2⁵ = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2, Just like with other numbers, we can do operations on the exponent. We have different theorems or rules of exponents with the same base. We are taught that anything to the power of 0 is 1. For example: 3⁰ = 1, 30⁰ = 1, and 300⁰ = 1. How could this be the case? After all, this means that a⁰ = 1 for any a such that a is a non-zero real number, making the solution indeterminate. But why is this the case? We can understand this using the theorem often referred to as the Quotient Rule. One of these rules says the following:

aⁿ ⁄ a^m= a ^n-m

With this idea in mind, we can look at a⁰=a^n-n, such that a is a non-zero real number and n is a real number. If this is the case, by the quotient rule, we can write this as the following:

a⁰=a^n-n=aⁿ ⁄ a^m

As commonly taught, any non-zero real number divided by itself is 1. Therefore, because a is defined as being a non-zero real number, and a^0is a non-zero real number to the power of n divided by itself, so a⁰=1 as long as a is a non-zero real number.

Zero to the Power of Zero:

What if, however, a was zero? What does 0⁰ equal? There is controversy about what this really equals, just like many other aspects of 0. However, I believe that 0⁰ is indeterminate. Let us begin the same way that we began above with using the quotient rule to break down 0⁰. By the quotient rule:

0⁰= 0^n-n = 0ⁿ⁄ 0 ⁿ

This seems to make sense, based on the quotient rule. Now, based on the definition of an exponent as being a number multiplied by itself n times such that n is an element of the real numbers. We know that one of the properties of 0 is that anything multiplied by 0 is 0, including 0 ∙ 0. Therefore, 0ⁿ= 0 when n is a non-zero element of the real numbers. This means the statement above can be written as the following:

0⁰= 0^n-n = 0⁄ 0

As discussed earlier, the solution to 0 ⁄ 0 is indeterminate because anything multiplied by 0=0, therefore, there could be infinitely many solutions to what this could mean. This proof however does not take into account the idea that n=0. If we allow n=0, then the results get more complicated:

0⁰=0⁰⁼⁰ = 0⁰ ⁄ 0 ⁰

This seems to become cyclic, due to the idea that we are proving what 0 ⁰ is based on 0 ⁰.However, as we look at the rules of multiplication, we have the following results:

0 ⁰ = 0⁰ ⁄ 0 ⁰
0 ⁰ ∙ 0 ⁰ = 0 ⁰

Because this is the case, we have the case that 0 ⁰ multiplied by itself equals itself. THis is only the case in two numbers that we know of, 0 and 1. Therefore, many people believe that the solution to 0 ⁰ is either 0 or it is 1.

Many people believe that 0 ⁰ = 1. This idea is based upon the idea that any number to the power of 0 should be 1. I mean, take a second to ask Google (search 0^0).
Google

Using more complex mathematics, such as the binomial coefficients of the n-th row of Pascal’s Triangle, when you expand (1-1)ⁿ using the binomial theorem, and the exception of k=1. There are also different proofs using things such as mapping elements with different sets, and the uses of limits as the value becomes infinite (Su).

Due to the uncertainty that occurs with 0, many mathematicians dedicate parts of their life to prove these abstract concepts as having a finite set of solution. There is this sense of discomfort that occurs with uncertainty and a sense of incompleteness with 0 and infinitely. This is what caused so many problems when it was first in existence (See the History of the Number Zero).