Zero is a complicated and fun concept of mathematics. It’s a fundamental thing to learn when you approach mathematics, and it has been studied for over a thousand years now. While it seems like the concept of zero is just a simple concept, however, the idea of zero is much more significant than you probably think. As a matter of fact, the discovery of zero can be considered the most important discovery in mathematics history by some mathematicians. This article, it will cover the world of zero—the concept of it and how it affects the world of mathematics.

What actually is zero?

Many dictionaries such as the Cambridge dictionary have considered the definition of zero as a number or nothing. Another definition of zero is temperature and pressure. Therefore, we can conclude that zero is a number, nothing, null or other words that represent nothing. It can also be considered as a temperature or pressure.

Origin of zero

Actually, zero has such a rich history in multiple civilizations and periods. Zero appeared for the first time in Mesopotamia around five-thousand years ago. According to the record, the Mayans invented zero and it was later devised in India. It spreads to Cambodia in the seventh century. China and Islamic countries received the concept of zero by the end of the eighth century. Western countries later received the concept in the twelfth century. After zero reached Europe, an Italian mathematician named “Fibonacci” helped introduce zero in mathematics concepts to the world. It also appeared in the important work of many mathematicians and scientists, zero also appeared in the calculus of Isaac Newton and Gottfried Wilhelm Leibniz. From the past to the present, the concept of “nothing” has continued to be used in physics, economics, engineering, and many more. Zero in mathematics and the philosophy of nothingness are related but not the exactly same.

Zero in Mathematics

Zero is very important to mathematics as it represents and explains concepts that do not have physical forms.

Multiplication with zero

As we all learn in school, any number multiplied by a number zero will get the result as zero, or in other words, anything times zero gets zero. But why? Here is a brief proof.

1. Let $\alpha$ be any number.
2. $0\cdot\alpha=\beta$ (Includes $\alpha=0$)
3. $\alpha-\alpha=0$ (Additive inverse)
4. $(\alpha-\alpha)\cdot\alpha=\beta$ (Substitute 3 into 2)
5. $\alpha\cdot\alpha-\alpha\cdot\alpha=\beta$ (Distributive property)
6. $\alpha^{2}-\alpha^{2}=\beta$ (Product Theorem)
7. $0=\beta$ (Additive inverse)
8. $\therefore0\cdot\alpha=0$

Power of zero

As we start working with the power of a number we know that if $\alpha^n$ means $\alpha$ multiplied themself $n$ times. For example

$\alpha^4=\alpha\times\alpha\times\alpha\times\alpha$

But, what about if $n$ equals zero? How do we suppose to multiply $\alpha$ zero times? Is it similar to multiplication with zero? The answer is the power of zero is not the same as multiplication with zero. If you have learned about the power theorem you will definitely know that $\alpha^0$ is equal to 1. Here’s proof to make you understand more about it. First, we let $\beta$ be a constant ($\alpha\neq0$) and have a power of $n$ and $m$.

$\beta^m=\underbrace{\beta \times \beta \times \beta \times \ldots \times \beta}_{\displaystyle m \, factors}$

$\beta^n=\underbrace{\beta \times \beta \times \beta \times \ldots \times \beta}_{\displaystyle n \, factors}$

If we use the quotient rule of exponents with the same base we get

$\frac{\beta^m}{\beta^n}=\beta^{m-n}$

As the matter of fact, according to additive inverse, we know that if $\beta$ has the exact same power and divide them we get

$\frac{\beta^m}{\beta^m}=\beta^{m-m}=\beta^0$

Since the quantities in both numerator and denominator are equal. We can cancel both of them out and get the answer of 1. Write down as an equation we get

$\frac{\beta^m}{\beta^m}=\beta^{m-m}=\beta^0=1$

In conclusion, $n^0$ is always equal to 1, however, if $n=0$. What will happen? Try it on your own calculators! Some of them might give you an error, undefined, or 0. In order to prove $0^0$, we can use the limit but it’s complicated to understand for most people so the most straightforward proof of $0^0$ will be...

$\frac{(\beta-\beta)^m}{(\beta-\beta)^m}=(\beta-\beta)^{m-m}=0^0=1$

And that’s proof of the power of zero, including the case that the base is equal to zero.

Zero factorial

Next, is zero factorials. The factorial function multiplies all whole numbers from our chosen number down to $1$. For example, let $n$ be a number that we chose but did not include the case that $n$ is equal to zero.

$n!=n\times(n-1)\times(n-2)\times...\times1$

Or in the shorter form as

$n!=n\times(n-1)!$

Now let’s see what if n is equal to zero. Why $0!$ is equal to $1$? Many of us might have a moment not to believe that $0!$ is actually equal to $1$. We know that $1! = 1$ where $n=1$ and substitute in the short form of factorial. We get

$n!=n\times(n-1)!$

$1!=1\times(1-1)!$

$1!=1\times(0)!$

Hence, in order to make this equation true we must recognize that $0!$ is equal to 1. This is the most straightforward proof for $0!$. Yes, $0!=1$ because mathematicians agreed to define $0!$ as 1.

Division with zero

We can express division in fraction terms. For example, $\frac{m}{n}=m\div n$. If $m=0$ and $n$ is a number, what is the value of $m \div n$? Word problems might help make this question easier. Try this, A man with zero pens, He wants to give it to 10 children equally. What is the answer now? The answer is zero because he has no pen. How did he suppose to give it away if he has no pen? That’s why $\frac{0}{m}=0$. Now let’s exchange the position to $\frac{m}{0}$. What is it equal to now? Let’s write down some equations to make it simple to understand. First, a simple fraction equation

$\frac{6}{3}=x$

$x\times3=6$

$x=2$

Equations above have an exact answer but let’s look at the equations below

$\frac{6}{0}=x$

$x\times0=6$

Since every number times zero is zero. Thus, no number can be put in this equation. Now take a look at this equation

$\frac{0}{0}=x$

$x\times0=0$

From the above paragraph, it states that every number times zero is zero. Therefore, $x$ can be any number in $0\times x=0$ equation. Thus, $x=\infty$.

To sum up, $\frac{0}{n}=0$ ($n\neq0$), $\frac{n}{0}=$ undefined ($n\neq0$), and $\frac{0}{0}=\infty$.

Conclusion

Zero is one of the foundation topics to learn in Mathematics and zero can be able to help a lot of people finish their job by using the concept of it. Zero has some special things such as multiplication or factorial with it. Other than Mathematics zero appears in many subjects in school. That’s the beauty of $ZERO$.

References

1. zero 1 number - Definition, pictures, pronunciation and usage notes | Oxford Advanced Learner’s Dictionary (n.d.). Oxford Learners Dictionaries. https://www.oxfordlearnersdictionaries.com/definition/english/zero_1
2. What is the origin of zero? How did we indicate nothingness before zero? (n.d.). Scientific American. https://www.scientificamerican.com/article/what-is-the-origin-of-zer/
3. Who invented zero? Story of nothingness (n.d.). History. https://www.history.com/news/who-invented-the-zero
4. D. (2021, February 3). Importance of Zero in Mathematics. Edulyte. https://www.edulyte.com/blog/importance-of-zero-in-mathematics/
5. Proof of Zero Power Rule (n.d.) Math Doubts https://www.mathdoubts.com/zero-power-rule-proof/
6. Zero to the power of zero (n.d.) Wikipedia https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero
7. Factorial ! (n.d.) Math is fun https://www.mathsisfun.com/numbers/factorial.html
8. Zero factorial (n.d.) Chilimath https://www.chilimath.com/lessons/intermediate-algebra/zero-factorial/
9. Division by zero (n.d.) Wikipedia https://en.wikipedia.org/wiki/Division_by_zero