Divisão pelo elemento neutro 0

(Video) Division Properties

by mathman1024

Here's a video by mathman1024 that goes over some properties of division.

$\large{\text{Summary}}$

A Number Divided by Itself $$\begin{align} \frac aa&=1\\ \text{where }a&\ne0 \end{align}$$ This just says that any number divided by itself will give you $1$. Incidentally, this is the multiplicative identity. It makes sense we get $1$. That's because dividing $\frac{a}{a}$ is the same as multiplying $a \cdot \frac{1}{a}$. This is just multiplying a number by its reciprocal.

A Number Divided by $1$ $$\frac a1=a$$ Just like multiplying by $1$, dividing any number by $1$ doesn't change the number at all.

$0$ Divided by a Number $$\frac0a=0$$ Dividing $0$ by any number gives us a zero. Zero will never change when multiplying or dividing any number by it.

A Number Divided by $0$

Finally, probably the most important rule is: $$\frac a0\text{ is undefined}$$ You cannot divide a number by zero! If you want to know why this is check out this awesome video by Numberphile!

Try these problems out on your own and then watch the video to see the answers! They can all be solved using the list of rules above. Good luck!

1. $\frac{19}{0}$
2. $\frac{85}{85}$
3. $-54\div1$
4. $0\div44$

Dividing by 0

Notice this is different from dividing zero by a number. That looks like

$$0 \div a \ \text{ or } \ \frac{0}{a}$$

and will equal $0$ as long as $a \neq 0$.

For example:

$$\begin{align} \dfrac{3}{0} \text{ is undefined}\\ \\ 25/0\text{ is undefined}\\ \\ 100 \div 0\text{ is undefined} \end{align}$$

Bonus

Want to know where this concept comes up a bunch in math? Vertical lines have undefined slopes because the change in their $x$-values is always $0.$

(Video) Problems with 0

by Numberphile

Here's a fantastic video by Numberphile explaining what goes on behind the scenes when you divide by $0$, have $0^0$, and have $\frac{0}{0}$.

Note: this video goes into topics that are at a much higher level than pre-algebra.

$\large{\text{Summary}}$

The most relevant part of this video is the explanation of the issues with dividing by $0$.

We know that division is the same as repeated subtraction. Think about $20\div 4$. $$20\div 4 = 20 - 4 - 4 - 4 - 4 - 4$$ We subtract $4$ from $20$ five times, so $20 \div 4 = 5$.

Dividing by $0$ would be repeatedly subtracting $0$ from a number. $$20\div 0 = 20-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0...$$ This would go on forever and we would never reach an answer. You might think that this means $20\div 0 = \infty$, but this has an issue.

Let's think about fractions for one second. If two fractions have the same denominator, the one with the bigger numerator is bigger. $$\begin{align} \frac31 &> \frac21 \\ \frac54 &> \frac34\\ \frac{35}{12} &> \frac{23}{12} \end{align}$$

The process we described means that every single fraction with a denominator has the same value. $$\begin{align} \frac10 &= \infty\\ \frac20 &= \infty\\ \frac{40}{0}&= \infty \end{align}$$

This is impossible as we know that the fraction with the higher numerator in this case should be the bigger number. The case we have outlined with division of $0$ would mean that $1 = 2 = 40$ and would be true for any other number.

Can you divide by $0$?

Think of the meaning of division. If you had a pizza and each person received $0$ slices, how many people could you feed?

The question doesn't even make sense! Not only does your pizza remain untouched, but nobody is even getting fed. No matter how many people come over to your (not so great) pizza party, it is impossible to completely split the pizza.

So dividing by $0$ is an inherently impossible thing.

Undefined vs Indeterminate

When something other than $0$ is divided by $0$, the result is undefined. But when $0$ is divided $0$, it is called indeterminate.

$$\frac{3}{0}=\text{undefined}\qquad \frac{0}{0}=\text{indeterminate}$$

Well, was that some random pedantic piece of terminology, or is there a point?

One difference between them is that undefined is no value, and indeterminate is any value.

With undefined, think back to the terrible pizza party. If you try to split $3$ pizzas and everybody gets $0$ slices, the pizza will not be split no matter how many people come. Since no number of people can split the pizza, undefined means no value.

But indeterminate is slightly different: there is no meaning to $\frac{0}{0}$ because we have competing approaches on how to solve this problem. We know that $0$ divided by any number is $0$, but we also know that any number divided by $0$ is undefined.

Which piece of information do we use? Neither. This conflict is why we call this form indeterminate. Replace the $3$ by some other positive integer, and the problem works out the same way.