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Explanation

# Division by zero

Consider the expression $\frac{5}{0}$ which shows dividing a number by zero. This division is considered undefined or not possible. This is because, there exists no number $x,$ real or otherwise, such that $x \cdot 0 = 5.$

Division indicates how many times the denominator "fits" into the numerator. If you calculate $\frac{5}{0}$ to determine how many times $0$ will fit in $5$, you will never reach $5$ no matter how many zeros you try. $\text{Dividing by 0 then becomes impossible.}$ Therefore, certain conditions must be specified when s denominator contains a variable. For example, $\dfrac{5}{a} \quad \quad \text{for } a \neq 0.$

### Extra

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An illustration

Another way to illustrate why division by $0$ is undefined is to study the graph of $y=\frac{1}{x}.$ By using a table of values some points on the curve can be found.

$x$ $\dfrac{1}{x}$ $=$
${\color{#0000FF}{\text{-}0.5}}$ $\dfrac{1}{{\color{#0000FF}{\text{-}0.5}}}$ ${\color{#FF0000}{\text{-}2}}$
${\color{#0000FF}{\text{-}0.25}}$ $\dfrac{1}{{\color{#0000FF}{\text{-}0.25}}}$ ${\color{#FF0000}{\text{-}4}}$
${\color{#0000FF}{\text{-}0.1}}$ $\dfrac{1}{{\color{#0000FF}{\text{-}0.1}}}$ ${\color{#FF0000}{\text{-}10}}$
${\color{#0000FF}{0.1}}$ $\dfrac{1}{{\color{#0000FF}{0.1}}}$ ${\color{#009600}{10}}$
${\color{#0000FF}{0.25}}$ $\dfrac{1}{{\color{#0000FF}{0.25}}}$ ${\color{#009600}{4}}$
${\color{#0000FF}{0.5}}$ $\dfrac{1}{{\color{#0000FF}{0.5}}}$ ${\color{#009600}{2}}$

Graphing the points on a coordinate plane allows the behavior of the function to be shown. The red points show what happens as $x$ approaches $0$ from the left, and the green points show what happens as $x$ approaches $0$ from the right.

The closer $x$ comes to $0,$ on both sides, the more extreme the values of the functions become. The quotient approaches $+ \infty$ when $x$ approaches $0$ from the right, and $\text{-} \infty$ when $x$ approaches from the left. Because $f(x)=\frac{1}{x}$ is a function, $f$ can only produce one value when $x=0.$ Therefore, division by $0$ is undefined.

### Extra

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The consequences of dividing by zero
Suppose dividing by zero was defined. Then, the "logic" below would be accepted as true. Let $a$ and $b$ be any real numbers/
$a=b$
$a-b=0$
$\dfrac{a-b}{a-b}=\dfrac{0}{a-b}$
$1=0$

Since both sides of the equation are divided by $a-b=0,$ and $a-b=0,$ the expression was validly manipulated to create a false statement.