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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 | info | |

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 | info | |

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.

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