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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Consider the expression $05 $ 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 $x⋅0=5.$

Division indicates how many times the denominator "fits" into the numerator. If you calculate $05 $ to determine how many times $0$ will fit in $5$, you will never reach $5$ no matter how many zeros you try. $Dividing by0then becomes impossible.$ Therefore, certain conditions must be specified when s denominator contains a variable. For example, $a5 fora =0.$

Another way to illustrate why division by $0$ is undefined is to study the graph of $y=x1 .$ By using a table of values some points on the curve can be found.

$x$ | $x1 $ | $=$ |
---|---|---|

$-0.5$ | $-0.51 $ | $-2$ |

$-0.25$ | $-0.251 $ | $-4$ |

$-0.1$ | $-0.11 $ | $-10$ |

$0.1$ | $0.11 $ | $10$ |

$0.25$ | $0.251 $ | $4$ |

$0.5$ | $0.51 $ | $2$ |

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 $+∞$ when $x$ approaches $0$ from the right, and $-∞$ when $x$ approaches from the left. Because $f(x)=x1 $ is a function, $f$ can only produce one value when $x=0.$ Therefore, division by $0$ is undefined.

Suppose dividing by zero was defined. Then, the "logic" below would be accepted as true. Let $a$ and $b$ be any real numbers/

$a=b$

SubEqn$LHS−b=RHS−b$

$a−b=0$

DivEqn$LHS/(a−b)=RHS/(a−b)$

$a−ba−b =a−b0 $

SimpQuotSimplify quotient

$1=0$

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.