Division by zero

Consider the expression 50\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,x, real or otherwise, such that x0=5. x \cdot 0 = 5.

Division indicates how many times the denominator "fits" into the numerator. If you calculate 50\frac{5}{0} to determine how many times 00 will fit in 55, you will never reach 55 no matter how many zeros you try. Dividing by 0 then becomes impossible. \text{Dividing by $0$ then becomes impossible.} Therefore, certain conditions must be specified when s denominator contains a variable. For example, 5afor a0. \dfrac{5}{a} \quad \quad \text{for } a \neq 0.


An illustration

Another way to illustrate why division by 00 is undefined is to study the graph of y=1x.y=\frac{1}{x}. By using a table of values some points on the curve can be found.

xx 1x\dfrac{1}{x} ==
-0.5{\color{#0000FF}{\text{-}0.5}} 1-0.5\dfrac{1}{{\color{#0000FF}{\text{-}0.5}}} -2{\color{#FF0000}{\text{-}2}}
-0.25{\color{#0000FF}{\text{-}0.25}} 1-0.25\dfrac{1}{{\color{#0000FF}{\text{-}0.25}}} -4{\color{#FF0000}{\text{-}4}}
-0.1{\color{#0000FF}{\text{-}0.1}} 1-0.1\dfrac{1}{{\color{#0000FF}{\text{-}0.1}}} -10{\color{#FF0000}{\text{-}10}}
0.1{\color{#0000FF}{0.1}} 10.1\dfrac{1}{{\color{#0000FF}{0.1}}} 10{\color{#009600}{10}}
0.25{\color{#0000FF}{0.25}} 10.25\dfrac{1}{{\color{#0000FF}{0.25}}} 4{\color{#009600}{4}}
0.5{\color{#0000FF}{0.5}} 10.5\dfrac{1}{{\color{#0000FF}{0.5}}} 2{\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 xx approaches 00 from the left, and the green points show what happens as xx approaches 00 from the right.

The closer xx comes to 0,0, on both sides, the more extreme the values of the functions become. The quotient approaches ++ \infty when xx approaches 00 from the right, and -\text{-} \infty when xx approaches from the left. Because f(x)=1xf(x)=\frac{1}{x} is a function, ff can only produce one value when x=0.x=0. Therefore, division by 00 is undefined.


The consequences of dividing by zero
Suppose dividing by zero was defined. Then, the "logic" below would be accepted as true. Let aa and bb be any real numbers/

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

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