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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.

Dividing by 0 then becomes impossible .

Therefore, certain conditions must be specified when s denominator contains a variable. For example,

\frac{5}{a} for a \neq = 0 .

Extra

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.

$x$	$\frac{1}{x}$	$=$
$- 0.5$	$\frac{1}{- 0 . 5}$	$- 2$
$- 0.25$	$\frac{1}{- 0 . 2 5}$	$- 4$
$- 0.1$	$\frac{1}{- 0 . 1}$	$- 10$
$0.1$	$\frac{1}{0 . 1}$	$10$
$0.25$	$\frac{1}{0 . 2 5}$	$4$
$0.5$	$\frac{1}{0 . 5}$	$2$

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 $- \infty$ 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

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

SubEqn

$LHS - b = RHS - b$

a - b = 0

DivEqn

$LHS / (a - b) = RHS / (a - b)$

\frac{a - b}{a - b} = \frac{0}{a - b}

SimpQuot

Simplify 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.

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