What Is 1 Divided By

This is part of a series on common misconceptions.

True or False?

$\frac10$ is undefined.

Why some people say it'due south true: Dividing by $0$ is not allowed.

Why some people say it'south false: $\frac10 = \infty.$

Can you meet which of these is the correct explanation?

The argument is $\color{#3D99F6}{\textbf{true}}$ .

Proof:

If $\frac10 = r$ were a real number, then $r\cdot 0 = i,$ simply this is incommunicable for any $r.$ See division by zero for more details. $_\square$

At that place are some mutual responses to this logic, but they all have various flaws.

Rebuttal: In calculus, $\frac10$ equals $\infty.$

Reply: This statement is incorrect for two reasons. First, infinity is not a existent number. The proof demonstrates that the quotient $\frac10$ is undefined over the real numbers.

It is true that, in some situations, the indeterminate form $\frac10$ can be interpreted as $\infty:$ for case, when taking limits of a quotient of functions. Only fifty-fifty this is not ever truthful, equally the following example shows:

Consider $\lim\limits_{x\to 0}\frac{1}{x}.$

Approaching from the right, $\lim\limits_{x \to 0^+} \frac{one}{ten} = + \infty.$
Approaching from the left, $\lim\limits_{x \to 0^-} \frac{1}{ten} = - \infty.$

In order for $\frac{1}{0}$ to be consistent, the limits from both directions should be equal, which is conspicuously non the case here.

Rebuttal: What most on the Riemann sphere?

Respond: For certain complex functions, it is convenient and consistent to extend their domain and range to ${\mathbb C} \cup \{\infty\}.$ This set has the geometric structure of a sphere, chosen the Riemann sphere. For instance, suppose $a,b,c,d$ are complex numbers such that $advertising-bc\ne 0.$ And so the function $f(z) = \frac{az+b}{cz+d}$ can exist extended by defining $f\left(-\frac dc\correct) = \infty$ and $f(\infty) = \frac ac$ $\big($ or $f(\infty) = \infty$ when $c=0\large).$ This makes $f$ a bijection on the Riemann sphere, with many nice properties.

Then there are situations where $\frac10$ is defined, only they are defined in a tightly controlled way. Information technology is still the example that $\frac10$ tin never be a real (or circuitous) number, so—strictly speaking—it is undefined.

See the consequences of assuming that $\frac{1}{0}$ is defined for yourself in the following problem:

Wrong factoring Multiplying unknown variables False substitution Introducing $-b^2$ Dividing by zero

What is wrong with the following "proof"?

Let $a = b=1$ , then $a+b=b.$

• Step 1: $a^2 = ab$
• Step two: $a^ii - b^2 = ab - b^2$
• Stride 3: $(a+b)(a-b) = b(a-b)$
• Step four: $a+b= \dfrac{b(a-b)}{a-b}$
• Step v: $a+b = b$

Conclusion: By substituting in $a = b = one,$ we have $ane+1 = i \implies 2 = 1.$

See Also

• Listing of Common Misconceptions

• Indeterminate Forms

What Is 1 Divided By,

Source: https://brilliant.org/wiki/what-is-1-0/

Posted by: berniertheirm.blogspot.com

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