Another 0=1 Proof

Algebra Level 1

Here's my proof that 0 = 1 0=1 .
In which of these steps did I first make a flaw in my logic?

Step 1 : Let's establish the following equations first, 0 × 0 × 0 × 0 = 0 0 × 0 × 0 = 0 \begin{aligned} 0 \times 0 \times 0 \times 0 &=& 0 \\ 0\times 0 \times 0 &=& 0 \end{aligned}

Step 2 : Since both these numbers are equal in value, then their ratio is equal to 1.

0 × 0 × 0 × 0 0 × 0 × 0 = 1 \dfrac{ 0 \times 0 \times 0 \times 0}{ 0\times 0 \times 0 } = 1

Step 3 : Cancel out common factors:

0 × 0 × 0 × 0 0 × 0 × 0 = 1 0 = 1 \begin{aligned} \dfrac{ 0 \times \xcancel0 \times \xcancel0 \times \xcancel0}{ \xcancel0\times \xcancel0 \times \xcancel0 } &=& 1 \\ 0 &=& 1 \end{aligned}

Step 1 Step 2 Step 3 No flaw at all, your logic is perfect!

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4 solutions

Sherwin D'souza
Mar 19, 2016

Step 2 says if value of 2 equations are equal their ratio is equal to 1, but that applies to non-zero values of the equations. Also in step 2 by dividing the 2 equations you are considering that

0 0 = 1 \frac {0}{0} = 1

Which is totally absurd or wrong, as

0 0 \frac {0}{0}

is indeterminate.

This is correct! Thankyouuu

Pi Han Goh - 5 years, 2 months ago
Matin Naseri
Feb 10, 2018

The first mistake is in s t e p t w o \color{#D61F06}{ \boxed{step~two}} .

0 × 0 × 0 × 0 0 × 0 × 0 = 1 \dfrac{ 0 \times 0 \times 0 \times 0}{ 0\times 0 \times 0 } = 1

It must be

0 × 0 × 0 × 0 0 × 0 × 0 = U n d e f i n e d \dfrac{ 0 \times 0 \times 0 \times 0}{ 0\times 0 \times 0 } = Undefined

Note

But why 0 0 \frac{0}{0} Is Undefined ?

Check out this Why 0 0 \frac{0}{0} Is undefined

It's called undefined not undefinite.

Pi Han Goh - 3 years, 4 months ago

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Thank's. I will edit it.

Matin Naseri - 3 years, 4 months ago
Munem Shahriar
Nov 25, 2017

The first was made in the step 2 \boxed{\text{step 2}} .

0 × 0 × 0 × 0 0 × 0 × 0 Is undefined \dfrac{ 0 \times 0 \times 0 \times 0}{ 0\times 0 \times 0 } \text{Is undefined}

Note: See this article.

Instead of saying that it is undefined, it would be better to explain why it is undefined.

Pi Han Goh - 3 years, 6 months ago

Division by 0 is not defined!

Then, wouldn't step 3 be the first wrong step?

Pi Han Goh - 5 years, 2 months ago

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0/0 is not 1.

A Former Brilliant Member - 5 years, 2 months ago

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