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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Hi,

I came across a proof today, which stated that 1=0. While I don’t agree with the statement, neither can I find the inaccuracy in the proof. Here it is:

-20=-20

16-36 = 25-45

4 ² - 9*4 = 5 ² - 9*5

4 ² - 9*4 + 81/4 = 5 ² - 9*5 + 81/4

(4 - 9/2) ² = (5 - 9/2) ²

4 = 5

4 + 0 = 4 + 1

0 = 1

Learning is fun - Exceptio probat regulam!

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 38,057

Hi CurlyBracket,

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,471

A quicker route to a similar contradiction would be:

Now the error is obvious. Every number has two square roots, so you have to take care when using 'rooting' in an algebraic manipulation.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Hi everyone,

Thank you for the replies.

From what I understood, then there can be 4 possibilities:

4=5

-4=-5

-4=5

4=-5

None of these are correct.

Am I on the right track?

Learning is fun - Exceptio probat regulam!

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 38,057

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,471

Up to this point all is ok. You can test this by evaluating the arithmetic.

This is ok because both sides evaluate to 1/4.

It's what happens next that is not correct.

If you just square root both sides you get

and that is not correct. It's the careless square rooting that causes the false result.

Because a square root can be + or - the equation is valid if one of the signs is changed.

Take it as a warning not to square root without thinking about what is happening.

Calculators will only give a positive square root because they are programmed to give a single answer. Some people regard the root sign as meaning this which is fine. But it's always worth thinking about the negative case, because that might give the answer you want.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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