Good Evening,
I came across this proof which proves (1=2).I found this one very interesting and so thought of sharing with you.
Let me warn you that though this proof seems to be correct,it is wrong.
This is perfect example of proving wrong things correctly.
Step Reason
1 . a = b Given
2 . a^2 = ab Multiplying both sides by a
3 . a^2 - b^2 = ab - b^2 Subtract b^2 from both sides of (2)
4. (a-b)(a+b) = b(a - b) Factor both sides of (3)
5. a+b = b Dividing both sides of (4) by (a - b)
6. 2b = b Replace a by b in (5) As per (1) a = b,Simplify
7. 2 = 1 Dividing both sides of (6) by b
This is how we can prove 2 = 1,which is wrong.
**This proof was mentioned in "Discrete Mathematics and Its Application by Kenneth H Rosen" under "Mistakes in Proof" section
Try to find the mistake and write it in comments.
I came across this proof which proves (1=2).I found this one very interesting and so thought of sharing with you.
Let me warn you that though this proof seems to be correct,it is wrong.
This is perfect example of proving wrong things correctly.
Step Reason
1 . a = b Given
2 . a^2 = ab Multiplying both sides by a
3 . a^2 - b^2 = ab - b^2 Subtract b^2 from both sides of (2)
4. (a-b)(a+b) = b(a - b) Factor both sides of (3)
5. a+b = b Dividing both sides of (4) by (a - b)
6. 2b = b Replace a by b in (5) As per (1) a = b,Simplify
7. 2 = 1 Dividing both sides of (6) by b
This is how we can prove 2 = 1,which is wrong.
**This proof was mentioned in "Discrete Mathematics and Its Application by Kenneth H Rosen" under "Mistakes in Proof" section
Try to find the mistake and write it in comments.
step 6 is wrong
ReplyDelete2b=b
than 2b-b=0
i.e. b=0
than only its possible
Actually Step 5 is wrong
ReplyDeleteAs a = b
a-b = 0
so we cant perform division by zero (a -b )
as equality will change there..
thanks for taking your time to find the error....
Hi,
ReplyDeleteI found a similar sort of proof on the net while surfing, which proves "2=3"
The proof is as follow :
6 = -6
9-15 = 4-10
adding 25/4 to both sides:
9-15+(25/4) = 4-10+(25/4 )
(this is just like : a2 – 2(a)(B) + b2 = (a-B)2 )
Here a = 3, b=5/2 for L.H.S
a =2, b=5/2 for R.H.S.
So it can be expressed as follows:
(3-5/2) 2 = (2-5/2)2
Taking positive square root on both sides:
3 - 5/2 = 2 - 5/2
3 = 2 ????????
@Aniket
ReplyDeletethe first statement itself is wrong
how come 6 = -6 ?
positive number is never equal to negative number..
sorry my typing mistake ..it was
ReplyDelete-6 = -6
@Aniket
ReplyDeletehow come you are assigning a = 2 for RHS and a = 3 for LHS..
Dont you think this step is wrong.
U read it wrongly, it was just for understanding.
ReplyDeleteSubstituting the values in the formula.
You can directly read it as :
9-15+(25/4) = 4-10+(25/4 ) =>
(3-(5/2))2 = (2-(5/2))2
hmmm..interesting...
ReplyDeletetake a look at step where you are taking postive square root of both sides
ReplyDelete(2-5/2) is not a positive square root..
ham.... u r right ..
ReplyDeleteafter all it a falcy proof..
In the 4th step you divided both side by (a-b) means divided by "0" as a=b from step 1. So in my opinion it is false at step 4.
ReplyDeletehere, a=b then,
ReplyDeleteat 2nd step we can put b's value as a coz a=b right?
then step 2nd will be like this,
a^2=ab as u mentioned but a=b then a^2 = a (a),
its a^2=a^2 means 1=1 only.
step 4 to 5 is not possible! a-b is 0. Thus can't be cancelled! (cancelling essentially means dividing both sides by the same number. Division by zero is illegal (a crime punishable by law! :p)!!) :-) :-)
ReplyDeleteSTEP 4 :: (a-b)(a+b) = b(a-b)
ReplyDeleteFrom this step you reach step 5 by dividing both side by (a-b)
STEP 5 : (a+b) = b
You are allowed division only when (a-b) is >0 because division by 0 is undefined.
Therefore you cannot reach step 5 from step 4 and hence the result is incorrect
Good and Correct Explation in my opinion.
DeleteNice Blog!!
ReplyDelete