# Can anyone spot the error in this fallacy

1. ### mjordan2nd

150
<prepared>a + b = t
<prepared>(a + b)(a - b) = t(a - b)
<prepared>a^2 - b^2 = ta - tb
<prepared>a^2 - ta = b^2 - tb
<prepared>a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4
<prepared>(a - t/2)^2 = (b - t/2)^2
<prepared>a - t/2 = b - t/2
<prepared>a = b

Therefore all numbers are the same!

2. ### Norman

922
When you take the square root from the 6th to 7th line.

3. ### mjordan2nd

150
Could you explain why that is erroneous?

4. ### Perau

17
the formulas seem legit, but if you put numbers in each, as i have done a =1 b =2 t =3

the 6th line shows inconsistency. BOTH 5th lines would appear to be -1.4375=-1.4375 and the next line you would expect the same, but one appears to be 0.5625 = 1.5625

i still dont know why is that. but i would like to know

5. ### Phrak

I'm perfectly happy with this. All numbers are equal.

6. ### Sorry!

571
You need to take +/- when you take a square root unless it's an absolute value.

7. ### Perau

17
lets just argue for sakes here that it is absolute value because it really doesn't matter, if you put the -/+ in front of the numbers no way will it still equal each other. the magnitude is still different.

ok might be a bit confusing, assume absolute for our case. the magnitudes should equal in theory but by placing numbers its not. but there is nothing wrong with the square root.

8. ### Integral

7,345
Staff Emeritus
Why in the world would think that?

This does not say anything about a and b beyond what you have done. To satisfy the process you have defined you must have a = b . It does not say anything in general about anything.

9. ### Sorry!

571
EDIT: I think you misread what I wrote. The solutions to a square root must be +/- or you just give ABSOLUTE VALUE.

I think my wording may have confused you. What I mean is that if you give an absolute value of 3 it can be +/-3. So I was saying that the solutions have to be +/- unless you give your answer as an absolute (which we are not doing in this with numbers.)

All that this proves as of now with letters is that a=b, if you assume that all letters must be different numbers then YES the square root is the error.

Last edited: Dec 3, 2009
10. ### Sorry!

571
I don't think you are doing your math correctly anyways (from what I read up above) You do remember BEDMAS correct?

$$(1-\frac{3}{2})^2=(2-\frac{3}{2})^2$$

We end up with this correct? Now what we have is (-0.5)^2=(0.5)^2 which is true.

But if we take the square root in the previous step we end up with:

$$1-\frac{3}{2}$$ which = -0.5 or we have $$-1+\frac{3}{2}$$ which gives us 0.5.

Same for the other side. If you end up with -0.5=0.5 Then you are wrong in your taking of the square root and this is not a valid solution.

11. ### Perau

17
oh okay, i see your point, but this still goes back to my first post on this, the calculation i done from 5th to 6th line goes haywire, where squareroot hasnt been applied yet.

12. ### Sorry!

571
You're not doing your math correctly.

13. ### Perau

17
a = 1 b = 2 t = 3

5th line
1^2 -3(1) + 3^2/4 = 2^2 - 3(2) + 3^2/4
0.25 = 0.25

6th line
(1-3/4)^2 = (2-3/4)^2
0.5625 = 1.5625

how is that wrong math?

14. ### Sorry!

571
5th line:
$$a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4$$

OR

$$1-3+1.5=4-6+1.5$$

Both sides equal -0.5

6th line:
$$(a - t/2)^2 = (b - t/2)^2$$

OR

$$(1-1.5)^2=(2-1.5)^2$$
$$(-0.5)^2=(0.5)^2$$

Both sides equal 0.25... This is way off what you got. If you post your steps maybe I can help you?

EDIT: I noticed you have it as 3/4... why? As well I'm quite certain (1-3/4)^2 does not equal 0.5625

Last edited: Dec 3, 2009
15. ### Perau

17
OH, i left my t at t/4 istead of t/2. my bad.

16. ### Sorry!

571
Well even so your answer for line 5 is still wrong and so is the one half of line 6... I'll help you out with that if you wanted.

17. ### Perau

17
haha cheers, but how i showed you is exactly how i calculated. should have brought a calculator. but i think i seem to know why it's wrong. thanks anyway!

18. ### Sorry!

571
Heh sure, I'm not trying to make you look 'stupid' or anything just if you need help with how to calculate it I'd be glad to show you. (I'm not sure how old you are but I'm assuming still in highschool? No offense.)

Welcome to the forums by the way Did you get hit with a fish yet?

19. ### sandbanana

8
Square root issues aside I do not see where you are saying anything other than a and b are the same number and t is any number which is the sum of those numbers.

20. ### Phrak

Without giving too much away,

a+b = t.

For two terms

a - t/2 and b - t/2,

either both terms are equal and a=b=t/2,

or one of 'a or 'b is less than t/2.