Spot the Mathematical Fallacy

[Tiberius]

Which line contains the fallacy in this mathematical proof?

1. Let a and b be equal non-zero quantities.
a = b

2. Multiply both sides by a.
a2 = ab

3. Subtract b2.
a2 - b2 = ab - b2

4. Factor both sides.
(a - b)(a + b) = b(a - b)

5. Divide out (a - b).
(a + b) = b

6. Substitute a = b.
b + b = b

7. Simplify.
2b = b

8. Divide by b.
2 = 1

Warning!!! Answer:

---

Line 5. Dividing by (a - b) is the equivalent of dividing by zero, and this is undefined in mathematics. This is an algebraic form of the following fallacious proof:

2 * 0 = 1 * 0

Divide each side by zero.

2 = 1

(Or substitute any numbers for 2 and 1)

---

[BioLogos]

Which line contains the fallacy in this mathematical proof?

1. Let a and b be equal non-zero quantities.
a = b

2. Multiply both sides by a.
a2 = ab

3. Subtract b2.
a2 - b2 = ab - b2

4. Factor both sides.
(a - b)(a + b) = b(a - b)

5. Divide out (a - b).
(a + b) = b

6. Substitute a = b.
b + b = b

7. Simplify.
2b = b

8. Divide by b.
2 = 1

Very famous and illuminating example.

[Rhizomorph13]

Line six

[Tiberius]

What's wrong with line 6? It's a substitution, and since a = b you can substitute it fine.

[theVOID]

Line 5 is divide by 0?

[Tiberius]

Well done

[theVOID]

Well done

It only took 1 and a half a4 pages of scribbling to work it out

[Tiberius]

Now I have to find another problem. Grrr.

[theVOID]

Give me a heads up, i need to go buy more pad paper for working them out.

[chris]

1. Let a and b be equal non-zero quantities.
a = b

Actually line 1 assumes that both non zero numbers are the same number, line 1 should read:
1. Let a and b be equal non-zero quantities.
a > b or a < b.
This then changes the rest of the proof :-)

All the best


Chris