Question

What is wrong with the following proof? Clearly we know that 1 does not equal 2. What rule of algebra did we break that makes this conclusion invalid? I've numbered the lines for you so you can refer to the line numbers in your explanation ​

Answer 1

An attempt to divide by zero gives a contradictory result

A rule of algebra broken is dividing by zero (leading to a contradiction) and stating a finite result

Reason:

The given calculation is presented as follows;

1. a > 0, b > 0 given

2. a = b given

3. a·b = b²

4. a·b - a² = b² - a²

5. a·(b - a) = (b + a)·(b - a)

6. a = b + a

7. 0 = b

8. b = 2·b

9. 1 = 2

From line 4, the result are;

4. a·b - a² = b² - a² = 0

5. a·(b - a) = (b + a)·(b - a) = 0

On line 6, both sides where divided by (b - a) = 0, which should given an infinite result

Therefore, one rule of algebra broken is dividing by zero to get a finite result

In line 7, we have;

7. 0 = b

From 6. a = b + a, and a = b, we have;

8. b = 2·b

Therefore, line 9 should be;

9. 0 = 2·0; 0 = 0, given that we have;

1 × 0 = 0

2 × 0 = 0

∴ 1 × 0 = 2 × 0

However

• 1 ≠ 2

In line 9., by dividing by b = 0, again, we have;

• 9. 1 = 2 (a contradiction)

Therefore, one rule of algebra that is broken is dividing by zero and having a finite result

Learn more about the rules of algebra here:

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