0  0 = 0 |
a 0 = a |
a·0 = 0·a = 0·0 = 0 | 0/a =0. |
We have introduced fractions, in order to execute all divisions without remainder. Fractions occur in the oldest documents, but the number zero occurred first with the Indians during the 4th Century. It perfected the position system. It allowed to write numbers like 1004 and perform the subtraction a - a = 0. It follows the Rules:
| 0 0 = 0 |
a 0 = a |
a·0 = 0·a = 0·0 = 0 | 0/a =0. |
What is the meaning of a/0? If a/0 is a number, it must yield a when multiplied by 0 which contradicts the above rules. 0/0 is an arbitrary number, since the product of 0 with any number yields 0. Hence both these expressions do not make sense and you have the Rule:
Do not divide by 0.
An equation of the form 0·x = a has no solution!
| Examples: | 2x + 3 = 2x = 4 | 2x - 2x = 4 - 3 | 0·x = 1 |
This equation involves a contradiction.
Finally, consider 0·x = 0. Any small or large value of x satisfies this equation. It is not a determining equation, but an identity.
| Examples: | 2x + a = a + 2x | 2x - 2x = a - a | 0·x = 0 |
This identity is due to the fact that the sequence of terms in a sum is arbitrary.
Tasks: 1. Find the mistake in the equation:
| (a²-a²)/4=(a²-a²)/4 | a(a-a)/4=(a+a)·(a-a)/4 | a·(a-a)/4=(a+a)·(a-a)/4 |
Divide both sides of the equation by (a - a)/4 and obtain
| a = a + a | a = 2a | 1 = 2 |
2. Find the mistake in the equation:
| 18x - 36 = 21x - 42 | 6(3x - 6) = 7(3x - 6) |
Divide both sides of the equation by 3x - 6 to obtain 6 = 7.
You can easily construct such tricks. Every time the same mistake is introduced: Division by zero. In the first example, it was division by (a-a)/4=0, in the second, division by 3x - 6 = 0. You confirm that 3x - 6 = 0 by bringing all terms to one side, when 0 = 21x - 42 - 18x + 36 or 3x - 6=0.