# Can we ever prove 1=0 or 1=2?

Sometimes when we observe a problem we observe it scientifically and then prove it mathematically and mathematics is such a branch of science in which we create and solve the problem. So it is possible to create a little problem to solve a major one. But we can never neglect the universal rules of mathematics. It means if we neglect the rules of mathematics we can get a result that is logically wrong. Also sometimes, we are not able to understand which rule of mathematics is breaking. It is due to lack of proper knowledge about the subject. For example, we can see the following operations:

Let             x = y ………. (i)

Multiplying both sides by ‘y’, we get

xy = y2

Subtracting ‘x2’ from both sides, we get

xy – x2 = y2 – x2

Now taking ‘x’ as common from right side, we get

x(y –x) = (y + x)(y – x)   ; Because  a2 – b2 = (a + b)(a – b)

By dividing both sides by (y – x), we get

x(y – x)/(y – x) = (y + x)(y – x)/(y – x)

By canceling (y – x) from both sides, we get

x = y + x

x = x + x     ; by Eq. (i)

x = 2x   ; General form

Now canceling ‘x’ from both sides, we get

1 = 2  …………. RESULT (a)

Or

By subtracting ‘x’ from both sides of General form, we have

x – x = 2x – x

0 = x

Now dividing both sides by ‘x’, we get

0 = 1  ……..…… RESULT (b) If we look at this solution, we see that all the steps that have been performed are according to the rules of mathematics and mathematics is a branch of science and science believes in reality, so how can we get an unreal result by using real rules of mathematics. If it is true then  General form means if we have \$1, it should be doubled as given in RESULT (a) that is impossible and according to RESULT (b), if we have \$1, we have nothing. So how it is possible or what is that rule of mathematics that has been broken to get these wrong RESULTS? The answer lies in the following step:

x(y – x)/(y – x) = (y + x)(y – x)/(y – x)

As x = y, according to Eq.(i), so y – x = 0 and the division by 0 (zero) is not possible in mathematics. We can also find other mistakes in this derivation. But this is the perfect reason near me. We did the operation that is impossible in mathematics, so the results that we derived is also impossible in reality. It means mathematics also believes in the real thing if anything is logically wrong no doubt it would be mathematically wrong.