The Mystery Behind the Proof of 1=2

You have probably seen, or even been challenged to explain the proof of 1 = 2 shown in the table below. If you have no idea why the proof seems valid, it is now your opportunity to look at it and probe about it. Try to see if you can identify what is wrong with it before proceeding.

We have discussed the division by 0 is undefined, and this operation is not permitted.

In step number 5, we divided (a + b)(a-b) = b(a-b) by a – b. Now from step 1, a =  b which means that a – b = 0. Hence, our division by a – b is illegal and that what makes the proof wrong.

Division by Zero

In studying mathematics, you have probably heard that division of zero is undefined. What does this mean?

Since we do not know exactly what is the answer when a number is divided by 0, it is probably reasonable for us to examine the quotient of a number that is divided by a number that is close to 0.

If we look at the number line, the numbers close to 0 are numbers numbers between –1 and 1.

Figure 1 – The number line showing the numbers close to 0.

For instance, several positive numbers close to 0 and less than 1 are 0.1, 0.01, 0.001 and so on. Similarly, negative numbers close to 0 but greater than – 1 are –0.1, -0.01, -0.001 and so on.

The table and the numbers below shows the quotient 1/x when 1 is divided by x, where the x’s are numbers close to 0.

Figure 2 – The value of 1/x as x approaches 0 from both sides.

In the graph, as x approaches 0 from the right (as x, where x are positive numbers, approach 0), the quotients of 1/x are getting larger and larger. On the other hand, as x approaches 0 from the left (as x, where x are negative numbers, approach 0), 1/x is getting smaller and smaller. Hence, there is no single number that 1/x approach as x approaches 0.  For this reason, we  say that 1/0 is undefined.

A simple analogy would also let us realize that allowing division by 0 will violate an important property of real numbers.  For example 8/4 = 2 because 2 x 4 = 8.  Assuming division of 0 is allowed. If 5/0 = n, then n x 0 = 5.  Now, that violates the property of a real number that any number multiplied by 0 is equal to 0.

Since division by 0 yields an answer which is not defined, the said operation is not allowed.

Related Posts

The Definition of “Undefined”

In learning mathematics, we often encounter terms that are not always clear. Example of such term is the word undefined. What do we mean by undefined?

The word undefined may slightly differ in meaning depending on the context. In plain language, it means something which has no sensible meaning. For instance, during the time when the negative numbers were not yet invented, the numerical expression 5 – 8 has no meaning. In our time, we can say that 5 – 8 is undefined in the set of positive integers.

Below are some examples of the different contexts where the different meanings of “undefined” can be drawn.

Numbers

Square Root of Negative Numbers. The \sqrt{-1} is undefined in the set of real numbers. This means that no real number exists that when multiply it by itself, the product is equal to -1. Note, however, that some operations may be undefined under some sets, but defined in other sets. We know from high school mathematics that square root of – 1 equals i in the set of complex numbers.

Algebra

Division by 0. Since we do not know the answer if a number divided by 0, let us examine the quotient of numbers when divided by numbers close to 0. To make it simple, let us try 1/0.

Figure 1 – The value of the 1/x as x approaches 0.

As we can see, as x approaches 0 from the right, the quotient of 1/x is getting larger and larger. On the other hand, as x approaches 0 from the left, 1/x is getting smaller and smaller. As a consequence, there is no single number that 1/x approaches as x approaches 0.   Therefore we can say that 1/0 is undefined.

Geometry

Intersection of two lines. In Euclidean Geometry, if we talk about the intersection of two lines, we can have three cases: intersecting lines have one intersection, coinciding lines have infinitely many intersections, and parallel lines have no intersection.

Figure 2 – The three cases, in terms of intersection, two lines can be places on a plane.

We can say that if two lines are parallel, no intersection exists. Algebraically, the solution to the system of equations of the two lines is their intersection.  Hence, the solution of the two systems of linear equations of parallel lines as graphs is undefined.

 

Matrices

Matrices with Different Sizes. If A is a 2 by 2 matrix and B is a 3 x 3 matrix, then A + B has no meaning since five of the entries of matrix B have no corresponding entries in matrix A. We can say that the sum of matrix A and matrix B is not well-defined.

Sets

Intersections of Sets. If E is the set of even integers and O be the set of odd inteers, then there is no common value to both sets.

Figure 3 – The Venn Diagram of the intersection of even and odd numbers.

In set theory, we call the common values the intersection, and in this example, the intersection is the empty set.  As a consequence, we can say that the intersection of set E and O is undefined.

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Although the word “undefined” has different meanings depending on the context, by now, you would have realized that the phrases “does not exist”, “without sensible meaning” and “cannot be determined” are somewhat synonymous to it. If the result of an operation yields no value  at all (note that 0 is a value and is not the same to no value), then it is more likely that it is undefined.

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