Sofisma Algebrico – Wikipedia

In mathematics, a Algebric sophism It is a demonstration or mathematical reasoning containing an error, which therefore leads to an incorrect or contradictory result. Usually these sophisms are used for educational purposes, to demonstrate the importance of rigor in mathematical demonstrations; For this reason, the errors present are generally very subtle and difficult to detect (in relation to the public they are intended) but in the end the reasoning has evidently erroneous conclusions. However, the history of mathematics records numerous cases of erroneous reasoning due to important mathematics.

Some classic examples of algebraic sophisms are reported below, divided according to the type of error that is introduced.

The second principle of equivalence states that, given an equation or equality, it is possible to obtain another equivalent by multiplying or dividing both its members by the same real value, which however must be different from scratch. ^[first] The incorrect application of this rule leads to incorrect results, as in the following example. They are

${displaystyle a}$

It is

${displaystyle b}$

Two real numbers do not null equal to each other:

${displaystyle a=b.}$

Multiplying both members of equality by

${displaystyle a}$

and subtracting

${displaystyle b^{2}}$

you get:

${displaystyle {begin{matrix}a^{2}&=&ab\a^{2}-b^{2}&=&ab-b^{2}.end{matrix}}}$

By breaking down both members of the equation in factors, it is obtained as a common factor

${displaystyle a-b}$

:

${displaystyle (a+b)(a-b)=b(a-b).}$

Dividing for

${displaystyle a-b}$

you get:

${displaystyle a+b=b.}$

Having placed as the initial condition

${displaystyle a=b}$

We can perform, without being incorrect, the replacement obtaining:

${displaystyle a+a=2a=a,}$

from which, however, by placing for example

${displaystyle a=1}$

, would have the incorrect conclusion

${Displaystyle 1 = 2}$

, that is, that a number is equal to its double. The incorrect passage consists in the division for

${displaystyle a-b}$

, which is the same as 0, since it was supposed

${displaystyle a=b,}$

And that, therefore, it cannot be done.

Application of rules outside the limits of validity [ change | Modifica Wikitesto ]

Another widespread error is the application of theorems and properties outside their validity limits, ^[2] as in the example below:

${displaystyle {frac {-1}{1}}={frac {1}{-1}}Rightarrow {sqrt {frac {-1}{1}}}={sqrt {frac {1}{-1}}}Rightarrow {frac {sqrt {-1}}{sqrt {1}}}={frac {sqrt {1}}{sqrt {-1}}}.}$

The last step shown is incorrect as the passage of the root on each element of the fraction

${displaystyle {sqrt {frac {a}{b}}}={frac {sqrt {a}}{sqrt {b}}}}$

It is valid only if

${displaystyle a}$

It is

${displaystyle b}$

They are positive numbers. Starting here, using complex numbers you get:

${displaystyle {frac {sqrt {-1}}{sqrt {1}}}={frac {sqrt {1}}{sqrt {-1}}}Rightarrow {frac {i}{1}}={frac {1}{i}}}$

.

Multiplying by the imaginary unit

${displaystyle i}$

Finally you have:

${displaystyle {frac {i^{2}}{1}}={frac {1i}{i}}Rightarrow -1=1}$

.

The series represent sums of infinite terms; The application of them characteristic of the finished sums can lead to erroneous results. ^[3] For example the large series.

${displaystyle sum _{k=0}^{infty }(-1)^{k}=1-1+1-1+ldots }$

It can be represented as

${displaystyle (1-1)+(1-1)+ldots =0+0+ldots =0}$

or

${displaystyle 1+(-1+1)+(-1+1)+ldots =1+0+0+ldots =1}$

.

From which it follows

${Displaystyle 0 = 1}$

. The error consists in this case in the use of the associative property, which is worth only if the series without parenthesis is convergent.

In the past, similar errors were also made by famous mathematicians, such as Guido Grandi, who even gave a philosophical aspect to the previous result, claiming that it was the way in which God created the world from nothing. The same great also obtained a third incorrect result in the calculation of the series: starting from the well -known formula for the geometric series:

${displaystyle sum _{k=0}^{infty }x^{k}=1+x+x^{2}+x^{3}+ldots ={frac {1}{1-x}}}$

,

valid only when

${displaystyle |x|<1}$

, Great extracted for

${displaystyle x=-1}$

the result

${displaystyle 1-1+1-1+ldots ={frac {1}{2}}}$

.

Eulero committed a similar mistake by placing

${displaystyle x=2}$

and getting

${displaystyle -1=1+2+4+8+ldots }$

,

That is, a succession of positive numbers whose sum is negative.

However, we observe that through the sum of Cesàro it is possible to give meaning to the case

${displaystyle x=-1}$

.