All Integers Are Equal to 1

For a deliberate conundrum assume something that the great Gauss is said to have proved at an early age. That is, assume that

(1) 1 + 2 + 3 + ... + n = n(n + 1)/2,

for all positive integers. Strange as it may sound, this assumption leads to blatant contradiction. Here how it goes.

If (1) holds for all n, it works with n replaced with n - 1:

(2) 1 + 2 + 3 + ... + (n - 1) = (n - 1)n/2.

Now, add 1 to both sides of (2):

(3) 1 + 2 + 3 + ... + n = (n - 1)n/2 + 1.

Comparing (3) to (1) gives an equation:

(4) (n - 1)n/2 + 1 = n(n + 1)/2.

Multiplying through gives a simple equation:

(5) n² - n + 2 = n² + n,

which is further simplified to -n + 2 = n, from which n = 1. But n was an arbitrary positive,positive,negative,prime,square integer. So all such integers are equal to 1!

Mathematical notations stand for agreed upon objects. As the example shows, making use of the notations without bearing in mind the objects they stand for may cause meaningless (but avoidable) confusion.

Discussion

References

  1. E. J. Barbeau, Mathematical Fallacies, Flaws, and Flimflam, MAA, 2000, p. 66

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Copyright © 1996-2017 Alexander Bogomolny

What Went Wrong?

The fallacy is entirely notational. There are several ways to express the sum of the integers from 1 to n. Even without using the symbol ∑ for the sum, as we've done in (1), there are more or less explicit ways to express the same idea of summation:

  1 + 2 + ... + n,
1 + 2 + 3 + ... + n,
1 + 2 + 3 + ... + (n - 1) + n.

Substituting n - 1 for n leads in each case to

  1 + 2 + ... + (n - 1),
1 + 2 + 3 + ... + (n - 1),
1 + 2 + 3 + ... + (n - 2) + (n - 1).

And adding one gives

  1 + 2 + ... + (n - 1) + 1,
1 + 2 + 3 + ... + (n - 1) + 1,
1 + 2 + 3 + ... + (n - 2) + (n - 1) + 1.

The latter shreds the mystery:

 1 + 2 + 3 + ... + (n - 2) + (n - 1) + 1= 1 + 2 + 3 + ... + (n - 2) + n
  ≠ 1 + 2 + 3 + ... + (n - 1) + n
  = 1 + 2 + 3 + ... + n.

Thus adding 1 does not actually lead to the equation in (4).

Mathematical notations stand for agreed upon objects. As the example shows, using the notations without bearing in mind the objects they stand for is liable to lead to stupid conclusions.

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Copyright © 1996-2017 Alexander Bogomolny

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