[Math Lair] Mathematical Fallacies

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Mathematical fallacies are errors, typically committed with an intent to deceive, that occur in a mathematical proof or argument. A fallacy in an argument doesn't necessarily mean that the conclusion is necessarily incorrect, only that the argument itself is wrong. However, fallacious arguments can have surprising conclusions, as shown below:

Division By Zero

1.Letx = 1
2.Multiply both side by xx² = x
3.Subtract 1 from both sidesx² − 1 = x − 1
4.Divide both sides by x − 1x² − 1x − 1 = 1
5.Simplify(x − 1)(x + 1)x − 1 = 1
x + 1 = 1
6.Subtract 1 from both sidesx = 0
7.Substitute the value of x from step 11 = 0

The fallacy here is subtle. In step 2, multiplying both sides by x introduces an extraneous solution to the equation of x = 0. Then, in step 4, there is a division by x − 1, which is an illegal operation because x − 1 = 0 and you can't divide by zero. This illegal operation has the effect of leaving the extraneous solution x = 0 as the only solution to the equation.

You can use a similar method to "show" that any number is equal to any other number. For example, to show that 7 = −4:

x = 7
x − 7 = 0
(x − 7)(x + 4) = 0 [multiply both sides by x + 4]
x + 4 = 0 [divide both sides by x − 7]
x = −4

Infinite Series Fallacies

If you've taken mathematics in high school or first-year university or college, you may have learned that an infinite geometric series has a sum if it is convergent; that is, if the ratio between any term and the previous term is less than 1 and greater than −1. Geometric series that are not convergent cannot be summed. If you ignore this, then you can come up with all kinds of strange results. For example:

S = 1 − 1 + 1 − 1 + 1 − 1 + 1 − ...
Grouping the terms, we get:
S = (1 − 1) + (1 − 1) + (1 − 1) + ...
S = 0 + 0 + 0 + 0 + ...
S = 0

However, we could also group the series differently:

S = 1 − 1 + 1 − 1 + 1 − 1 + 1 − ...
S = 1 + (1 − 1) + (1 − 1) + (1 − 1) + ...
S = 1 + 0 + 0 + 0 + 0 + ...
S = 1

Since S is equal to both 1 and 0, then 1 = 0.

Here's another example:

S = 1 + 2 + 4 + 8 + 16 + 32 + ...
S − 1 = 2 + 4 + 8 + 16 + 32 + ...

Now, we can multiply the original equation by 2, getting:

2S = 2 + 4 + 8 + 16 + 32 + ...

Both the second and third equation have the same sum, so:

2S = S − 1
S = −1

But it is absurd that the sum of this series could be negative, since all of the terms are positive. The fallacy lies in assuming that a divergent series has a sum.

Fallacies With Units

10¢ is equal to 110 of a dollar. If we square both numbers, we get 100¢ = 1100 of a dollar. But 1100 of a dollar is just 1¢! Where did the other nine cents go? Well, we didn't treat the units properly. If we square 10¢, the units are not ¢ and $, but ¢² (cents squared) and $² (dollars squared). The conversion factor between square dollars and square cents is 10,000, not 100, so the paradox vanishes.

You may also be interested in logical fallacies or in mathematical paradoxes. You may also be interested in a "proof" that you never go to school.