Proving 3 = 2 and 4 = 3

One way of grabbing students' attention, according to John Keller, the proponent of the ARCS Model of Motivational Design, is to use incongruity and conflict. This can be done by presenting students with statements that go against their previous knowledge. Here are two fallacies that will surely catch students' attention.

(1) Prove that 3 = 2.

Proof:
Let a and b be equal non-zero quantities:
a = b

Multiply both sides by a²:
a³ = a²b

Subtract b³:
a³ - b³ = a²b - b³

Factor both sides:
(a - b)(a² + ab + b²) = b(a² - b²)
(a - b)(a² + ab + b²) = b(a + b)(a - b)

Divide both sides by (a - b):
a² + ab + b² = b(a + b)

Multiply terms on the right side:
a² + ab + b² = ab + b²

Since a = b, both sides may be expressed in terms of b:
b² + b² + b² = b² + b²

Combine like terms on both sides:
3b² = 2b²

Divide by the non-zero b²:
3 = 2

Q.E.D.

(2) Prove that 4 = 3.

Proof:
Let a and b be equal non-zero quantities:
a = b

Multiply both sides by a³:
a⁴ = a³b

Subtract b⁴:
a⁴ - b⁴ = a³b - b⁴

Factor both sides:
(a² + b²)(a² - b²) = b(a³ - b³)
(a² + b²)(a + b)(a - b) = b(a - b)(a² + ab + b²)

Divide both sides by (a - b):
(a² + b²)(a + b)= b(a² + ab + b²)

Multiply terms on both sides:
a³ + ab² + a²b + b³ = a²b + ab² + b³

Since a = b, both sides may be expressed in terms of b:
b³ + b³ + b³ + b³ = b³ + b³ + b³

Combine like terms on both sides:
4b³ = 3b³

Divide by the non-zero b³:
4 = 3

Q.E.D.