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(1)

The square root of 2 ain’t rational

A Casual Talk By

Pete Agoras

(2)
(3)
(4)

The square root of 2 ain’t rational

Demonstration

The problem

a b

=

2

(

a b

)

2

= 2

a2 b2

= 2

a

2

= 2b

2

(2k)

2

= 2b

2

4k

2

= 2b

2

2k

2

= b

2 a b

6=

2

(5)

Pete Agoras

Some centuries B.C.

All fractions are reducible

Suppose

c

d

is a rational number. If c and d have no

common factor, then a = b and b = d. If they have a

common factor, divide both by their greatest common

divisor. The result is

ab

, with no common factor.

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The square root of 2 ain’t rational

Appendices

An even square has an even root

An even number, by definition, is expressible in the form 2k, where k is any integer. On the other hand, an odd number is expressible by

2k + 1 Thus the square of an odd number is

(2k + 1)2 i.e.

4k2+ 4k + 1 i.e.

2 × 2(k2+ k) + 1

which is of the form 2k + 1 with 2(k2+ k) as k. Hence, an odd number produces an odd square, and thus if a square is even its root is even too.

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