Learning Mathematics - Enjoy Pure Mathematics

Understanding Real Numbers M. Velrajan To understand real numbers click the following : Understanding Real numbers If it is useful kindly share with students / friends and to get notification of such future publication follow my blog. Thank you M. Velrajan

    View Differently-Enjoy Pure Maths M. Velrajan This illustrates how viewing pure mathematics differently and asking related questions lead to more interesting results. We start with Theorem (Cantor) Cardinality of the set of real numbers ℝ, Card( ℝ ) is not the same as Card( ℕ ), in fact Card( ℕ ) < Card( ℝ ) and ℝ is uncountable. Proof   Since the map x → x from ℕ to ℝ is 1-1, card( ℕ ) ≤ card( ℝ). We prove Card ( ℝ ) is not the same as Card ( ℕ ) by contrapositive way, i.e. prove by contradiction, by a variation of Cantor’s diagonalization argument . Suppose there is a one-one correspondence between ℕ and ℝ . Then the real numbers can be arranged in a sequence as x 1 , x 2 , . . . , x n , . . .  . Consider all the real numbers  x 1 , x 2 , . . . , x n , . . .  in decimal form. x 1 :  x 10 . x 11 x 12 . . . x 1n . . . . . x 2 :  x 20 . x 21 x 22 . . . x 2n . . . . . . . . x n :  x n0 . x n1 x n2 . . . x nn . . . ....