a > b

      taking the reciprocal of both side causes the > to change to <.

      reciprocal of  a > b  is  a^-1 < b^-1

2 - Apply Reciprocal Operation of Inequality to the Inequality of  ¥ ^{-1 > 0}

      ¥ ^-1 > 0 

      the reciprocal of  ¥ ^-1 > 0  is  ¥ < 0 ^-1

      if the inequality  ¥ < 0 ^-1  is true, then 0 ^-1 is a new number, I, and exists outside the set of Real numbers or Hyper-infinite.

      the number I is analogous to the number i = Ö-1

3 - Some other additional inequalities and equalities

      0 ¹ ¥ ^-1

      0 ^-1 ¹ ¥

      I = 0 ^-1  Hyper-infinite number

      0 ´ I = 1

      1 ´ I = I

      n ´ I = nI

      0 ´ nI = n

      a is Real and bI is Hyper-infinite, then the sum of a and bI is complex and is  a + bI

4 -  If ¥ ^-1 = 0 , then 0 + ¥ ^-1 = 0

      0 + ¥ ^-1 = 0

      0 + ( 1 / ¥ ) = 0

      0 ( ¥ / ¥ ) + ( 1 / ¥ ) = 0

      ( ( 0 * ¥ ) / ¥ ) + ( 1 / ¥ ) = 0

      ( 0 * ¥ + 1 ) / ¥ = 0

      0 * ¥ + 1 = 0 * ¥

      ( 0 * ¥ ) - ( 0 * ¥ ) + 1 = ( 0 * ¥ ) - ( 0 * ¥ )

      0 + 1 = 0

      1 = 0

      1 = 0 , is false \  ¥ ^-1 ¹ 0

5 -  If ¥ ^-1 > 0 , then 0 + ¥ ^-1 > 0

      0 + ¥ ^-1 > 0

      0 + ( 1 / ¥ ) > 0

      0 ( ¥ / ¥ ) + ( 1 / ¥ ) > 0

      ( ( 0 * ¥ ) / ¥ ) + ( 1 / ¥ ) > 0

      ( 0 * ¥ + 1 ) / ¥ > 0

      0 * ¥ + 1 > 0 * ¥

      ( 0 * ¥ ) - ( 0 * ¥ ) + 1 > ( 0 * ¥ ) - ( 0 * ¥ )

      0 + 1 > 0

      1 > 0

      1 > 0 , is true \  ¥ ^-1 > 0

6 -   n ¥ = ¥

          ( n ¥ ) / ¥ = ¥ / ¥

          n ( ¥ / ¥ ) = ¥ / ¥

          n ( 1 ) = 1

        n = 1, \  n ¥ = ¥  true when  n = 1

7 -  n ¥ ¹ ¥

        ( n ¥ ) / ¥ ¹ ¥ / ¥

        n ( ¥ / ¥ ) ¹ ¥ / ¥

        n ( 1 ) ¹ 1

        n ¹ 1, \  n ¥ ¹ ¥  true when  n ¹ 1

8 -  n ¥ = -¥

        ( n ¥ ) / ¥ = -¥ / ¥

        n ( ¥ / ¥ ) = -¥ / ¥

      n ( 1 ) = -1

      n = -1, \  n ¥ = -¥  true when  n = -1 

9 -  n ¥ ¹ -¥

      ( n ¥ ) / ¥ ¹ -¥ / ¥

      n ( ¥ / ¥ ) ¹ -¥ / ¥

      n ( 1 ) ¹ -1

      n ¹ -1, \  n ¥ ¹ -¥  true when  n ¹ -1

10 -  0 * ¥  = n

      ( 0 * ¥ ) / ¥ = n / ¥

      0 ( ¥ / ¥ ) = n / ¥

      0 ( 1 ) = n / ¥

      0 = n / ¥ , if and only if n = 0 * ¥

11 -  0 * ¥ = n

      ( 0 * ¥ ) / 0 = n / 0

      ¥ ( 0 / 0 ) = n / 0

      ¥ ( 1 ) = n / 0

      ¥ = n / 0

      ¥ = n / 0 , if and only if n = 0 * ¥

12 -  If X / X = 1, then -¥ / -¥ = 1 

        -¥ / -¥ = 1

        -¥ = 1 * -¥

        -¥ = -¥, true \ -¥ / -¥ = 1

13 -  If X / X ¹ 1, then -¥ / -¥ ¹ 1

        -¥ / -¥ ¹ 1

        -¥ ¹ 1 * -¥

        -¥ ¹ -¥, false \ -¥ / -¥ = 1

14 -  If X / X = 1, then 0 / 0 = 1 

        0 / 0 = 1

        0 = 1 * 0

        0 = 0, true \ 0 / 0 = 1

15 -  If X / X ¹ 1, then 0 / 0 ¹ 1

        0 / 0 ¹ 1

        0 ¹ 1 * 0

        0 ¹ 0, false \ 0 / 0 = 1

16 -  If X / X = 1, then ¥ / ¥ = 1 

        ¥ / ¥ = 1

        ¥ = 1 * ¥

        ¥ = ¥, true \ ¥ / ¥ = 1

17 -  If X / X ¹ 1, then ¥ / ¥ ¹ 1

        ¥ / ¥ ¹ 1

        ¥ ¹ 1 * ¥

        ¥ ¹ ¥, false \ ¥ / ¥ = 1

18 -  If Y = 1 and is continuous for all Real values in the function Y = f(X) and f(X) = X / X, then Y = 1 = X / X and is continuous for all Real values

      Y = 1 = X / X

      Y * X = 1 * X = X

      1 * X = 1 * X = X

      X = X = X, true if and only if Y = 1

      -¥ = -¥ = -¥, true if and only if Y = 1

      0 = 0 = 0, true if and only if Y = 1

      ¥ = ¥ = ¥, true if and only if Y = 1

The mathematical operation of X / X = 1 must be continuous for all Real values of X; including {-¥, 0, ¥}.

When dealing with 0 and ¥ it is better to treat them as if they were variables first and do normal math operations first, then work the final equation or inequality. There are times when 0² ¹ 0 and ¥² ¹ ¥ , much like X² does not exactly equal X. 

Also,  0 * ¥   is a non-reducible number much like 2i is a non-reducible number, it is what it is, a number all by itself.

In addition, ¥ ^-1  is the infinitesimal unit and is exactly and only exactly the first number after 0.