Real Number and its properties

Real Numbers
All real numbers correspond to points on the number line and all points on the number line correspond to real numbers. All real numbers except zero are either positive or negative.
Here are some properties of real numbers that are used frequently. If x , y , and z are real numbers, then
(1) x + y = y + x and xy = yx .
For example, 8 + 3 = 3 + 8 = 11, and (l7)(5) = (5)(l7) = 85.
(2) ( x + y ) + z = x + ( y + z ) and ( xy ) z = x ( yz ).
For example, (7 + 5) + 2 = 7 + (5 + 2) = 7 + (7) = 14, and (5 3)( 3) = (5)( 3 3) = (5)(3) = 15.
(3) xy + xz = x ( y + z ).
For example, 718(36) + 718(64) = 718(36 + 64) = 718(l00) = 71,800.
(4) If x and y are both positive, then x + y and xy are positive.
(5) If x and y are both negative, then x + y is negative and xy is positive.
(6) If x is positive and y is negative, then xy is negative.
(7) If xy = 0, then x = 0 or y = 0. For example, 3 y = 0 implies y = 0.
(8) | x + y | ≤ | x | + | y |. For example, if x = 10 and y = 2, then | x + y | = |12| = 12 = | x | + | y |;
and if x = 10 and y = −2, then | x + y | = |8| = 8 < 12 = | x | + | y |.