Basic algebraic properties of real numbers, Mathematics

These can be expressed in terms of two fundamental operations of addition and multiplication.

If a, b and c are any three real numbers, then;

    1.          i.  a + b = b + a

This property is called commutative property of addition. According to this property, addition can be carried out in any order and irrespective of this we obtain the same result.

  1. a.b = b.a

This property is called commutative property of multiplication.

    2.          i.   (a + b) + c = a + ( b + c)

This property is referred to as associative property of addition. According to this property, elements can be grouped according to any manner and irrespective of the grouping we obtain the same result.

  1. (a.b).c = a.(b.c)

This property is referred to as the associative property of multiplication.

      3.           a.(b + c) = a.b + a.c or (a + b).c = a.c + b.c

This property is referred to as distributive property. This is generally employed to expand a product into a sum or the other way round. That is, to rewrite a sum as a product.

      4.        i.   a + 0 = 0 + a = a 

This property is referred to as identity property under addition. That is, 0 when added to a real number returns back the number itself which is same or identical to itself. Thus 0 is the identity element under addition.

  1. a.1 = 1.a = a   

This property is referred to as identity property under multiplication. That is, when a real number is multiplied by 1, we get back the same number.

Thus the element 1 is the multiplicative identity.

         5.    i.    a + (-a) = (-a) + a = 0

This property is referred to as inverse property under addition. According to this property, for every element a, there exists another element - a such that the addition of the both gives us zero. The element - a is referred to as the additive inverse of the element a. On a number line, an element and its additive inverse lie at equi-distant from the origin.

  1.  1212_algebraic properties.png

This property is referred to as inverse property under multiplication. According to this property for every element a, a ≠ 0, there exists another element 1/a such that the multiplication of a and 1/a results in 1. The element 1/a is referred to as multiplicative inverse element.

     6.        i.   If a + x = a + y, then x = y.   

This property is referred to as the cancelation property. According to this property a constant quantity when present on both sides of the equation can be canceled without disturbing the balance which exists between the expressions.

  1. If a≠0 and ax = ay, then x = y.

This property is referred to as the cancelation property under multiplication.

     7.          i.   a.0 = 0.a = 0

This property is referred to as the zero factor property. According to this property any real number a, if multiplied by zero would yield a zero. This can be also put as: if one of the factors happens to be zero, irrespective of other factors, the product of all these factors would yield a zero.

  1. If a.b = 0, then a = 0 or b = 0 or both.

According to this property, the product of any two real numbers a and b is zero if one of them happens to be zero, that is either a = 0 or b = 0 or both of them happen to be equal to zero.

Posted Date: 9/13/2012 2:08:35 AM | Location : United States







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