We are all familiar with some properties of real numbers. Real numbers are commutative, associative, and closed under addition and multiplication. We have also discussed that multiplication of real numbers is distributive over addition.
In this post, we formalize our knowledge of these properties of real numbers and add two more to the list: the identity and inverse properties. Notice also that almost all properties under addition have their corresponding equivalents under multiplication.
Properties of Real Numbers
For any real numbers a, b, and c:
Addition: a + b is a real number
Multiplication: ab is a real number
Addition: a + b = b + a
Multiplication: ab = ba
Addition: a + (b + c) = (a + b) + c
Multiplication: a(bc) = (ab)c
Addition: There is a number 0 such that a + 0 = 0 + a = a
Multiplication: There is a number 1 such that a(1) = 1(a) = a
Addition: There is a number –a such that a + –a = –a + a = 0.
Multiplication: If a is not 0, there is a number 1/a such that (1/a)(a) = (a)(1/a) = 1
a(b + c) = ab + ac
The latest addition to the properties are the identity and inverse properties. In multiplication, there exists an identity element is 1 and in addition the identity element is 0. The identity element does not change the value of an element when a certain operation is performed. For example, in addition is performed, 8 + 0 is still 8 and 3 times 1 is still 3.
In addition, if an element is added to its inverse, the result is the identity element 0. The inverse element of 5 is – 5, and 5 + -5 equals the identity element 0. In multiplication, if an element is multiplied to its inverse, the result is the identity element 1. The inverse element of 3 is 1/3 and (1/3)(3) = 1.
In mathematics, a mathematical system that has all the properties stated above is called a field.