What are the different properties of real numbers?

Answer 1

Properties of real numbers

In this lesson we look at some properties that apply to all real numbers. If you learn these properties, they will help you solve problems in algebra. Let's look at each property in detail, and apply it to an algebraic expression.

#1. Commutative properties

The commutative property of addition says that we can add numbers in any order. The commutative property of multiplication is very similar. It says that we can multiply numbers in any order we want without changing the result.

addition

5a + 4 = 4 + 5a

multiplication

3 x 8 x 5b = 5b x 3 x 8

#2. Associative properties

Both addition and multiplication can actually be done with two numbers at a time. So if there are more numbers in the expression, how do we decide which two to "associate" first? The associative property of addition tells us that we can group numbers in a sum in any way we want and still get the same answer. The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.

addition

(4x + 2x) + 7x = 4x + (2x + 7x)

multiplication

2x2(3y) = 3y(2x2)

#3. Distributive property

The distributive property comes into play when an expression involves both addition and multiplication. A longer name for it is, "the distributive property of multiplication over addition." It tells us that if a term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside.

2x(5 + y) = 10x + 2xy

Even though order of operations says that you must add the terms inside the parenthesis first, the distributive property allows you to simplify the expression by multiplying every term inside the parenthesis by the multiplier. This simplifies the expression.

#4. Density property

The density property tells us that we can always find another real number that lies between any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612, 5.613 and so forth.

Between 5.612 and 5.613, there is 5.6121, 5.6122 ... and an endless list of other numbers!

#5. Identity property

The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."

Addition

5y + 0 = 5y

Multiplication

2c × 1 = 2c

* * * * *

The above is equally true of the set of rational numbers.

One of the main differences between the two, which was used by Dedekind in defining real numbers is that a non-empty set of real numbers that is bounded above has a least upper bound. This is not necessarily true of rational numbers.

Answer 2

To start with, the Real number system is a Group. This means that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element which is also in the set. The Group satisfies four axioms: closure, associativity, identity and invertibility. In addition, it is a Ring. A ring is an Abelian group (that is, addition is commutative) and it has a second binary operation (multiplication) that is defined on its elements. This second operation is distributive over the first. And finally, a Field is a Ring over which division - by non-zero numbers - is defined. There are several mathematical terms above which have been left undefined to keep the answer to a manageable size. All these algebraic structures are more than a term's worth of studying. You can find out more about them using Wikipedia but be sure to select the hit that has "mathematical" in it! There are examples in the answer given below.