Real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced.

EXAMPLES: Closure Properties of Real Numbers

1) The sum of any two real numbers is a real number.

(In other words, if a and b are real, then so is a + b.)

2) The product of any two real numbers is a real number.

(In other words, if a and b are real, then so is ab.)

EXAMPLES OF PROPERTIES

#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...

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