Real numbers are all the points on a number line, and they have several kinds of properties. The four properties governing calculations with real numbers are the commutative, associative, distributive properties and closure. Other properties include real number characteristics such as a number being prime, even, rational or negative and properties expressed as rules. These properties explain how real numbers work, but the four calculation properties are important because they govern how to get the right answers in basic arithmetic.
TL;DR (Too Long; Didn't Read)
The properties of real numbers that govern calculations are the commutative, associative, distributive properties and closure, defined as follows:
- Commutative means that adding or multiplying real numbers gives the same answer no matter in which order the numbers are added or multiplied.
- Associative means that if three numbers are added or multiplied, the answer is the same whether the first two are added or multiplied and then the third or the last two and then the first.
- Distributive means that multiplying the sum of two numbers gives the same answer as multiplying each number and adding.
- Closure means that calculations with real numbers result in real numbers, with the exception of division by zero.
Examples of Real Number Calculation Properties
The commutative property applies to addition and multiplication. It means that numbers can be commuted or switched around in any sequence and the answers will be the same. For example, 2 x 3 x 7 gives the same answer as 7 x 2 x 3, 42, and 4 + 9 + 1 + 3 gives the same answer as 9 + 4 + 3 +1, 17. This property does not apply to subtraction or division because 6 - 4 is not the same as 4 - 6, and 4 ÷ 2 is not the same as 2 ÷ 4.
The associative property also applies to addition and multiplication and means that it doesn't matter in which order the calculations are performed. In the above multiplication example, "associating" 2 and 3 by multiplying them first, and then multiplying the result by 7 gives the same answer as multiplying 3 and 7 first and then multiplying by 2. The same is true for addition but again not for subtraction or division.
The distributive property applies to multiplication and addition. It means that multiplication can be "distributed" across a series of added numbers or the numbers can be multiplied individually for the same answer. For example 3 x (4 + 2 + 7) = 39 and 3 x 4 plus 3 x 2 plus 3 x 7 also equals 39.
Closure as a property of real numbers means that all arithmetical calculations including multiplication, addition, subtraction and division give answers that are also real numbers. This is true of all the examples given above. The exception is division by zero, an operation that is not defined within the real number space.
Other Properties of Real Numbers
Real numbers have other properties that give information about specific numbers but are not needed to get the right answers in calculations. For example, if a number is even, it can be divided by 2. If a number is prime, it can't be broken down into smaller factors. If a number is rational, it can be expressed as a fraction. If a number is negative, multiplying it by a negative number gives a positive result. These are descriptive properties.
Additional properties are rules-based and apply to all real numbers. For example, if a real number is not a prime, it can be expressed as the product of primes. A set of unequal numbers can always be placed in order of size. The real numbers have no largest and no smallest number no matter what number is chosen, there is always a larger and a smaller one. There are many such descriptive and rules-based properties of real numbers, but the most important for getting the right answers in calculations are the four dealing with arithmetic.