The closure property in algebra applies to equations dealing with multiplication and division. This property states that a real number added to or multiplied by a second real number will result in another real number. No unreal numbers can come into an addition or multiplication problem that does not already have an unreal number in it. The closure property also covers closed sets, in which the operation of two numbers within a set result in another number that meets the requirements to make belong to the same set.
The closure property covers all real numbers. A real number can be found on the number line. One, two, three, four and every other whole number is a real number. Fractions and decimal amounts are also real numbers, as are irrational numbers like pi and square root amounts. Real numbers may be negative, positive or zero. Imaginary numbers, which are excluded from the closure property, include infinity and the square root of negative one. These numbers cannot be the result of the addition or multiplication of only real numbers .
The closure property can also be seen in the addition of even numbers. Any even number added to another even number results in another even number. This means that the set of all even numbers is closed when they undergo the operation of addition. An odd number can never enter that set using addition. By contrast, the set of even numbers is not closed when they undergo the operation of division. Although many even number division problems result in even numbers, equations like 100 divided by four result in the number 25, which is odd. Because an odd number can enter the set, that set is not closed.
Binary tables are another example of closed sets. The numbers of a given binary table are listed horizontally and vertically outside of the table. The numbers listed inside the table are limited to the numbers outside. If the table's numbers are one, two, three and four outside of a table, they must also be the same inside. No other numbers can be included in the table's determined operation. Therefore, the table consists of a closed set of numbers under that operation.