# Are natural numbers closed under division

## The Closure Property The natural numbers are "closed" under addition and multiplication. The division of two natural numbers does NOT necessarily create another natural number.

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Mathematicians are often interested in whether or not certain sets have particular properties under a given operation. One reason that mathematicians were interested in this was so that they could determine when equations would have solutions. If a set under a given operation has certain general properties, then we can solve linear equations in that set, for example. There are several important properties that a set may or may not satisfy under a particular operation. A property is a certain rule that holds if it is true for all elements of a set under the given operation and a property does not hold if there is at least one pair of elements that do not follow the property under the given operation. In this lecture, we will learn about the closure property. A set has the closure property under a particular operation if the result of the operation is always an element in the set.

Suppose if you want to count the trees in the garden, you will definitely start with one and go on to two, three, four and so on. You have zero trees now, and you just used whole numbers. Let us now study the properties of whole and natural numbers in detail. Let us do it right away. Natural numbers are the counting numbers. These are the numbers which start with 1.

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Teacher professional development and classroom resources across the curriculum. Solutions for Session 2, Part A. See solutions for Problems: A1 A2 A3. Counting Numbers: This set is closed only under addition and multiplication. In other words, we can solve all addition and multiplication problems, but not all subtraction and division problems are solvable. Rational Numbers: This set is closed under addition, subtraction, multiplication, and division with the exception of division by 0. Irrational Numbers: This set is closed for none of the operations e.

## Properties of Whole & Natural Numbers

The natural numbers are closed under division.,

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