![]() We can cancel the shared factors of □ and □ since these are nonzero, giving This is called the distribution of multiplication over addition, and it holds for ![]() To see where this comes from, we can recall that, for integers There is one final property that links the multiplication and addition of rational □ □ (except 0) have a multiplicative inverse given by The fact that multiplying any rational number by 0 gives 0 is called In other words, we cannot find a rational number that multiplies with 0 to We cannot find a multiplicative inverse for 0 since We need □ to be nonzero otherwise, the reciprocal □ □ That is,Ī rational number that, when multiplied by our original rational number, will give Next, we want to consider whether all rational numbers have an inverse. So, 1 is the multiplicative identity for the rational numbers. That is, a rational number that leaves all rational numbers unchanged Let’s next consider whether there is a multiplicative There are three more properties we can find by considering the properties ofĪddition we already know. Hence, the product of rational numbers is associative. Next, we note that the product of integers is associative, so we can rewrite this as To do this, we first evaluate the product of three rational numbers as follows: Let’s consider whether the multiplication of rational numbers is associative. We can also recall that the addition of rational numbers is associative. □, and □ are integers and the multiplication of integers isĬommutative, so □ □ = □ □ and □ □ = □ □. We can also show that the multiplication of rational numbers is commutative. ![]() Together, the result is also a rational number. In other words, if we add two rational numbers ![]() For example, we can recall that the addition of rational ![]() These definitions allow us to show many properties that multiplication and addition Multiplication operation in the set of rational numbers.Īnd □ are nonzero so that □ □ and □ □Īre rational numbers, then we can add and multiply these numbers by using the In this explainer, we will learn how to identify the properties of the ![]()
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