We are all familiar with some properties of real numbers. Real numbers are commutative, associative, and closed under addition and multiplication. We have also discussed that multiplication of real numbers is distributive over addition.
In this post, we formalize our knowledge of these properties of real numbers and add two more to the list: the identity and inverse properties. Notice also that almost all properties under addition have their corresponding equivalents under multiplication. » Read more
In the previous discussion, we have shown that addition of two integers a and b is commutative by considering a and b as lengths of two segments. Joining the two segments is the same as adding two integers. We have observed that a + b equals b + a.
We have also shown that multiplication of two integers a and b is commutative by considering a and b as side lengths of a rectangle. Getting the area of a rectangle with side lengths a and b is the same as multiplying a and b. We have observed that ab equals ba. However, we argued about commutativity that any segment does not change its length and the movement of rectangles does not change its area. This reasoning is only limited to positive integers because we cannot represent negative integers as length or area. To explore further, we use the addition and multiplication tables to see if the addition and multiplication of integers are both commutative.
» Read more
One of the elementary observations about real numbers is that if we add the two numbers, regarding of the order, their sum is unique. We know that 4 + 3 = 3 + 4. This property is also the same when we multiply. If we have two numbers, regarding of the order, their product is also unique. For example, 6 x 3 = 3 x 6.
In general if we have real numbers a and b, we have a + b is always equal to b + a. Also, if we have real numbers c and d, c x d is always equal to d x c. These are called the commutative property of addition and multiplication, two of the axioms of algebra.
Although axioms, as we have discussed, are statements that we accept without proof, some axioms can be explained intuitively. In this post, we are going to discuss the two axioms mentioned above intuitively using geometric representations.
The geometry of a + b
One way of understanding why a + b is always equal to b + a is by representing the two numbers as lengths. For instance, we can represent the addition of integers above (4+3 = 3 + 4) as the sum of the lengths of two segments. In geometric representation, reversing the order of the numbers in the operation is just like rotating the segment 180 degrees about the center as shown below. Of course, rotating the segment way will never change its length.
Now, since that we can substitute any positive numbers to a and b we are sure that the commutative property is true for all positive numbers. » Read more