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
For the past two years, we have talked a lot about real numbers. We have talked about integers and its operations (addition, subtraction, multiplication, and division), we have discussed about rational and irrational numbers, and we have talked about their properties, structure, and wonders. In this post, we are going to summarize what we have learned about them.
Figure 1 - The Number Line
The set of real numbers is the collection of all rational and irrational numbers. By convention, real numbers are represented by a line infinitely long where the positive real numbers are situated at the right hand side of 0, while the negative are at the left hand side. It is also important to note that for each point on the number line, there exists a corresponding real number equivalent to it, and for each real number, there is a corresponding point on the line that represents it. » 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